Add Batch 6ce879d8-bede-4222-8ea6-bbd748472dda data

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@@ -3088,3 +3088,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
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2025/Towards Interpreting Visual Information Processing in Vision-Language Models/600bd478-2d40-4b3d-957a-a9f71760f4ee_content_list.json

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+    {
+        "type": "text",
+        "text": "TOWARDS INTERPRETING VISUAL INFORMATION PROCESSING IN VISION-LANGUAGE MODELS",
+        "text_level": 1,
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+        "text": "Clement Neo $^{1*}$ , Luke Ong $^{1}$ , Philip Torr $^{2}$ , Mor Geva $^{3}$ , David Krueger $^{4}$ , Fazl Berez $^{2,5}$",
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+    {
+        "type": "text",
+        "text": "$^{1}$ Nanyang Technological University  $^{2}$ University of Oxford  $^{3}$ Tel Aviv University",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "4MILATangentic",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+    {
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+        "text": "Vision-Language Models (VLMs) are powerful tools for processing and understanding text and images. We study the processing of visual tokens in the language model component of LLaVA, a prominent VLM. Our approach focuses on analyzing the localization of object information, the evolution of visual token representations across layers, and the mechanism of integrating visual information for predictions. Through ablation studies, we demonstrated that object identification accuracy drops by over  $70\\%$  when object-specific tokens are removed. We observed that visual token representations become increasingly interpretable in the vocabulary space across layers, suggesting an alignment with textual tokens corresponding to image content. Finally, we found that the model extracts object information from these refined representations at the last token position for prediction, mirroring the process in text-only language models for factual association tasks. These findings provide crucial insights into how VLMs process and integrate visual information, bridging the gap between our understanding of language and vision models, and paving the way for more interpretable and controllable multimodal systems.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
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+    },
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+        "text": "Vision-Language Models (VLMs) take an image and text as input, and generate a text output. They have become powerful tools in processing and understanding text and images, enabling a wide range of applications from question answering to image captioning (Liu et al., 2023b; Li et al., 2023; Alayrac et al., 2022). Among VLM architectures, the \"adapter\" style has demonstrated impressive state-of-the-art performance, notable for its simplicity. These models combine a pre-trained image encoder, a pre-trained language model (LM), and a learned adapter network that maps image encoder outputs to soft prompts for LM inputs (Merullo et al., 2022; Liu et al., 2023b;a). Despite their importance and potential, the inner workings of VLMs remain poorly understood as compared to their text-only counterparts. While significant progress has been made in understanding language models (Elhage et al., 2021; Wang et al., 2023; Bricken et al., 2023) and, to a lesser extent, vision transformers (Palit et al., 2023; Vilas et al., 2023; Pan et al., 2024), there is a notable gap in our understanding of VLMs, where both modalities operate in the same space. A deeper comprehension of VLMs could lead to practical advances in safety, robustness, and functionality, similar to the progress made in language models, such as model editing (Meng et al., 2022) or identifying specific components responsible for certain behaviors (Arditi et al., 2024).",
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+        "text": "In this paper, we investigate the LM component of VLMs as they serve as the \"brain\" of the system, comprising more than  $95\\%$  of the total parameters for models like LLaVA 1.5 7B. LLMs pre-trained on large text datasets are likely to remain the foundation of VLMs because of their powerful reasoning capabilities. Therefore, understanding how these LMs process and integrate visual information is crucial for advancing VLM interpretability.",
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+        "text": "Many open questions remain regarding the inner workings of VLMs. The nature and structure of visual representations fed into the LM remain unclear, as these visual inputs are soft prompts that",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "*Work done during ERA-Krueger AI Safety Lab internship. Author contributions detailed in §6. Correspondence to Clement Neo <clement@clementneo.com>.",
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+    {
+        "type": "page_number",
+        "text": "1",
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+    {
+        "type": "text",
+        "text": "do not correspond to language tokens and cannot be interpreted as such (Lester et al., 2021; Merullo et al., 2022). This raises questions about how visual information is encoded in these representations. Furthermore, the spatial distribution of information within the image encoder's feature maps remains unclear, leaving open the question of whether and how much object-specific details are localized or dispersed across the entire representation.",
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+        "type": "text",
+        "text": "Furthermore, the mechanisms by which the language model processes these visual inputs are not well understood. A significant modality gap exists between visual and textual input (Jiang et al., 2024), and it is unclear whether our mechanistic understanding of text processing in language models generalizes to the processing of visual input in VLMs.",
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+        "type": "text",
+        "text": "To understand the representations in the visual inputs for VLM and how the VLM processes them, we study LLaVA 1.5 7B (Liu et al., 2023a), a popular open-source VLM with competitive performance that uses CLIP (Radford et al., 2021) as an image encoder and Vicuna 13B (Chiang et al., 2023) as its language model. We also test on LLaVA-  $\\mathsf{Phi}^{\\dagger}$  and Qwen2-VL (Wang et al., 2024). We focus on a set of object identification tasks and our main contributions are as follows:",
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+        "list_items": [
+            "1. Using ablation techniques, we demonstrate that the information for an object is highly localized to the token positions corresponding to their original location in the image.",
+            "2. By extending the logit lens technique primarily used for language models (Nostalgebraist, 2020), we find that the representations of the visual input in the LM are refined towards the embedding of interpretable tokens in the vocabulary, despite the LM not being explicitly trained to do so.",
+            "3. By blocking the attention flow between tokens, we show that the model extracts object information from the object tokens in the middle to late layers."
+        ],
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+    {
+        "type": "text",
+        "text": "Our findings are first steps in understanding the internal mechanisms of VLMs, paving the way for more interpretable and controllable multimodal systems. The code for our experiments is available at https://github.com/clemneo/llava-interp.",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Transformer Architecture. Transformer-based autoregressive Large Language Models (LLMs) process sequences of input tokens to predict the next token in the sequence (Vaswani et al., 2017). These models achieved state-of-the-art performance across various natural language processing tasks (Brown et al., 2020), even though they are merely trained on next-token prediction (Radford et al., 2019). A LLM takes an input sequence  $X = (x_{1},\\dots,x_{n})$  and outputs a probability distribution over the vocabulary  $V$  to predict the next token  $x_{n + 1}$ .",
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+    {
+        "type": "text",
+        "text": "To do so, the model refines token representations layer by layer through a series of computations. Each token  $x_{i}$  is initially represented by an embedding vector  $h_{i}^{0}$  of dimension  $d_{m}$ , obtained from a lookup operation in an embedding matrix  $W_{E} \\in \\mathbb{R}^{|V| \\times d_{m}}$ . This representation is then updated through successive layers using multi-head self-attention (MHSA) and feed-forward (FF) sublayers:",
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh _ {i} ^ {l} = h _ {i} ^ {l - 1} + a _ {i} ^ {l} + f _ {i} ^ {l} \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $h_i^l$  is the representation of token  $x_i$  at layer  $l$ ,  $a_i^l$  is the output from the MHSA sublayer, and  $f_i^l$  is the output from the FF sublayer. All vectors  $h_i^l, a_i^l, f_i^l \\in \\mathbb{R}^{d_m}$ .",
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+        "type": "text",
+        "text": "The MHSA sublayer consists of  $H$  attention heads working in parallel, each implemented as follows:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {A t t e n t i o n} (Q, K, V) = \\operatorname {s o f t m a x} \\left(\\frac {Q K ^ {T}}{\\sqrt {d _ {k}}} + M\\right) V \\tag {2}\n$$\n",
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+    {
+        "type": "text",
+        "text": "where  $Q, K, V \\in \\mathbb{R}^{n \\times d_k}$  are query, key, and value matrices derived from learned linear projections of the input, and  $M \\in \\mathbb{R}^{n \\times n}$  is a causal mask. The causal mask  $M$  is set to  $-\\infty$  in the upper right triangle, meaning  $M_{ij} = 0$  if  $i \\geq j$ , else  $-\\infty$ . This ensures that each position can only attend to previous positions and itself, thus preserving the autoregressive property in language models. These",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "Sourced from the xtuner HuggingFace model.",
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+    },
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+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "image",
+        "img_path": "images/4b691183acdfd479a0d5c4e86f753e356bbe9b572fbc0cd48df65f809ba3b70f.jpg",
+        "image_caption": [
+            "Figure 1: In adapter-style Vision-Language Models (VLMs), the visual tokens (in green) are soft prompts for the language model (LM), and are not interpretable through the vocabulary embedding. Through a set of object identification tasks, we find that (1) object information can be localized to a subset of visual tokens, (2) the representations of the visual tokens evolve towards interpretable text embeddings, and (3) the model extracts some information from the visual tokens to the last token position in the middle to late layers, to identify the object."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "outputs are then concatenated and linearly projected to produce the final output of the multi-head attention sublayer.",
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+        "type": "text",
+        "text": "The FF sublayer applies two linear transformations with an element-wise non-linear function  $\\sigma$  between them:",
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+    {
+        "type": "equation",
+        "text": "\n$$\nf _ {i} ^ {l} = W _ {v} ^ {l} \\sigma \\left(W _ {k} ^ {l} a _ {i} ^ {l} + b _ {k} ^ {l}\\right) + b _ {v} ^ {l}, \\tag {3}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $W_{v}^{l}, W_{k}^{l}, b_{k}^{l}$ , and  $b_{v}^{l}$  are learned parameters. Finally, the representation of the last token  $x_{n}$  at the final layer,  $h_{n}^{L}$ , is projected into a probability distribution over  $V$  using an unembedding matrix  $W_{U} \\in \\mathbb{R}^{d \\times |V|}$  and a softmax operation.",
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+    },
+    {
+        "type": "text",
+        "text": "Image Encoders. The transformer architecture has been successfully adapted for state-of-the-art image encoders, as demonstrated by Dosovitskiy et al. (2021). In this approach, input images are divided into fixed-size patches, flattened, and linearly embedded before being processed as a sequence of tokens, analogous to word embeddings in NLP tasks. Unlike text-based models, the attention mechanism in these image encoders is bidirectional (i.e., no masking is applied). The input sequence includes a special class token, whose final embedding is typically used for image classification tasks. The representations at the other token positions form what is called the feature map of the image.",
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+    {
+        "type": "text",
+        "text": "Building upon this foundation, Radford et al. (2021) introduced Contrastive Language-Image Pretraining (CLIP), which jointly trains a vision transformer and a textTransformer to produce similar embeddings for matching image-text pairs. CLIP's image encoder is currently being used for state-of-the-art performance on many downstream tasks like zero-shot image classification and vision-language models like LLaVA.",
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+    },
+    {
+        "type": "text",
+        "text": "Connecting Text and Image Models. Tsimpoukelli et al. (2021) pioneered the concept of training an image encoder such that its output can be used by a frozen language model for multimodal few-shot reasoning. Merullo et al. (2022) advanced this approach by freezing both the pre-trained image encoder and pre-trained language model, training only a linear map that converts image features into inputs for the text model. This introduced the \"adapter style\" approach, utilizing pre-trained components for both modalities.",
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
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+    {
+        "type": "text",
+        "text": "Building on this foundation, Liu et al. (2023b) introduced LLaVA, which combined a CLIP ViT-L/14 image encoder with a Vicuna-13B language model, connected by a trained linear projection, or an MLP for improved performance (Liu et al., 2023a). Crucially, LLaVA's training process focuses solely on fine-tuning for multimodal conversation, without any next-token pretraining on image-text pairs. This approach allows LLaVA to leverage the strong pre-trained capabilities of both the vision and language models while efficiently adapting to multimodal tasks.",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "2.1 NOTATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe how an image and query is processed through LLaVA.",
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+    {
+        "type": "text",
+        "text": "Image Processing. The input image  $I$  is processed by first being cropped into a square, then suitably resized before being divided into 576 image patches. The CLIP ViT-L/14 image encoder  $f_{\\mathrm{CLIP}}$  processes this image to produce a feature map  $E_I = f_{\\mathrm{CLIP}}(I) \\in \\mathbb{R}^{N \\times d_{\\mathrm{CLIP}}}$ , where  $N = 576$  is the number of image patches and  $d_{\\mathrm{CLIP}}$  is the dimensionality of CLIP's embeddings. An adapter network  $A$  then maps these CLIP embeddings to the language model's input space, yielding a set of visual tokens  $E_A = A(E_I) \\in \\mathbb{R}^{N \\times d}$ , where  $d_m$  is the dimensionality of the language model's input embeddings. We call  $E_A$  visual tokens to distinguish them from the image tokens.",
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+    {
+        "type": "text",
+        "text": "Text Processing and Combined Input. For the text input, given a tokenized prompt sequence  $T = (t_{1}, \\ldots, t_{M})$ , the embedding layer of the language model  $E_{\\mathrm{LM}}$  maps these tokens to embeddings:  $E_{T} = E_{\\mathrm{LM}}(T) \\in \\mathbb{R}^{M \\times d_{\\mathrm{LM}}}$ . The full input to the language model is the concatenation of the adapted image embeddings and the text embeddings:  $X = [E_{A}; E_{T}] \\in \\mathbb{R}^{(N + M) \\times d_{\\mathrm{LM}}}$ , from which the language model can generate output autoregressively.",
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+    {
+        "type": "text",
+        "text": "2.2 OPEN QUESTIONS AND HYPOTHESES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One might expect that the output of the adapter  $E_{A}$  would produce embeddings corresponding to image information. For example, if the image contained a car, we might expect the adapter to produce tokens that correspond to the embedding of \"car\". However, the output of the adapter forms soft prompts that have no semantic meaning when decoded through the vocabulary (Merullo et al., 2022; Bailey et al., 2023).",
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+    {
+        "type": "text",
+        "text": "2.2.1 LOCALIZATION OF OBJECT-SPECIFIC INFORMATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A key question of our investigation is: Where is object-specific information located in visual tokens? Two conflicting hypotheses emerge from recent research:",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Object-Centric Localization: Joseph & Nanda (2024) found that in classical vision transformers, token representations corresponding to specific object locations tended to align with those objects' class embeddings in the late layers, even without explicitly training for this behavior. This suggest that object information might be localized to the tokens corresponding to the spatial location of the object in the original image. However, their findings were on a vision transformer trained on ImageNet classes, and not the vision transformer in CLIP.",
+            "2. Global Information in Register Tokens: In contrast, Darcet et al. (2024) identified register tokens in vision transformers, including a CLIP variant. These are background patch tokens with unusually high norms that appear to encode global image features. It follows that the LM in an adapter-style VLM could primarily rely on these information-rich register tokens for its output."
+        ],
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We investigate these hypotheses in Section 3 using ablation techniques in a set of object identification tasks.",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.2 PROCESSING OF OBJECT-SPECIFIC REPRESENTATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The processing of visual tokens may differ from text tokens. Soft prompts are very different from normal prompts (Bailey et al., 2023), and there is a modality gap between textual and visual representations in VLMs (Jiang et al., 2024). Furthermore, LLaVA is fine-tuned exclusively on visual question-answering (VQA) prompt-response pairs, without pre-training on next-token prediction.",
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "This may affect how the model processes visual information as compared to text. In Section 4.1, we attempt the logit lens method to see how the representations of visual tokens evolve through the layers.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recent studies have shed light on how LMs process information. Geva et al. (2023) identified a refine-then-extract process in factual association prompts, where the model first refines information at subject tokens through MLP sublayers before transferring it to the last token position via attention sublayers. Similar mechanisms have been observed in arithmetic tasks (Stolfo et al., 2023), suggesting that this (sequencing of functions) might be a general mechanism in LMs.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, Basu et al. (2024) adapted the factual association setup to VLMs and found preliminary evidence that the LM may first summarize image information in text tokens, instead of extracting the image information to the last token position from the image tokens directly. While we do not study factual association – rather we choose to focus on a simpler object identification task, we study how information flows from the image tokens in Section 4.2.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 INVESTIGATING THE REPRESENTATIONS IN MAPPED VISUALTOKENS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we use ablation experiments to test if object information is concentrated in specific visual tokens. By ablating selected tokens and observing degradations in the model's ability to identify the object, we map how object information is distributed across the visual tokens.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Dataset. We use images from the COCO Detection Training set (Lin et al., 2014). To ensure the reliability of our results, we employ two filtering steps:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(1) Choosing Simpler Images. As a heuristic to focus on simpler images, we choose images where (a) there is an object whose size is between 1,000-2,000 square pixels, or about  $2 - 4\\%$  of the image by area, (b) only one instance of the that object is present in the image, and (c) there are less than 4 annotated objects. We present a few examples of the images in Figure 4.",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(2) Controlling for Hallucination. Model can sometimes hallucinate objects based on context, even when the object's visual information is removed. To address this, we create two versions of each image: the original and one where the target object is masked with noise. We only keep images where the model correctly identifies the object in the original image but fails to identify it in the masked version. This ensures that the model's object identification relies on the object's visual information.",
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+        "type": "text",
+        "text": "After applying both filtering steps, our final dataset comprises 4,318 images.",
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+        "type": "text",
+        "text": "Method. We present a high level overview of our methodology in Figure 2. Let  $E_A = \\{e_1, \\dots, e_N\\}$  be the set of visual tokens. We define a subset  $S \\subset \\{1, \\dots, N\\}$  consisting of the indices of tokens hypothesized to contain information about a particular object  $o$  in the image.",
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+    {
+        "type": "text",
+        "text": "For each ablation experiment, we create a modified set of embeddings  $E_A'$ :",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne _ {i} ^ {\\prime} = \\left\\{ \\begin{array}{l l} \\bar {e} & \\text {i f} i \\in S \\\\ e _ {i} & \\text {o t h e r w i s e} \\end{array} \\right.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where  $\\bar{e} = \\frac{1}{N}\\sum_{i=1}^{N} e_i$  is the mean embedding across all visual tokens from 50,000 images from the ImageNet validation split (Deng et al., 2009). This replaces the hypothesized object-relevant tokens with an average token, effectively ablating their specific information. We do mean ablation to preserve the norm of the image token and keep them in-distribution, as their norms are typically much higher than the norm of text tokens (Bailey et al., 2023). We then evaluate the impact of token ablation using three methods:",
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+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Generative Description: We prompt the model with \"Describe the image\" using both the original  $(E_A)$  and ablated embeddings  $(E_A^{\\prime})$ . We then ascertain whether the target object  $o$  is produced verbatim in the model's generated description before and after ablation. If  $o$  is mentioned with  $E_A$  and not  $E_A^\\prime$ , it indicates that the ablated tokens were crucial for the model to identify and include the object in its description.",
+            "2. Binary Polling: We ask the model \"Is there a [o] in this image?\" using both  $E_A$  and  $E_A'$ . We then compare whether the model's next generated token changes from \"Yes\" to"
+        ],
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/4d4afe056bb1dc431c372f1b9db3b213e973e755d8688c929fa8490b44748315.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our ablation experiments. (1) We ablate some visual tokens that potentially contain object-specific information, (2) prompt the model to describe the image, or answer object-specific questions, then (3) measure the impact of token ablation by calculating the percentage of initially correct object identifications that become incorrect after ablation."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "\"No\" after ablation, which indicates that the ablated tokens were crucial for identifying the object.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3. Visual Question-Answering: We manually curate a set of 100 images with specific questions about objects, such as \"What is on the bed?\" We choose questions with unambiguous answers and avoid objects directly related to the question (e.g., avoiding \"pillow\" for a bed-related question). We prefill the model's response with \"It is a\" and compare the next generated token before and after ablation. A set of examples is in Figure 5.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We choose the subset  $S$  of tokens to ablate in five ways:",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(1) Object Tokens. We choose the visual tokens whose positions correspond to the image patches that originally contained that object. (2) Object Tokens with Buffer. In addition to the object tokens, we include neighboring tokens. We test with two buffer sizes: 1 Buffer (including immediately adjacent tokens) and 2 Buffer (including tokens up to 2 positions away from object tokens). (3) Register Tokens. We select visual tokens with norms more than 2 standard deviations from the mean norm. These correspond to the register tokens identified by Darcet et al. (2024), which are thought to encode global image features. (4) Random Tokens. As a baseline, we ablate  $n$  random tokens. (5) High-Gradient Tokens. As a stronger baseline, we use the Integrated Gradients attribution method (Sundararajan et al., 2017) to identify tokens most important for the model's decision. For a given object  $o$ , we prompt the model with \"Is there a [o] in the image?\" and compute integrated gradients with respect to the logit for the \"Yes\" token over 50 steps. We then ablate the  $n$  tokens with the highest attribution scores.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. Our results are presented in Table 1. We find that abating the object tokens significantly impairs the model's ability to identify the object. For comparable numbers of ablated tokens, object token ablation consistently results in larger performance decreases across all settings as compared to the gradient-based and random baselines. This suggests that the information about that object is localised to the region of the object token. Furthermore, these findings are consistent across all three experimental settings and applicable to both LLaVA 1.5 and LLaVA-Phi.",
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+    {
+        "type": "text",
+        "text": "4 INVESTIGATING VISUAL INFORMATION PROCESSING",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 ANALYZING THE RESIDUAL STREAM THROUGH EMBEDDING SPACE",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Can we interpret how the representations at the visual token positions evolve through the layers? To do so, we employ the logit lens technique (Nostalgebraist, 2020): for each layer, we decode the activation at each token position using the unembedding. Formally, for each hidden state  $h_i^l$  at each token position  $i$  and layer  $l$ , we project it into a probability distribution over  $V$  using  $W_U$  and take the token with the highest logit. We do this for LLaVA 1.5.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    {
+        "type": "table",
+        "img_path": "images/22b0ecace33905ba9387ba15b18cb4a8a4ce0618be76e03142afb2bc15e5c6f4.jpg",
+        "table_caption": [
+            "Table 1: Performance degradation after token ablation. A lower percentage indicates a greater impact of the ablation, meaning the model is more likely to answer incorrectly, thus suggesting that the ablated tokens contain more localized object information. Average token counts apply to both the generative and polling settings for LLaVA 1.5. We find that ablating the object tokens with adjacent tokens (33.4 tokens on average, in **bold**) significantly causes object identification performance to drop, more so than the integrated gradients and random baselines for a comparable number of tokens (40 tokens, in **bold**)."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ablation Type (Avg Token Count)</td><td>Generative Decrease (%)</td><td>LLaVA 1.5 Polling Decrease (%)</td><td>VQA Decrease (%)</td><td>Generative Decrease (%)</td><td>LLaVA-Phi Polling Decrease (%)</td><td>VQA Decrease (%)</td></tr><tr><td>Object (12.6)</td><td>33.33</td><td>15.38</td><td>33.33</td><td>47.46</td><td>28.48</td><td>41.51</td></tr><tr><td>+1 Buffer (33.4)</td><td>71.79</td><td>51.28</td><td>86.67</td><td>82.89</td><td>86.48</td><td>96.23</td></tr><tr><td>+2 Buffer (60)</td><td>76.92</td><td>64.10</td><td>92.22</td><td>89.56</td><td>95.49</td><td>100.00</td></tr><tr><td>Register tokens (3.1)</td><td>5.13</td><td>0.00</td><td>3.33</td><td>7.63</td><td>0.00</td><td>1.89</td></tr><tr><td>Int. Gradients (5)</td><td>2.56</td><td>0.00</td><td>6.67</td><td>17.11</td><td>1.23</td><td>28.30</td></tr><tr><td>Int. Gradients (10)</td><td>20.51</td><td>2.56</td><td>6.67</td><td>25.26</td><td>4.30</td><td>24.53</td></tr><tr><td>Int. Gradients (20)</td><td>33.33</td><td>20.51</td><td>17.78</td><td>41.14</td><td>9.63</td><td>37.74</td></tr><tr><td>Int. Gradients (40)</td><td>48.72</td><td>35.90</td><td>50.00</td><td>61.49</td><td>26.64</td><td>56.60</td></tr><tr><td>Int. Gradients (60)</td><td>58.97</td><td>48.72</td><td>71.11</td><td>70.61</td><td>40.98</td><td>69.81</td></tr><tr><td>Int. Gradients (100)</td><td>76.92</td><td>51.28</td><td>84.44</td><td>81.23</td><td>61.68</td><td>83.02</td></tr><tr><td>Int. Gradients (250)</td><td>76.92</td><td>69.23</td><td>97.78</td><td>91.93</td><td>90.16</td><td>98.11</td></tr><tr><td>Random (5)</td><td>0.00</td><td>0.00</td><td>1.11</td><td>3.68</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (10)</td><td>4.88</td><td>0.00</td><td>0.00</td><td>3.95</td><td>0.00</td><td>1.89</td></tr><tr><td>Random (20)</td><td>2.44</td><td>0.00</td><td>2.22</td><td>5.88</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (40)</td><td>2.44</td><td>0.00</td><td>0.00</td><td>8.51</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (60)</td><td>7.32</td><td>0.00</td><td>1.11</td><td>9.39</td><td>0.41</td><td>3.77</td></tr><tr><td>Random (100)</td><td>7.32</td><td>0.00</td><td>4.44</td><td>11.23</td><td>0.20</td><td>7.55</td></tr><tr><td>Random (250)</td><td>24.39</td><td>0.00</td><td>16.67</td><td>20.61</td><td>1.84</td><td>15.09</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Results. We find that the activations in the late layers at each visual token position correspond to token embeddings that describe its original patch object in its corresponding image token. We provide some case studies in Figure 3, and an interactive demo on more examples.",
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+    {
+        "type": "text",
+        "text": "To quantify this phenomenon, we analyzed 170 COCO validation images with objects of sizes between 20,000 and 30,000 square pixels (approximately 1/2 width and 1/3 height of the image). We found that in the best-performing layer for each image, an average of  $23.7\\%$  of the object patch token positions correspond to the correct object class token. The best-performing layer occurs on average at layer 25.7 (out of 33 layers), confirming that mid-to-late layers tend to develop the strongest object-token correspondences (Figure 6).",
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+    {
+        "type": "text",
+        "text": "We found that the tokens decoded often correspond to concepts more specific than the object-level. For example, in Figure 3a, the tokens go beyond identifying the sweater worn (\"swe\") and describe the specific patterns on the sweater with tokens like \"diam\"(ond) and \"cross\".",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In some instances, tokens correspond to their corresponding concepts in other languages, like in Figure 3c where the tokens correspond to \"year\" and \"month\" but in Chinese and Korean instead. This is somewhat surprising given that previous work has shown that Llama used for non-English languages tend to have English tokens in their intermediate activations when interpreted through the logit lens (Wendler et al., 2024).",
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+    {
+        "type": "text",
+        "text": "Overall, it is surprising that the logit lens work on VLMs! This technique works on auto-regressive LLMs because they are pretrained on next-token prediction, meaning their activations will be iteratively refined towards the unembedding of the predicted next token. However, VLMs are merely fine-tuned on multimodal tasks. The effectiveness of the logit lens here suggests that the hypothesis that transformers build predictions by promoting concepts in vocabulary space (Geva et al., 2022) may generalise to VLMs, even when only fine-tuned on multimodal tasks.",
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+    {
+        "type": "text",
+        "text": "Testing Other Models. We extended our logit lens analysis to Qwen2VL-2B to verify if these findings generalize beyond the LLaVA family. Following the same methodology, we analyzed 253 COCO validation images and found that in the best-performing layer, an average of  $6.5\\%$  of the visual tokens map to the correct object class token. The best-performing layer occurs at layer 25.1 (out of 29 layers), confirming the middle-to-late layer pattern observed in LLaVA (Figure 7). Inter",
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        "page_idx": 6
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+        "text": "7",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e6360f78f03960ad403b04644acb22b0bc42d77edb77e48f9f91d95ac419ae5.jpg",
+        "image_caption": [
+            "(a) An image of a lady in the sweater. The logit lens identifies tokens that correspond to specific detail of the sweater, such as \"pattern\" and \"diam\"(ond)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fc71183ccd89bf7e0776fd60825f5729393c66c7aded76dbf4f8be08ba7461f4.jpg",
+        "image_caption": [
+            "(b) An image of a child in a go-kart. The representations sometimes encode specific details, such as \"look\" and \"gaz\"(e) instead of just \"face\"."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6cd5c88723571b7fc1c586607647a14fa9be8ff8d4461d09cb6e7c717e9206e0.jpg",
+        "image_caption": [
+            "(c) An image of a switch. In the intermediate layers, the year and month tokens are encoded in non-English characters."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/933596ab884a1b13165eeebbb3fe428e1a451a14c54104ab1b5dbad6dbf2494b.jpg",
+        "image_caption": [
+            "(d) An image of a bunch of apples. Global features, like count, show up in background tokens, though this may be an artifact of the LM processing.",
+            "Figure 3: Examples of tokens and their positions that the logit lens yields in the late layers. The labelled tokens come from a range of layers around the middle-to-late layers. In our labelling for an image, tokens that are part of a word are completed with an educated guess e.g. \"diam\"(ond)"
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "estingly, Qwen2VL shows non-zero correspondence even in early layers, possibly due to its crossattention adapter providing better initial mapping compared to LLaVA's simpler MLP adapter. Both models demonstrate increasingly aligned token representations with object tokens in later layers, suggesting this refinement pattern is a general property of adapter-style VLMs.",
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+    },
+    {
+        "type": "text",
+        "text": "Case Study: Global Features. We sometimes find global-level features, such as numbers possibly referring to object counts, appearing in unexpected background tokens (Figure 3d). Ablating these specific tokens doesn't change the model's description, and the numbers persist in the logit lens even after ablation. This suggests these global features may be artifacts of the language model's processing rather than direct information from visual tokens—potentially explaining why CLIP-based VLMs often underperform CLIP itself on classification tasks (Zhang et al., 2024).",
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+    },
+    {
+        "type": "text",
+        "text": "Potential Applications. This may be useful for getting coarse segmentation maps for little to no extra computation, which could lead to downstream applications. For example, Liu et al. (2024) suggests to adjust the attention weights towards the visual tokens in the LM to reduce hallucinations. We speculate that directing attention towards specific objects may improve this method, though we leave further exploration to future work.",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 TRACING ATTENTION FLOW",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To understand how visual information is processed and integrated within the language model, we investigate the flow of critical information through the network. Specifically, we examine whether the",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        ],
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+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "text",
+        "text": "model directly extracts information from relevant visual tokens, or whether it aggregates information from visual tokens before transferring it to the final output token.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Method. We employ the attention knockout technique introduced by Geva et al. (2023). This method involves selectively blocking attention between specific tokens at different layers of the model, allowing us to assess the importance of various connections for the model to identify the object. Our process is as follows:",
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(1) For a given input and target layer  $\\ell$ , we artificially block specific attention connections in the multi-head self-attention (MHSA) sublayer by setting the corresponding attention mask values to negative infinity:  $M_{rc}^{\\ell + 1, j} = -\\infty \\forall j \\in [1, H]$  where  $M^{\\ell + 1, j}$  is the attention mask for head  $j$  at layer  $\\ell + 1$ , and  $r$  and  $c$  are the indices of the tokens between which we want to block attention.",
+            "(2) We apply this blocking over a window of consecutive layers, including early layers (L1-10), early to middle layers (L5-14), middle layers (L11-20), middle to late layers (L15-24), and late layers (L21-31).",
+            "(3) For each window, we measure the decrease in accuracy and prediction probability for the correct token when blocking attention under three scenarios: (i) From object tokens and their surrounding buffer to the final token. (ii) From all non-object tokens to the final token. (iii) From all visual tokens except the last row to the last row of visual tokens $^2$ ."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We conduct this experiment using LLaVA 1.5 on our curated Visual Question Answering (VQA) dataset of 100 images.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Findings. Our results are presented in Table 2.",
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We find that blocking attention from the object tokens (and their buffers) to the final token in mid-late layers leads to noticeable performance degradation. This suggests that the model directly extracts object-specific information in these later stages. Blocking attention from non-object tokens to the final token in early layers also causes some performance drop. This indicates that contextual information from the broader image is processed and integrated in the earlier stages of the model.",
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+        "type": "text",
+        "text": "Interestingly, blocking attention from visual tokens to the last row of visual tokens has minimal effect on performance. This contrasts with findings by Basu et al. (2024), who suggested that the model might summarize image information in the last row of visual tokens before using it. Our results indicate that, at least for our object identification tasks, the model doesn't rely on such a summarization step.",
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+        "type": "table",
+        "img_path": "images/5c18dbb6aad548169aeb5a31bf148a2fc1df3cf3f69c1dcd1fca97528df32713.jpg",
+        "table_caption": [
+            "Table 2: Results of blocking attention between different token groups across various layers of the LLaVA 1.5 model. The values represent the relative performance on the correct token being predicted when blocking attention, with 1.00 being no impact and 0.00 being all questions wrong post-ablation."
+        ],
+        "table_footnote": [
+            "\\*From:  $\\mathrm{O} =$  Object Tokens,  $\\mathrm{O + n = O + n}$  Buffer, I-  $(\\mathrm{O + 1}) =$  All visual tokens except  $\\mathrm{O + 1}$  , I-LVR  $=$  All visual tokens except last row  $\\dagger$  To: LTP  $=$  Last Token Position, LVR  $=$  Last Visual Token Row"
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Attention Blocking</td><td colspan=\"6\">Layer</td></tr><tr><td>From*</td><td>To†</td><td>Early</td><td>Early-Mid</td><td>Mid</td><td>Mid-Late</td><td>Late</td><td>All</td></tr><tr><td>O</td><td>LTP</td><td>1.00</td><td>1.00</td><td>0.96</td><td>0.88</td><td>0.93</td><td>0.82</td></tr><tr><td>O+1</td><td>LTP</td><td>1.00</td><td>0.99</td><td>0.90</td><td>0.82</td><td>0.89</td><td>0.67</td></tr><tr><td>O+2</td><td>LTP</td><td>1.00</td><td>1.00</td><td>0.91</td><td>0.80</td><td>0.91</td><td>0.68</td></tr><tr><td>I-(O+1)</td><td>LTP</td><td>0.88</td><td>1.00</td><td>0.97</td><td>0.96</td><td>0.98</td><td>0.82</td></tr><tr><td>O+1</td><td>LVR</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>I-LVR</td><td>LVR</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr></table>",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "2We do this as recent work by Basu et al. (2024) suggests that the model may summarise visual information in the last row of visual tokens, though they study this in a different VQA task. If the model does use the last row of visual tokens as a 'working summary', then blocking the attention from all the visual tokens to the last row should prevent it from answering the question.",
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+    {
+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
+        "text": "Testing Other Models. To verify that our attention blocking results are not specific to LLaVA-1.5, we conducted additional experiments with LLaVA-Phi-3. Rather than using fixed layer groupings, we employed a sliding window of 10 layers to obtain finer-grained results across the network. We find that the results (Figure 8) closely mirror our original results: blocking attention from object tokens to the last token position shows the strongest effect in middle-to-late layers, while blocking attention from visual tokens to the last row continues to show minimal impact.",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Explainability. Classical techniques like GRAD-Cam (Selvaraju et al., 2017) and Integrated Gradients (Sundararajan et al., 2017), focus on input attribution, while recent tools like LVLM-Interpret (Stan et al., 2024) extend these methods to VLMs. Our work complements these by investigating the model's internal processing mechanisms.",
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+    },
+    {
+        "type": "text",
+        "text": "Mechanistic Interpretability (MI) for LLMs. Techniques such as causal tracing (Meng et al., 2022) and sparse coding (Bricken et al., 2023; Huben et al., 2023; Marks et al., 2024) have advanced our understanding of how LLMs perform tasks despite being primarily trained for next-token prediction, but VLMs are not trained on next-token prediction for visual inputs (Liu et al., 2023b). Our work extends some of these techniques to understand visual information processing in multimodal contexts.",
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+    },
+    {
+        "type": "text",
+        "text": "MI for Multimodal Models. In CLIP, researchers have discovered multimodal neurons (Goh et al., 2021), decomposed image representations (Gandelsman et al., 2023; 2024), identified interpretable network subgraphs (Rajaram et al., 2024), and extracted interpretable features using sparse coding (Rao et al., 2024; Daujotas, 2024; Fry, 2024). In contrast, our work focuses on generative VLMs.",
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+    },
+    {
+        "type": "text",
+        "text": "For generative VLMs, Schwettmann et al. (2023) found neurons corresponding to visual concepts, while Palit et al. (2023) adapted causal tracing for BLIP. While these studies help us understand the role of specific components in the VLM, we investigate the representations of the visual inputs.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Basu et al. (2024) used causal tracing and attention contributions in a visual question-answering setup to find that VLMs such as LLaVA retrieve information from earlier causal layers of the LM (i.e. layers 1-4 vs layers 4-7 in an LLM) when answering a multi-modal question. Their analysis suggests that the visual information may be summarised in a consistent subset of late visual tokens. In contrast, we study VLMs in a more straightforward object-identification setting. We also examine their information transfer hypothesis through our attention blocking experiments in Section 4.2.",
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+    },
+    {
+        "type": "text",
+        "text": "Parallel Work. Concurrent research by Jiang et al. (2025) independently arrived at similar logit lens findings and developed a hallucination reduction method based on these insights. Wu et al. (2024) observed comparable logit lens patterns across both visual and audio modalities.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Our investigation into visual information processing in LLaVA reveals that object information is highly localized within visual tokens corresponding to the object's spatial location, challenging the hypothesis that VLMs primarily rely on global features in register tokens. Visual token representations evolve to align with interpretable textual concepts across layers, suggesting VLMs refine visual information toward language-like representations even without next-token prediction pretraining for visual inputs. Our attention blocking experiments suggest that VLMs may extract object information from relevant visual tokens in mid-to-late layers. These findings bridge the gap between our understanding of language and vision models, providing a foundation for more interpretable multimodal systems.",
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+    },
+    {
+        "type": "text",
+        "text": "Limitations & Future Work. We focused on LLaVA-type models as a starting point for interpreting more complex VLMs, but our results may not generalize to significantly different architectures. Our study concentrates on object identification tasks, providing a simpler setup compared to multi-step reasoning. While we tested object identification in three ways, our findings may not fully represent VLM behavior in complex tasks like reasoning or open-ended question answering. Future work should explore practical applications in reducing hallucinations and enhancing factual accuracy, while examining whether these findings extend to broader task types and model architectures.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
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+    },
+    {
+        "type": "text",
+        "text": "AUTHOR CONTRIBUTIONS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Clement Neo conceived the study, designed and performed the experiments, and led the writing of the manuscript. Luke Ong contributed to the writing and literature review. Philip Torr, Mor Geva and David Krueger provided advice and comments. Fazl Berez served as the primary advisor for this work and helped significantly with the design and write-up of the paper.",
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We thank the Torr Vision Group for providing compute and hosting Clement Neo during the internship.",
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+    },
+    {
+        "type": "text",
+        "text": "We thank Sarah Schwettmann, Ashkan Khakzar, William Rudman, Joseph Miller, Alex Spies for discussing and reading drafts of the paper.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            329,
+            287,
+            344
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in neural information processing systems, 35:23716-23736, 2022.",
+            "Andy Arditi, Oscar Obeso, Aaquib Syed, Daniel Paleka, Nina Panickssery, Wes Gurnee, and Neel Nanda. Refusal in language models is mediated by a single direction, 2024. URL https://arxiv.org/abs/2406.11717.",
+            "Luke Bailey, Gustaf Ahdritz, Anat Kleiman, Siddharth Swaroop, Finale Doshi-Velez, and Weiwei Pan. Soft prompting might be a bug, not a feature. 2023.",
+            "Samyadeep Basu, Martin Grayson, Cecily Morrison, Besmira Nushi, Soheil Feizi, and Daniela Massiceti. Understanding Information Storage and Transfer in Multi-modal Large Language Models, jun 2024. URL http://arxiv.org/abs/2406.04236.",
+            "Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Conerly, Nick Turner, Cem Anil, Carson Denison, Amanda Askell, Robert Lasenby, Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, Brayden McLean, Josiah E Burke, Tristan Hume, Shan Carter, Tom Henighan, and Christopher Olah. Towards monoseismicity: Decomposing language models with dictionary learning. Transformer Circuits Thread, 2023. https://transformercircuits.pub/2023/monoseismic-features/index.html.",
+            "Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 1877-1901. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper_files/paper/2020/file/1457c0d6bfbcb4967418bf8ac142f64a-Paper.pdf.",
+            "Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E Gonzalez, et al. Vicuna: An open-source chatbot impressing gpt-4 with  $90\\%$  * chatgpt quality. See https://vicuna.lmsys.org (accessed 14 April 2023), 2(3):6, 2023.",
+            "Timothee Darcet, Maxime Oquab, Julien Mairal, and Piotr Bojanowski. Vision transformers need registers. In The Twelfth International Conference on Learning Representations, 2024."
+        ],
+        "bbox": [
+            171,
+            353,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
+            490,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Gytis Daujotas. Case study: Interpreting, manipulating, and controlling clip with sparse autoencoders, 2024. URL https://www.lesswrong.com/posts/iYFuZo9BMvr6GgMs5/case-study-interpreting-manipulating-and-controlling-clip. Accessed: 2024-09-24.",
+            "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248-255, 2009. doi: 10.1109/CVPR.2009.5206848.",
+            "Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=YicbFdNTTy.",
+            "Nelson Elhage, Neel Nanda, Catherine Olsson, Tom Henighan, Nicholas Joseph, Ben Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Nova DasSarma, Dawn Drain, Deep Ganguli, Zac Hatfield-Dodds, Danny Hernandez, Andy Jones, Jackson Kernion, Liane Lovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish, and Chris Olah. A mathematical framework for transformer circuits. Transformer Circuits Thread, 2021. https://transformer-circuits.pub/2021/framework/index.html.",
+            "Hugo Fry. Towards multimodal interpretability: Learning sparse interpretable features in vision transformers, 2024. URL https://www.lesswrong.com/posts/bCtbuWraqYTDtuARg/towards-multimodal-interpretability-learning-sparse-2#Future_Work. Accessed: 2024-06-28.",
+            "Yossi Gandelsman, Alexei A. Efros, and Jacob Steinhardt. Interpreting clip's image representation via text-based decomposition, 2023.",
+            "Yossi Gandelsman, Alexei A. Efros, and Jacob Steinhardt. Interpreting the second-order effects of neurons in clip, 2024.",
+            "Mor Geva, Avi Caciularu, Kevin Wang, and Yoav Goldberg. Transformer feed-forward layers build predictions by promoting concepts in the vocabulary space. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pp. 30-45, 2022.",
+            "Mor Geva, Jasmijn Bastings, Katja Filippova, and Amir Globerson. Dissecting recall of factual associations in auto-regressive language models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pp. 12216-12235, 2023.",
+            "Gabriel Goh, Nick Cammarata †, Chelsea Voss †, Shan Carter, Michael Petrov, Ludwig Schubert, Alec Radford, and Chris Olah. Multimodal neurons in artificial neural networks. *Distill*, 2021. doi: 10.23915/distill.00030. https://distill.pub/2021/multimodal-neurons.",
+            "Robert Huben, Hoagy Cunningham, Logan Riggs Smith, Aidan Ewart, and Lee Sharkey. Sparse autoencoders find highly interpretable features in language models. In *The Twelfth International Conference on Learning Representations*, 2023.",
+            "Chaoya Jiang, Haiyang Xu, Mengfan Dong, Jiaxing Chen, Wei Ye, Ming Yan, Qinghao Ye, Ji Zhang, Fei Huang, and Shikun Zhang. Hallucination augmented contrastive learning for multimodal large language model, 2024. URL https://arxiv.org/abs/2312.06968.",
+            "Nick Jiang, Anish Kachinthaya, Suzie Petryk, and Yossi Gandelsman. Interpreting and editing vision-language representations to mitigate hallucinations, 2025. URL https://arxiv.org/abs/2410.02762.",
+            "Sonia Joseph and Neel Nanda. Laying the foundations for vision and multimodal mechanistic interpretability & open problems, 2024. URL https://wwwalignmentforum.org/posts/kobJymvvcvhbjWFKe/ laying-the-foundations-for-vision-and-multimodal-mechanistic. Accessed: 2024-06-28."
+        ],
+        "bbox": [
+            171,
+            103,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Brian Lester, Rami Al-Rfou, and Noah Constant. The power of scale for parameter-efficient prompt tuning. In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wentau Yih (eds.), Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pp. 3045-3059, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.emnlp-main.243. URL https://aclanthology.org/2021.emnlp-main.243.",
+            "Junnan Li, Dongxu Li, Silvio Savarese, and Steven Hoi. Blip-2: Bootstrapping language-image pre-training with frozen image encoders and large language models. In International conference on machine learning, pp. 19730–19742. PMLR, 2023.",
+            "Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dóllár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In Computer Vision-ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13, pp. 740-755. Springer, 2014.",
+            "Haotian Liu, Chunyuan Li, Yuheng Li, and Yong Jae Lee. Improved baselines with visual instruction tuning, 2023a.",
+            "Haotian Liu, Chunyuan Li, Qingyang Wu, and Yong Jae Lee. Visual instruction tuning. In NeurIPS, 2023b.",
+            "Shi Liu, Kecheng Zheng, and Wei Chen. Paying more attention to image: A training-free method for alleviating hallucination in lvlms, 2024. URL https://arxiv.org/abs/2407.21771.",
+            "Samuel Marks, Can Rager, Eric J Michaud, Yonatan Belinkov, David Bau, and Aaron Mueller. Sparse feature circuits: Discovering and editing interpretable causal graphs in language models. arXiv preprint arXiv:2403.19647, 2024.",
+            "Kevin Meng, David Bau, Alex Andonian, and Yonatan Belinkov. Locating and editing factual associations in GPT. Advances in Neural Information Processing Systems, 36, 2022. arXiv:2202.05262.",
+            "Jack Merullo, Louis Castricato, Carsten Eickhoff, and Ellie Pavlick. Linearly mapping from image to text space. In The Eleventh International Conference on Learning Representations, 2022.",
+            "Nostalgebraist. Interpreting gpt: The logit lens. https://wwwalignmentforum.org/ posts/AcKRB8wDpdaN6v6ru/interpreting-gpt-the-logit-lens, Aug 2020. Accessed: 23 Sep 2024.",
+            "Vedant Palit, Rohan Pandey, Aryaman Arora, and Paul Pu Liang. Towards vision-language mechanistic interpretability: A causal tracing tool for blip. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2856-2861, 2023.",
+            "Xu Pan, Aaron Philip, Ziqian Xie, and Odelia Schwartz. Dissecting query-key interaction in vision transformers. ICML MI Workshop, 2024.",
+            "Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.",
+            "Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision, 2021. URL https://arxiv.org/abs/2103.00020.",
+            "Achyuta Rajaram, Neil Chowdhury, Antonio Torralba, Jacob Andreas, and Sarah Schwettmann. Automatic discovery of visual circuits, 2024. URL https://arxiv.org/abs/2404.14349.",
+            "Sukrut Rao, Sweta Mahajan, Moritz Böhle, and Bernt Schiele. Discover-then-name: Task-agnostic concept bottlenecks via automated concept discovery, 2024. URL https://arxiv.org/abs/2407.14499.",
+            "Sarah Schwettmann, Neil Chowdhury, Samuel Klein, David Bau, and Antonio Torralba. Multimodal neurons in pretrained text-only transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2862-2867, 2023."
+        ],
+        "bbox": [
+            171,
+            102,
+            826,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pp. 618-626, 2017.",
+            "Gabriela Ben Melech Stan, Estelle Aflalo, Raanan Yehezkel Rohekar, Anahita Bhiwandiwalla, ShaoYen Tseng, Matthew Lyle Olson, Yaniv Gurwicz, Chenfei Wu, Nan Duan, and Vasudev Lal. Lvlm-intrepret: An interpretability tool for large vision-language models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8182-8187, 2024.",
+            "Alessandro Stolfo, Yonatan Belinkov, and Mrinmaya Sachan. A mechanistic interpretation of arithmetic reasoning in language models using causal mediation analysis. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pp. 7035-7052, 2023.",
+            "Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In International conference on machine learning, pp. 3319-3328. PMLR, 2017.",
+            "Maria Tsimpoukelli, Jacob L Menick, Serkan Cabi, SM Eslami, Oriol Vinyals, and Felix Hill. Multimodal few-shot learning with frozen language models. Advances in Neural Information Processing Systems, 34:200-212, 2021.",
+            "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.",
+            "Martina G Vilas, Timothy Schaumlöffel, and Gemma Roig. Analyzing vision transformers for image classification in class embedding space. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.",
+            "Kevin Ro Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt. Interpretability in the wild: a circuit for indirect object identification in gpt-2 small. In *The Eleventh International Conference on Learning Representations*, 2023.",
+            "Peng Wang, Shuai Bai, Sinan Tan, Shijie Wang, Zhihao Fan, Jinze Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, et al. Qwen2-vl: Enhancing vision-language model's perception of the world at any resolution. arXiv preprint arXiv:2409.12191, 2024.",
+            "Chris Wendler, Veniamin Veselovsky, Giovanni Monea, and Robert West. Do llamas work in English? on the latent language of multilingual transformers. In Lun-Wei Ku, Andre Martins, and Vivek Srikumar (eds.), Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 15366-15394, Bangkok, Thailand, August 2024. Association for Computational Linguistics. doi: 10.18653/v1/2024.acl-long.820. URL https://aclanthology.org/2024.acl-long.820.",
+            "Zhaofeng Wu, Xinyan Velocity Yu, Dani Yogatama, Jiasen Lu, and Yoon Kim. The semantic hub hypothesis: Language models share semantic representations across languages and modalities, 2024. URL https://arxiv.org/abs/2411.04986.",
+            "Yuhui Zhang, Alyssa Unell, Xiaohan Wang, Dhruba Ghosh, Yuchang Su, Ludwig Schmidt, and Serena Yeung-Levy. Why are visually-grounded language models bad at image classification? arXiv preprint arXiv:2405.18415, 2024."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
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+        ],
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+    },
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+    {
+        "type": "image",
+        "img_path": "images/5ea2e162ae25ec2b4099e8da5244b514048acd70a256f861df0d4468b2839f61.jpg",
+        "image_caption": [
+            "A FILTERED DATASET EXAMPLES"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            147,
+            470,
+            299
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
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+        "image_caption": [
+            "Figure 4: Example of dataset images for the object identification tasks."
+        ],
+        "image_footnote": [],
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+        "img_path": "images/6bd63076bd059ed82fb53a0905a217e4c1a9557186137f1a72682bd9a2b27266.jpg",
+        "image_caption": [],
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
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+    },
+    {
+        "type": "page_number",
+        "text": "15",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B CURATED IMAGES AND QUESTIONS EXAMPLES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            101,
+            606,
+            119
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb5b6f9f27f8004d1f4fd510ff514f0d8d0a7db1b9a39a89c110575874be4119.jpg",
+        "image_caption": [
+            "(a) \"What is on the bench? It is a\"  $\\rightarrow$  book"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            166,
+            493,
+            469
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e551b2609818e75fb34f48ab0d4cd1044fb03d3e9e74c02469e8303cd61fdac.jpg",
+        "image_caption": [
+            "(b) \"What is below the street sign? It is a\"  $\\rightarrow$  bottle"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            503,
+            282,
+            823,
+            469
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/03e9580f5ba4c85b2c31072badfaf876675ff3de31108670dc07ca4e31ad5e15.jpg",
+        "image_caption": [
+            "(c) \"What is on the truck? It is a\"  $\\rightarrow$  dog"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            500,
+            493,
+            686
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b566fa7956d4d0b36d726ebeed8f470646daa713e5df8ad46469403e480b8bb0.jpg",
+        "image_caption": [
+            "(d) \"What is on the seat? It is a\"  $\\rightarrow$  cup",
+            "Figure 5: Examples of curated images and their questions. We ask the model the question and prefill its answer with \"It is a\". The correct answer is underlined."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            503,
+            520,
+            823,
+            686
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_number",
+        "text": "16",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C OBJECT CORRESPONDENCES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            450,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c526e66913a0c1455c2acaf26c6aa499671ba16612eaa7b598837c0e5cbf5336.jpg",
+        "image_caption": [
+            "Figure 6: Token-to-object correspondence in LLaVA-1.5 across transformer layers. Analysis of 170 COCO images shows peak correspondence (23.7%) occurs in later layers (average layer 25.7/33)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            148,
+            763,
+            330
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/649abe2e0f27e5d60c391c5cd808e2704e96783f686087642d3e35885251586b.jpg",
+        "image_caption": [
+            "Figure 7: Token-to-object correspondence in Qwen2VL-2B showing increasing alignment in deeper layers, peaking at layer 25.1/29 with  $6.5\\%$  correct mapping. Unlike LLaVA, Qwen exhibits non-zero correspondence in early layers."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            405,
+            818,
+            595
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
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+            490,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D LLAVA-PHI-3 BLOCKING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            101,
+            426,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/00d3abc71a437ee6bd62820f57d5f5b6684145fbb0b47df49c1d9b10c8736984.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            138,
+            790,
+            256
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1efd0bc4878b55c7dde09ae394f1e7744aaa2b1c33942192ff0de3120823003.jpg",
+        "image_caption": [],
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+            207,
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+    {
+        "type": "image",
+        "img_path": "images/4203283f1e67817cd019b318f5e9d2cb897284aaaa6f65a093e86a6fe61b309b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+            207,
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+            476
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1247937820d1280504c529e1c9bd71a0f8813fa28549bf569c62eb724bf30a93.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+            207,
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+            585
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/6d0ac4b34ad4f749b6e4ef20364e0b1b995da540b2fc8423d582cf7c3827f451.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            207,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/287cce761d01a645bfb8270fab8c28b6fa5a14ff87070fc8b580b8bad0ee757b.jpg",
+        "image_caption": [
+            "Figure 8: Attention blocking results for LLaVA-Phi-3 using a sliding window of 10 layers. We find that similar to LLaVA-1.5, blocking attention from object tokens to the last token position has the strongest effect in middle-to-late layers, while blocking attention from visual tokens to the last visual row shows minimal impact throughout the network."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            699,
+            789,
+            806
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
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+        ],
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+    },
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+]

2025/Towards Interpreting Visual Information Processing in Vision-Language Models/600bd478-2d40-4b3d-957a-a9f71760f4ee_origin.pdf

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2025/Towards Interpreting Visual Information Processing in Vision-Language Models/full.md

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+# TOWARDS INTERPRETING VISUAL INFORMATION PROCESSING IN VISION-LANGUAGE MODELS
+
+Clement Neo $^{1*}$ , Luke Ong $^{1}$ , Philip Torr $^{2}$ , Mor Geva $^{3}$ , David Krueger $^{4}$ , Fazl Berez $^{2,5}$
+
+$^{1}$ Nanyang Technological University  $^{2}$ University of Oxford  $^{3}$ Tel Aviv University
+
+4MILATangentic
+
+# ABSTRACT
+
+Vision-Language Models (VLMs) are powerful tools for processing and understanding text and images. We study the processing of visual tokens in the language model component of LLaVA, a prominent VLM. Our approach focuses on analyzing the localization of object information, the evolution of visual token representations across layers, and the mechanism of integrating visual information for predictions. Through ablation studies, we demonstrated that object identification accuracy drops by over  $70\%$  when object-specific tokens are removed. We observed that visual token representations become increasingly interpretable in the vocabulary space across layers, suggesting an alignment with textual tokens corresponding to image content. Finally, we found that the model extracts object information from these refined representations at the last token position for prediction, mirroring the process in text-only language models for factual association tasks. These findings provide crucial insights into how VLMs process and integrate visual information, bridging the gap between our understanding of language and vision models, and paving the way for more interpretable and controllable multimodal systems.
+
+# 1 INTRODUCTION
+
+Vision-Language Models (VLMs) take an image and text as input, and generate a text output. They have become powerful tools in processing and understanding text and images, enabling a wide range of applications from question answering to image captioning (Liu et al., 2023b; Li et al., 2023; Alayrac et al., 2022). Among VLM architectures, the "adapter" style has demonstrated impressive state-of-the-art performance, notable for its simplicity. These models combine a pre-trained image encoder, a pre-trained language model (LM), and a learned adapter network that maps image encoder outputs to soft prompts for LM inputs (Merullo et al., 2022; Liu et al., 2023b;a). Despite their importance and potential, the inner workings of VLMs remain poorly understood as compared to their text-only counterparts. While significant progress has been made in understanding language models (Elhage et al., 2021; Wang et al., 2023; Bricken et al., 2023) and, to a lesser extent, vision transformers (Palit et al., 2023; Vilas et al., 2023; Pan et al., 2024), there is a notable gap in our understanding of VLMs, where both modalities operate in the same space. A deeper comprehension of VLMs could lead to practical advances in safety, robustness, and functionality, similar to the progress made in language models, such as model editing (Meng et al., 2022) or identifying specific components responsible for certain behaviors (Arditi et al., 2024).
+
+In this paper, we investigate the LM component of VLMs as they serve as the "brain" of the system, comprising more than  $95\%$  of the total parameters for models like LLaVA 1.5 7B. LLMs pre-trained on large text datasets are likely to remain the foundation of VLMs because of their powerful reasoning capabilities. Therefore, understanding how these LMs process and integrate visual information is crucial for advancing VLM interpretability.
+
+Many open questions remain regarding the inner workings of VLMs. The nature and structure of visual representations fed into the LM remain unclear, as these visual inputs are soft prompts that
+
+do not correspond to language tokens and cannot be interpreted as such (Lester et al., 2021; Merullo et al., 2022). This raises questions about how visual information is encoded in these representations. Furthermore, the spatial distribution of information within the image encoder's feature maps remains unclear, leaving open the question of whether and how much object-specific details are localized or dispersed across the entire representation.
+
+Furthermore, the mechanisms by which the language model processes these visual inputs are not well understood. A significant modality gap exists between visual and textual input (Jiang et al., 2024), and it is unclear whether our mechanistic understanding of text processing in language models generalizes to the processing of visual input in VLMs.
+
+To understand the representations in the visual inputs for VLM and how the VLM processes them, we study LLaVA 1.5 7B (Liu et al., 2023a), a popular open-source VLM with competitive performance that uses CLIP (Radford et al., 2021) as an image encoder and Vicuna 13B (Chiang et al., 2023) as its language model. We also test on LLaVA-  $\mathsf{Phi}^{\dagger}$  and Qwen2-VL (Wang et al., 2024). We focus on a set of object identification tasks and our main contributions are as follows:
+
+1. Using ablation techniques, we demonstrate that the information for an object is highly localized to the token positions corresponding to their original location in the image.  
+2. By extending the logit lens technique primarily used for language models (Nostalgebraist, 2020), we find that the representations of the visual input in the LM are refined towards the embedding of interpretable tokens in the vocabulary, despite the LM not being explicitly trained to do so.  
+3. By blocking the attention flow between tokens, we show that the model extracts object information from the object tokens in the middle to late layers.
+
+Our findings are first steps in understanding the internal mechanisms of VLMs, paving the way for more interpretable and controllable multimodal systems. The code for our experiments is available at https://github.com/clemneo/llava-interp.
+
+# 2 BACKGROUND
+
+Transformer Architecture. Transformer-based autoregressive Large Language Models (LLMs) process sequences of input tokens to predict the next token in the sequence (Vaswani et al., 2017). These models achieved state-of-the-art performance across various natural language processing tasks (Brown et al., 2020), even though they are merely trained on next-token prediction (Radford et al., 2019). A LLM takes an input sequence  $X = (x_{1},\dots,x_{n})$  and outputs a probability distribution over the vocabulary  $V$  to predict the next token  $x_{n + 1}$ .
+
+To do so, the model refines token representations layer by layer through a series of computations. Each token  $x_{i}$  is initially represented by an embedding vector  $h_{i}^{0}$  of dimension  $d_{m}$ , obtained from a lookup operation in an embedding matrix  $W_{E} \in \mathbb{R}^{|V| \times d_{m}}$ . This representation is then updated through successive layers using multi-head self-attention (MHSA) and feed-forward (FF) sublayers:
+
+$$
+h _ {i} ^ {l} = h _ {i} ^ {l - 1} + a _ {i} ^ {l} + f _ {i} ^ {l} \tag {1}
+$$
+
+where  $h_i^l$  is the representation of token  $x_i$  at layer  $l$ ,  $a_i^l$  is the output from the MHSA sublayer, and  $f_i^l$  is the output from the FF sublayer. All vectors  $h_i^l, a_i^l, f_i^l \in \mathbb{R}^{d_m}$ .
+
+The MHSA sublayer consists of  $H$  attention heads working in parallel, each implemented as follows:
+
+$$
+\operatorname {A t t e n t i o n} (Q, K, V) = \operatorname {s o f t m a x} \left(\frac {Q K ^ {T}}{\sqrt {d _ {k}}} + M\right) V \tag {2}
+$$
+
+where  $Q, K, V \in \mathbb{R}^{n \times d_k}$  are query, key, and value matrices derived from learned linear projections of the input, and  $M \in \mathbb{R}^{n \times n}$  is a causal mask. The causal mask  $M$  is set to  $-\infty$  in the upper right triangle, meaning  $M_{ij} = 0$  if  $i \geq j$ , else  $-\infty$ . This ensures that each position can only attend to previous positions and itself, thus preserving the autoregressive property in language models. These
+
+![](images/4b691183acdfd479a0d5c4e86f753e356bbe9b572fbc0cd48df65f809ba3b70f.jpg)  
+Figure 1: In adapter-style Vision-Language Models (VLMs), the visual tokens (in green) are soft prompts for the language model (LM), and are not interpretable through the vocabulary embedding. Through a set of object identification tasks, we find that (1) object information can be localized to a subset of visual tokens, (2) the representations of the visual tokens evolve towards interpretable text embeddings, and (3) the model extracts some information from the visual tokens to the last token position in the middle to late layers, to identify the object.
+
+outputs are then concatenated and linearly projected to produce the final output of the multi-head attention sublayer.
+
+The FF sublayer applies two linear transformations with an element-wise non-linear function  $\sigma$  between them:
+
+$$
+f _ {i} ^ {l} = W _ {v} ^ {l} \sigma \left(W _ {k} ^ {l} a _ {i} ^ {l} + b _ {k} ^ {l}\right) + b _ {v} ^ {l}, \tag {3}
+$$
+
+where  $W_{v}^{l}, W_{k}^{l}, b_{k}^{l}$ , and  $b_{v}^{l}$  are learned parameters. Finally, the representation of the last token  $x_{n}$  at the final layer,  $h_{n}^{L}$ , is projected into a probability distribution over  $V$  using an unembedding matrix  $W_{U} \in \mathbb{R}^{d \times |V|}$  and a softmax operation.
+
+Image Encoders. The transformer architecture has been successfully adapted for state-of-the-art image encoders, as demonstrated by Dosovitskiy et al. (2021). In this approach, input images are divided into fixed-size patches, flattened, and linearly embedded before being processed as a sequence of tokens, analogous to word embeddings in NLP tasks. Unlike text-based models, the attention mechanism in these image encoders is bidirectional (i.e., no masking is applied). The input sequence includes a special class token, whose final embedding is typically used for image classification tasks. The representations at the other token positions form what is called the feature map of the image.
+
+Building upon this foundation, Radford et al. (2021) introduced Contrastive Language-Image Pretraining (CLIP), which jointly trains a vision transformer and a textTransformer to produce similar embeddings for matching image-text pairs. CLIP's image encoder is currently being used for state-of-the-art performance on many downstream tasks like zero-shot image classification and vision-language models like LLaVA.
+
+Connecting Text and Image Models. Tsimpoukelli et al. (2021) pioneered the concept of training an image encoder such that its output can be used by a frozen language model for multimodal few-shot reasoning. Merullo et al. (2022) advanced this approach by freezing both the pre-trained image encoder and pre-trained language model, training only a linear map that converts image features into inputs for the text model. This introduced the "adapter style" approach, utilizing pre-trained components for both modalities.
+
+Building on this foundation, Liu et al. (2023b) introduced LLaVA, which combined a CLIP ViT-L/14 image encoder with a Vicuna-13B language model, connected by a trained linear projection, or an MLP for improved performance (Liu et al., 2023a). Crucially, LLaVA's training process focuses solely on fine-tuning for multimodal conversation, without any next-token pretraining on image-text pairs. This approach allows LLaVA to leverage the strong pre-trained capabilities of both the vision and language models while efficiently adapting to multimodal tasks.
+
+# 2.1 NOTATION
+
+In this section, we describe how an image and query is processed through LLaVA.
+
+Image Processing. The input image  $I$  is processed by first being cropped into a square, then suitably resized before being divided into 576 image patches. The CLIP ViT-L/14 image encoder  $f_{\mathrm{CLIP}}$  processes this image to produce a feature map  $E_I = f_{\mathrm{CLIP}}(I) \in \mathbb{R}^{N \times d_{\mathrm{CLIP}}}$ , where  $N = 576$  is the number of image patches and  $d_{\mathrm{CLIP}}$  is the dimensionality of CLIP's embeddings. An adapter network  $A$  then maps these CLIP embeddings to the language model's input space, yielding a set of visual tokens  $E_A = A(E_I) \in \mathbb{R}^{N \times d}$ , where  $d_m$  is the dimensionality of the language model's input embeddings. We call  $E_A$  visual tokens to distinguish them from the image tokens.
+
+Text Processing and Combined Input. For the text input, given a tokenized prompt sequence  $T = (t_{1}, \ldots, t_{M})$ , the embedding layer of the language model  $E_{\mathrm{LM}}$  maps these tokens to embeddings:  $E_{T} = E_{\mathrm{LM}}(T) \in \mathbb{R}^{M \times d_{\mathrm{LM}}}$ . The full input to the language model is the concatenation of the adapted image embeddings and the text embeddings:  $X = [E_{A}; E_{T}] \in \mathbb{R}^{(N + M) \times d_{\mathrm{LM}}}$ , from which the language model can generate output autoregressively.
+
+# 2.2 OPEN QUESTIONS AND HYPOTHESES
+
+One might expect that the output of the adapter  $E_{A}$  would produce embeddings corresponding to image information. For example, if the image contained a car, we might expect the adapter to produce tokens that correspond to the embedding of "car". However, the output of the adapter forms soft prompts that have no semantic meaning when decoded through the vocabulary (Merullo et al., 2022; Bailey et al., 2023).
+
+# 2.2.1 LOCALIZATION OF OBJECT-SPECIFIC INFORMATION
+
+A key question of our investigation is: Where is object-specific information located in visual tokens? Two conflicting hypotheses emerge from recent research:
+
+1. Object-Centric Localization: Joseph & Nanda (2024) found that in classical vision transformers, token representations corresponding to specific object locations tended to align with those objects' class embeddings in the late layers, even without explicitly training for this behavior. This suggest that object information might be localized to the tokens corresponding to the spatial location of the object in the original image. However, their findings were on a vision transformer trained on ImageNet classes, and not the vision transformer in CLIP.  
+2. Global Information in Register Tokens: In contrast, Darcet et al. (2024) identified register tokens in vision transformers, including a CLIP variant. These are background patch tokens with unusually high norms that appear to encode global image features. It follows that the LM in an adapter-style VLM could primarily rely on these information-rich register tokens for its output.
+
+We investigate these hypotheses in Section 3 using ablation techniques in a set of object identification tasks.
+
+# 2.2.2 PROCESSING OF OBJECT-SPECIFIC REPRESENTATION
+
+The processing of visual tokens may differ from text tokens. Soft prompts are very different from normal prompts (Bailey et al., 2023), and there is a modality gap between textual and visual representations in VLMs (Jiang et al., 2024). Furthermore, LLaVA is fine-tuned exclusively on visual question-answering (VQA) prompt-response pairs, without pre-training on next-token prediction.
+
+This may affect how the model processes visual information as compared to text. In Section 4.1, we attempt the logit lens method to see how the representations of visual tokens evolve through the layers.
+
+Recent studies have shed light on how LMs process information. Geva et al. (2023) identified a refine-then-extract process in factual association prompts, where the model first refines information at subject tokens through MLP sublayers before transferring it to the last token position via attention sublayers. Similar mechanisms have been observed in arithmetic tasks (Stolfo et al., 2023), suggesting that this (sequencing of functions) might be a general mechanism in LMs.
+
+However, Basu et al. (2024) adapted the factual association setup to VLMs and found preliminary evidence that the LM may first summarize image information in text tokens, instead of extracting the image information to the last token position from the image tokens directly. While we do not study factual association – rather we choose to focus on a simpler object identification task, we study how information flows from the image tokens in Section 4.2.
+
+# 3 INVESTIGATING THE REPRESENTATIONS IN MAPPED VISUALTOKENS
+
+In this section, we use ablation experiments to test if object information is concentrated in specific visual tokens. By ablating selected tokens and observing degradations in the model's ability to identify the object, we map how object information is distributed across the visual tokens.
+
+Dataset. We use images from the COCO Detection Training set (Lin et al., 2014). To ensure the reliability of our results, we employ two filtering steps:
+
+(1) Choosing Simpler Images. As a heuristic to focus on simpler images, we choose images where (a) there is an object whose size is between 1,000-2,000 square pixels, or about  $2 - 4\%$  of the image by area, (b) only one instance of the that object is present in the image, and (c) there are less than 4 annotated objects. We present a few examples of the images in Figure 4.
+
+(2) Controlling for Hallucination. Model can sometimes hallucinate objects based on context, even when the object's visual information is removed. To address this, we create two versions of each image: the original and one where the target object is masked with noise. We only keep images where the model correctly identifies the object in the original image but fails to identify it in the masked version. This ensures that the model's object identification relies on the object's visual information.
+
+After applying both filtering steps, our final dataset comprises 4,318 images.
+
+Method. We present a high level overview of our methodology in Figure 2. Let  $E_A = \{e_1, \dots, e_N\}$  be the set of visual tokens. We define a subset  $S \subset \{1, \dots, N\}$  consisting of the indices of tokens hypothesized to contain information about a particular object  $o$  in the image.
+
+For each ablation experiment, we create a modified set of embeddings  $E_A'$ :
+
+$$
+e _ {i} ^ {\prime} = \left\{ \begin{array}{l l} \bar {e} & \text {i f} i \in S \\ e _ {i} & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+where  $\bar{e} = \frac{1}{N}\sum_{i=1}^{N} e_i$  is the mean embedding across all visual tokens from 50,000 images from the ImageNet validation split (Deng et al., 2009). This replaces the hypothesized object-relevant tokens with an average token, effectively ablating their specific information. We do mean ablation to preserve the norm of the image token and keep them in-distribution, as their norms are typically much higher than the norm of text tokens (Bailey et al., 2023). We then evaluate the impact of token ablation using three methods:
+
+1. Generative Description: We prompt the model with "Describe the image" using both the original  $(E_A)$  and ablated embeddings  $(E_A^{\prime})$ . We then ascertain whether the target object  $o$  is produced verbatim in the model's generated description before and after ablation. If  $o$  is mentioned with  $E_A$  and not  $E_A^\prime$ , it indicates that the ablated tokens were crucial for the model to identify and include the object in its description.  
+2. Binary Polling: We ask the model "Is there a [o] in this image?" using both  $E_A$  and  $E_A'$ . We then compare whether the model's next generated token changes from "Yes" to
+
+![](images/4d4afe056bb1dc431c372f1b9db3b213e973e755d8688c929fa8490b44748315.jpg)  
+Figure 2: Overview of our ablation experiments. (1) We ablate some visual tokens that potentially contain object-specific information, (2) prompt the model to describe the image, or answer object-specific questions, then (3) measure the impact of token ablation by calculating the percentage of initially correct object identifications that become incorrect after ablation.
+
+"No" after ablation, which indicates that the ablated tokens were crucial for identifying the object.
+
+3. Visual Question-Answering: We manually curate a set of 100 images with specific questions about objects, such as "What is on the bed?" We choose questions with unambiguous answers and avoid objects directly related to the question (e.g., avoiding "pillow" for a bed-related question). We prefill the model's response with "It is a" and compare the next generated token before and after ablation. A set of examples is in Figure 5.
+
+We choose the subset  $S$  of tokens to ablate in five ways:
+
+(1) Object Tokens. We choose the visual tokens whose positions correspond to the image patches that originally contained that object. (2) Object Tokens with Buffer. In addition to the object tokens, we include neighboring tokens. We test with two buffer sizes: 1 Buffer (including immediately adjacent tokens) and 2 Buffer (including tokens up to 2 positions away from object tokens). (3) Register Tokens. We select visual tokens with norms more than 2 standard deviations from the mean norm. These correspond to the register tokens identified by Darcet et al. (2024), which are thought to encode global image features. (4) Random Tokens. As a baseline, we ablate  $n$  random tokens. (5) High-Gradient Tokens. As a stronger baseline, we use the Integrated Gradients attribution method (Sundararajan et al., 2017) to identify tokens most important for the model's decision. For a given object  $o$ , we prompt the model with "Is there a [o] in the image?" and compute integrated gradients with respect to the logit for the "Yes" token over 50 steps. We then ablate the  $n$  tokens with the highest attribution scores.
+
+Results. Our results are presented in Table 1. We find that abating the object tokens significantly impairs the model's ability to identify the object. For comparable numbers of ablated tokens, object token ablation consistently results in larger performance decreases across all settings as compared to the gradient-based and random baselines. This suggests that the information about that object is localised to the region of the object token. Furthermore, these findings are consistent across all three experimental settings and applicable to both LLaVA 1.5 and LLaVA-Phi.
+
+# 4 INVESTIGATING VISUAL INFORMATION PROCESSING
+
+# 4.1 ANALYZING THE RESIDUAL STREAM THROUGH EMBEDDING SPACE
+
+Can we interpret how the representations at the visual token positions evolve through the layers? To do so, we employ the logit lens technique (Nostalgebraist, 2020): for each layer, we decode the activation at each token position using the unembedding. Formally, for each hidden state  $h_i^l$  at each token position  $i$  and layer  $l$ , we project it into a probability distribution over  $V$  using  $W_U$  and take the token with the highest logit. We do this for LLaVA 1.5.
+
+Table 1: Performance degradation after token ablation. A lower percentage indicates a greater impact of the ablation, meaning the model is more likely to answer incorrectly, thus suggesting that the ablated tokens contain more localized object information. Average token counts apply to both the generative and polling settings for LLaVA 1.5. We find that ablating the object tokens with adjacent tokens (33.4 tokens on average, in **bold**) significantly causes object identification performance to drop, more so than the integrated gradients and random baselines for a comparable number of tokens (40 tokens, in **bold**).  
+
+<table><tr><td>Ablation Type (Avg Token Count)</td><td>Generative Decrease (%)</td><td>LLaVA 1.5 Polling Decrease (%)</td><td>VQA Decrease (%)</td><td>Generative Decrease (%)</td><td>LLaVA-Phi Polling Decrease (%)</td><td>VQA Decrease (%)</td></tr><tr><td>Object (12.6)</td><td>33.33</td><td>15.38</td><td>33.33</td><td>47.46</td><td>28.48</td><td>41.51</td></tr><tr><td>+1 Buffer (33.4)</td><td>71.79</td><td>51.28</td><td>86.67</td><td>82.89</td><td>86.48</td><td>96.23</td></tr><tr><td>+2 Buffer (60)</td><td>76.92</td><td>64.10</td><td>92.22</td><td>89.56</td><td>95.49</td><td>100.00</td></tr><tr><td>Register tokens (3.1)</td><td>5.13</td><td>0.00</td><td>3.33</td><td>7.63</td><td>0.00</td><td>1.89</td></tr><tr><td>Int. Gradients (5)</td><td>2.56</td><td>0.00</td><td>6.67</td><td>17.11</td><td>1.23</td><td>28.30</td></tr><tr><td>Int. Gradients (10)</td><td>20.51</td><td>2.56</td><td>6.67</td><td>25.26</td><td>4.30</td><td>24.53</td></tr><tr><td>Int. Gradients (20)</td><td>33.33</td><td>20.51</td><td>17.78</td><td>41.14</td><td>9.63</td><td>37.74</td></tr><tr><td>Int. Gradients (40)</td><td>48.72</td><td>35.90</td><td>50.00</td><td>61.49</td><td>26.64</td><td>56.60</td></tr><tr><td>Int. Gradients (60)</td><td>58.97</td><td>48.72</td><td>71.11</td><td>70.61</td><td>40.98</td><td>69.81</td></tr><tr><td>Int. Gradients (100)</td><td>76.92</td><td>51.28</td><td>84.44</td><td>81.23</td><td>61.68</td><td>83.02</td></tr><tr><td>Int. Gradients (250)</td><td>76.92</td><td>69.23</td><td>97.78</td><td>91.93</td><td>90.16</td><td>98.11</td></tr><tr><td>Random (5)</td><td>0.00</td><td>0.00</td><td>1.11</td><td>3.68</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (10)</td><td>4.88</td><td>0.00</td><td>0.00</td><td>3.95</td><td>0.00</td><td>1.89</td></tr><tr><td>Random (20)</td><td>2.44</td><td>0.00</td><td>2.22</td><td>5.88</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (40)</td><td>2.44</td><td>0.00</td><td>0.00</td><td>8.51</td><td>0.00</td><td>0.00</td></tr><tr><td>Random (60)</td><td>7.32</td><td>0.00</td><td>1.11</td><td>9.39</td><td>0.41</td><td>3.77</td></tr><tr><td>Random (100)</td><td>7.32</td><td>0.00</td><td>4.44</td><td>11.23</td><td>0.20</td><td>7.55</td></tr><tr><td>Random (250)</td><td>24.39</td><td>0.00</td><td>16.67</td><td>20.61</td><td>1.84</td><td>15.09</td></tr></table>
+
+Results. We find that the activations in the late layers at each visual token position correspond to token embeddings that describe its original patch object in its corresponding image token. We provide some case studies in Figure 3, and an interactive demo on more examples.
+
+To quantify this phenomenon, we analyzed 170 COCO validation images with objects of sizes between 20,000 and 30,000 square pixels (approximately 1/2 width and 1/3 height of the image). We found that in the best-performing layer for each image, an average of  $23.7\%$  of the object patch token positions correspond to the correct object class token. The best-performing layer occurs on average at layer 25.7 (out of 33 layers), confirming that mid-to-late layers tend to develop the strongest object-token correspondences (Figure 6).
+
+We found that the tokens decoded often correspond to concepts more specific than the object-level. For example, in Figure 3a, the tokens go beyond identifying the sweater worn ("swe") and describe the specific patterns on the sweater with tokens like "diam"(ond) and "cross".
+
+In some instances, tokens correspond to their corresponding concepts in other languages, like in Figure 3c where the tokens correspond to "year" and "month" but in Chinese and Korean instead. This is somewhat surprising given that previous work has shown that Llama used for non-English languages tend to have English tokens in their intermediate activations when interpreted through the logit lens (Wendler et al., 2024).
+
+Overall, it is surprising that the logit lens work on VLMs! This technique works on auto-regressive LLMs because they are pretrained on next-token prediction, meaning their activations will be iteratively refined towards the unembedding of the predicted next token. However, VLMs are merely fine-tuned on multimodal tasks. The effectiveness of the logit lens here suggests that the hypothesis that transformers build predictions by promoting concepts in vocabulary space (Geva et al., 2022) may generalise to VLMs, even when only fine-tuned on multimodal tasks.
+
+Testing Other Models. We extended our logit lens analysis to Qwen2VL-2B to verify if these findings generalize beyond the LLaVA family. Following the same methodology, we analyzed 253 COCO validation images and found that in the best-performing layer, an average of  $6.5\%$  of the visual tokens map to the correct object class token. The best-performing layer occurs at layer 25.1 (out of 29 layers), confirming the middle-to-late layer pattern observed in LLaVA (Figure 7). Inter
+
+![](images/0e6360f78f03960ad403b04644acb22b0bc42d77edb77e48f9f91d95ac419ae5.jpg)  
+(a) An image of a lady in the sweater. The logit lens identifies tokens that correspond to specific detail of the sweater, such as "pattern" and "diam"(ond).
+
+![](images/fc71183ccd89bf7e0776fd60825f5729393c66c7aded76dbf4f8be08ba7461f4.jpg)  
+(b) An image of a child in a go-kart. The representations sometimes encode specific details, such as "look" and "gaz"(e) instead of just "face".
+
+![](images/6cd5c88723571b7fc1c586607647a14fa9be8ff8d4461d09cb6e7c717e9206e0.jpg)  
+(c) An image of a switch. In the intermediate layers, the year and month tokens are encoded in non-English characters.
+
+![](images/933596ab884a1b13165eeebbb3fe428e1a451a14c54104ab1b5dbad6dbf2494b.jpg)  
+(d) An image of a bunch of apples. Global features, like count, show up in background tokens, though this may be an artifact of the LM processing.  
+Figure 3: Examples of tokens and their positions that the logit lens yields in the late layers. The labelled tokens come from a range of layers around the middle-to-late layers. In our labelling for an image, tokens that are part of a word are completed with an educated guess e.g. "diam"(ond)
+
+estingly, Qwen2VL shows non-zero correspondence even in early layers, possibly due to its crossattention adapter providing better initial mapping compared to LLaVA's simpler MLP adapter. Both models demonstrate increasingly aligned token representations with object tokens in later layers, suggesting this refinement pattern is a general property of adapter-style VLMs.
+
+Case Study: Global Features. We sometimes find global-level features, such as numbers possibly referring to object counts, appearing in unexpected background tokens (Figure 3d). Ablating these specific tokens doesn't change the model's description, and the numbers persist in the logit lens even after ablation. This suggests these global features may be artifacts of the language model's processing rather than direct information from visual tokens—potentially explaining why CLIP-based VLMs often underperform CLIP itself on classification tasks (Zhang et al., 2024).
+
+Potential Applications. This may be useful for getting coarse segmentation maps for little to no extra computation, which could lead to downstream applications. For example, Liu et al. (2024) suggests to adjust the attention weights towards the visual tokens in the LM to reduce hallucinations. We speculate that directing attention towards specific objects may improve this method, though we leave further exploration to future work.
+
+# 4.2 TRACING ATTENTION FLOW
+
+To understand how visual information is processed and integrated within the language model, we investigate the flow of critical information through the network. Specifically, we examine whether the
+
+model directly extracts information from relevant visual tokens, or whether it aggregates information from visual tokens before transferring it to the final output token.
+
+Method. We employ the attention knockout technique introduced by Geva et al. (2023). This method involves selectively blocking attention between specific tokens at different layers of the model, allowing us to assess the importance of various connections for the model to identify the object. Our process is as follows:
+
+(1) For a given input and target layer  $\ell$ , we artificially block specific attention connections in the multi-head self-attention (MHSA) sublayer by setting the corresponding attention mask values to negative infinity:  $M_{rc}^{\ell + 1, j} = -\infty \forall j \in [1, H]$  where  $M^{\ell + 1, j}$  is the attention mask for head  $j$  at layer  $\ell + 1$ , and  $r$  and  $c$  are the indices of the tokens between which we want to block attention.  
+(2) We apply this blocking over a window of consecutive layers, including early layers (L1-10), early to middle layers (L5-14), middle layers (L11-20), middle to late layers (L15-24), and late layers (L21-31).  
+(3) For each window, we measure the decrease in accuracy and prediction probability for the correct token when blocking attention under three scenarios: (i) From object tokens and their surrounding buffer to the final token. (ii) From all non-object tokens to the final token. (iii) From all visual tokens except the last row to the last row of visual tokens $^2$ .
+
+We conduct this experiment using LLaVA 1.5 on our curated Visual Question Answering (VQA) dataset of 100 images.
+
+Findings. Our results are presented in Table 2.
+
+We find that blocking attention from the object tokens (and their buffers) to the final token in mid-late layers leads to noticeable performance degradation. This suggests that the model directly extracts object-specific information in these later stages. Blocking attention from non-object tokens to the final token in early layers also causes some performance drop. This indicates that contextual information from the broader image is processed and integrated in the earlier stages of the model.
+
+Interestingly, blocking attention from visual tokens to the last row of visual tokens has minimal effect on performance. This contrasts with findings by Basu et al. (2024), who suggested that the model might summarize image information in the last row of visual tokens before using it. Our results indicate that, at least for our object identification tasks, the model doesn't rely on such a summarization step.
+
+Table 2: Results of blocking attention between different token groups across various layers of the LLaVA 1.5 model. The values represent the relative performance on the correct token being predicted when blocking attention, with 1.00 being no impact and 0.00 being all questions wrong post-ablation.  
+
+<table><tr><td colspan="2">Attention Blocking</td><td colspan="6">Layer</td></tr><tr><td>From*</td><td>To†</td><td>Early</td><td>Early-Mid</td><td>Mid</td><td>Mid-Late</td><td>Late</td><td>All</td></tr><tr><td>O</td><td>LTP</td><td>1.00</td><td>1.00</td><td>0.96</td><td>0.88</td><td>0.93</td><td>0.82</td></tr><tr><td>O+1</td><td>LTP</td><td>1.00</td><td>0.99</td><td>0.90</td><td>0.82</td><td>0.89</td><td>0.67</td></tr><tr><td>O+2</td><td>LTP</td><td>1.00</td><td>1.00</td><td>0.91</td><td>0.80</td><td>0.91</td><td>0.68</td></tr><tr><td>I-(O+1)</td><td>LTP</td><td>0.88</td><td>1.00</td><td>0.97</td><td>0.96</td><td>0.98</td><td>0.82</td></tr><tr><td>O+1</td><td>LVR</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>I-LVR</td><td>LVR</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr></table>
+
+\*From:  $\mathrm{O} =$  Object Tokens,  $\mathrm{O + n = O + n}$  Buffer, I-  $(\mathrm{O + 1}) =$  All visual tokens except  $\mathrm{O + 1}$  , I-LVR  $=$  All visual tokens except last row  $\dagger$  To: LTP  $=$  Last Token Position, LVR  $=$  Last Visual Token Row
+
+Testing Other Models. To verify that our attention blocking results are not specific to LLaVA-1.5, we conducted additional experiments with LLaVA-Phi-3. Rather than using fixed layer groupings, we employed a sliding window of 10 layers to obtain finer-grained results across the network. We find that the results (Figure 8) closely mirror our original results: blocking attention from object tokens to the last token position shows the strongest effect in middle-to-late layers, while blocking attention from visual tokens to the last row continues to show minimal impact.
+
+# 5 RELATED WORK
+
+Explainability. Classical techniques like GRAD-Cam (Selvaraju et al., 2017) and Integrated Gradients (Sundararajan et al., 2017), focus on input attribution, while recent tools like LVLM-Interpret (Stan et al., 2024) extend these methods to VLMs. Our work complements these by investigating the model's internal processing mechanisms.
+
+Mechanistic Interpretability (MI) for LLMs. Techniques such as causal tracing (Meng et al., 2022) and sparse coding (Bricken et al., 2023; Huben et al., 2023; Marks et al., 2024) have advanced our understanding of how LLMs perform tasks despite being primarily trained for next-token prediction, but VLMs are not trained on next-token prediction for visual inputs (Liu et al., 2023b). Our work extends some of these techniques to understand visual information processing in multimodal contexts.
+
+MI for Multimodal Models. In CLIP, researchers have discovered multimodal neurons (Goh et al., 2021), decomposed image representations (Gandelsman et al., 2023; 2024), identified interpretable network subgraphs (Rajaram et al., 2024), and extracted interpretable features using sparse coding (Rao et al., 2024; Daujotas, 2024; Fry, 2024). In contrast, our work focuses on generative VLMs.
+
+For generative VLMs, Schwettmann et al. (2023) found neurons corresponding to visual concepts, while Palit et al. (2023) adapted causal tracing for BLIP. While these studies help us understand the role of specific components in the VLM, we investigate the representations of the visual inputs.
+
+Basu et al. (2024) used causal tracing and attention contributions in a visual question-answering setup to find that VLMs such as LLaVA retrieve information from earlier causal layers of the LM (i.e. layers 1-4 vs layers 4-7 in an LLM) when answering a multi-modal question. Their analysis suggests that the visual information may be summarised in a consistent subset of late visual tokens. In contrast, we study VLMs in a more straightforward object-identification setting. We also examine their information transfer hypothesis through our attention blocking experiments in Section 4.2.
+
+Parallel Work. Concurrent research by Jiang et al. (2025) independently arrived at similar logit lens findings and developed a hallucination reduction method based on these insights. Wu et al. (2024) observed comparable logit lens patterns across both visual and audio modalities.
+
+# 6 CONCLUSION
+
+Our investigation into visual information processing in LLaVA reveals that object information is highly localized within visual tokens corresponding to the object's spatial location, challenging the hypothesis that VLMs primarily rely on global features in register tokens. Visual token representations evolve to align with interpretable textual concepts across layers, suggesting VLMs refine visual information toward language-like representations even without next-token prediction pretraining for visual inputs. Our attention blocking experiments suggest that VLMs may extract object information from relevant visual tokens in mid-to-late layers. These findings bridge the gap between our understanding of language and vision models, providing a foundation for more interpretable multimodal systems.
+
+Limitations & Future Work. We focused on LLaVA-type models as a starting point for interpreting more complex VLMs, but our results may not generalize to significantly different architectures. Our study concentrates on object identification tasks, providing a simpler setup compared to multi-step reasoning. While we tested object identification in three ways, our findings may not fully represent VLM behavior in complex tasks like reasoning or open-ended question answering. Future work should explore practical applications in reducing hallucinations and enhancing factual accuracy, while examining whether these findings extend to broader task types and model architectures.
+
+# AUTHOR CONTRIBUTIONS
+
+Clement Neo conceived the study, designed and performed the experiments, and led the writing of the manuscript. Luke Ong contributed to the writing and literature review. Philip Torr, Mor Geva and David Krueger provided advice and comments. Fazl Berez served as the primary advisor for this work and helped significantly with the design and write-up of the paper.
+
+# ACKNOWLEDGMENTS
+
+We thank the Torr Vision Group for providing compute and hosting Clement Neo during the internship.
+
+We thank Sarah Schwettmann, Ashkan Khakzar, William Rudman, Joseph Miller, Alex Spies for discussing and reading drafts of the paper.
+
+# REFERENCES
+
+Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in neural information processing systems, 35:23716-23736, 2022.  
+Andy Arditi, Oscar Obeso, Aaquib Syed, Daniel Paleka, Nina Panickssery, Wes Gurnee, and Neel Nanda. Refusal in language models is mediated by a single direction, 2024. URL https://arxiv.org/abs/2406.11717.  
+Luke Bailey, Gustaf Ahdritz, Anat Kleiman, Siddharth Swaroop, Finale Doshi-Velez, and Weiwei Pan. Soft prompting might be a bug, not a feature. 2023.  
+Samyadeep Basu, Martin Grayson, Cecily Morrison, Besmira Nushi, Soheil Feizi, and Daniela Massiceti. Understanding Information Storage and Transfer in Multi-modal Large Language Models, jun 2024. URL http://arxiv.org/abs/2406.04236.  
+Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Conerly, Nick Turner, Cem Anil, Carson Denison, Amanda Askell, Robert Lasenby, Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, Brayden McLean, Josiah E Burke, Tristan Hume, Shan Carter, Tom Henighan, and Christopher Olah. Towards monoseismicity: Decomposing language models with dictionary learning. Transformer Circuits Thread, 2023. https://transformercircuits.pub/2023/monoseismic-features/index.html.  
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 1877-1901. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper_files/paper/2020/file/1457c0d6bfbcb4967418bf8ac142f64a-Paper.pdf.  
+Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E Gonzalez, et al. Vicuna: An open-source chatbot impressing gpt-4 with  $90\%$  * chatgpt quality. See https://vicuna.lmsys.org (accessed 14 April 2023), 2(3):6, 2023.  
+Timothee Darcet, Maxime Oquab, Julien Mairal, and Piotr Bojanowski. Vision transformers need registers. In The Twelfth International Conference on Learning Representations, 2024.
+
+Gytis Daujotas. Case study: Interpreting, manipulating, and controlling clip with sparse autoencoders, 2024. URL https://www.lesswrong.com/posts/iYFuZo9BMvr6GgMs5/case-study-interpreting-manipulating-and-controlling-clip. Accessed: 2024-09-24.  
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248-255, 2009. doi: 10.1109/CVPR.2009.5206848.  
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=YicbFdNTTy.  
+Nelson Elhage, Neel Nanda, Catherine Olsson, Tom Henighan, Nicholas Joseph, Ben Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Nova DasSarma, Dawn Drain, Deep Ganguli, Zac Hatfield-Dodds, Danny Hernandez, Andy Jones, Jackson Kernion, Liane Lovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish, and Chris Olah. A mathematical framework for transformer circuits. Transformer Circuits Thread, 2021. https://transformer-circuits.pub/2021/framework/index.html.  
+Hugo Fry. Towards multimodal interpretability: Learning sparse interpretable features in vision transformers, 2024. URL https://www.lesswrong.com/posts/bCtbuWraqYTDtuARg/towards-multimodal-interpretability-learning-sparse-2#Future_Work. Accessed: 2024-06-28.  
+Yossi Gandelsman, Alexei A. Efros, and Jacob Steinhardt. Interpreting clip's image representation via text-based decomposition, 2023.  
+Yossi Gandelsman, Alexei A. Efros, and Jacob Steinhardt. Interpreting the second-order effects of neurons in clip, 2024.  
+Mor Geva, Avi Caciularu, Kevin Wang, and Yoav Goldberg. Transformer feed-forward layers build predictions by promoting concepts in the vocabulary space. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pp. 30-45, 2022.  
+Mor Geva, Jasmijn Bastings, Katja Filippova, and Amir Globerson. Dissecting recall of factual associations in auto-regressive language models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pp. 12216-12235, 2023.  
+Gabriel Goh, Nick Cammarata †, Chelsea Voss †, Shan Carter, Michael Petrov, Ludwig Schubert, Alec Radford, and Chris Olah. Multimodal neurons in artificial neural networks. *Distill*, 2021. doi: 10.23915/distill.00030. https://distill.pub/2021/multimodal-neurons.  
+Robert Huben, Hoagy Cunningham, Logan Riggs Smith, Aidan Ewart, and Lee Sharkey. Sparse autoencoders find highly interpretable features in language models. In *The Twelfth International Conference on Learning Representations*, 2023.  
+Chaoya Jiang, Haiyang Xu, Mengfan Dong, Jiaxing Chen, Wei Ye, Ming Yan, Qinghao Ye, Ji Zhang, Fei Huang, and Shikun Zhang. Hallucination augmented contrastive learning for multimodal large language model, 2024. URL https://arxiv.org/abs/2312.06968.  
+Nick Jiang, Anish Kachinthaya, Suzie Petryk, and Yossi Gandelsman. Interpreting and editing vision-language representations to mitigate hallucinations, 2025. URL https://arxiv.org/abs/2410.02762.  
+Sonia Joseph and Neel Nanda. Laying the foundations for vision and multimodal mechanistic interpretability & open problems, 2024. URL https://wwwalignmentforum.org/posts/kobJymvvcvhbjWFKe/ laying-the-foundations-for-vision-and-multimodal-mechanistic. Accessed: 2024-06-28.
+
+Brian Lester, Rami Al-Rfou, and Noah Constant. The power of scale for parameter-efficient prompt tuning. In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wentau Yih (eds.), Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pp. 3045-3059, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.emnlp-main.243. URL https://aclanthology.org/2021.emnlp-main.243.  
+Junnan Li, Dongxu Li, Silvio Savarese, and Steven Hoi. Blip-2: Bootstrapping language-image pre-training with frozen image encoders and large language models. In International conference on machine learning, pp. 19730–19742. PMLR, 2023.  
+Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dóllár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In Computer Vision-ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13, pp. 740-755. Springer, 2014.  
+Haotian Liu, Chunyuan Li, Yuheng Li, and Yong Jae Lee. Improved baselines with visual instruction tuning, 2023a.  
+Haotian Liu, Chunyuan Li, Qingyang Wu, and Yong Jae Lee. Visual instruction tuning. In NeurIPS, 2023b.  
+Shi Liu, Kecheng Zheng, and Wei Chen. Paying more attention to image: A training-free method for alleviating hallucination in lvlms, 2024. URL https://arxiv.org/abs/2407.21771.  
+Samuel Marks, Can Rager, Eric J Michaud, Yonatan Belinkov, David Bau, and Aaron Mueller. Sparse feature circuits: Discovering and editing interpretable causal graphs in language models. arXiv preprint arXiv:2403.19647, 2024.  
+Kevin Meng, David Bau, Alex Andonian, and Yonatan Belinkov. Locating and editing factual associations in GPT. Advances in Neural Information Processing Systems, 36, 2022. arXiv:2202.05262.  
+Jack Merullo, Louis Castricato, Carsten Eickhoff, and Ellie Pavlick. Linearly mapping from image to text space. In The Eleventh International Conference on Learning Representations, 2022.  
+Nostalgebraist. Interpreting gpt: The logit lens. https://wwwalignmentforum.org/ posts/AcKRB8wDpdaN6v6ru/interpreting-gpt-the-logit-lens, Aug 2020. Accessed: 23 Sep 2024.  
+Vedant Palit, Rohan Pandey, Aryaman Arora, and Paul Pu Liang. Towards vision-language mechanistic interpretability: A causal tracing tool for blip. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2856-2861, 2023.  
+Xu Pan, Aaron Philip, Ziqian Xie, and Odelia Schwartz. Dissecting query-key interaction in vision transformers. ICML MI Workshop, 2024.  
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.  
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision, 2021. URL https://arxiv.org/abs/2103.00020.  
+Achyuta Rajaram, Neil Chowdhury, Antonio Torralba, Jacob Andreas, and Sarah Schwettmann. Automatic discovery of visual circuits, 2024. URL https://arxiv.org/abs/2404.14349.  
+Sukrut Rao, Sweta Mahajan, Moritz Böhle, and Bernt Schiele. Discover-then-name: Task-agnostic concept bottlenecks via automated concept discovery, 2024. URL https://arxiv.org/abs/2407.14499.  
+Sarah Schwettmann, Neil Chowdhury, Samuel Klein, David Bau, and Antonio Torralba. Multimodal neurons in pretrained text-only transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2862-2867, 2023.
+
+Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pp. 618-626, 2017.  
+Gabriela Ben Melech Stan, Estelle Aflalo, Raanan Yehezkel Rohekar, Anahita Bhiwandiwalla, ShaoYen Tseng, Matthew Lyle Olson, Yaniv Gurwicz, Chenfei Wu, Nan Duan, and Vasudev Lal. Lvlm-intrepret: An interpretability tool for large vision-language models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8182-8187, 2024.  
+Alessandro Stolfo, Yonatan Belinkov, and Mrinmaya Sachan. A mechanistic interpretation of arithmetic reasoning in language models using causal mediation analysis. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pp. 7035-7052, 2023.  
+Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In International conference on machine learning, pp. 3319-3328. PMLR, 2017.  
+Maria Tsimpoukelli, Jacob L Menick, Serkan Cabi, SM Eslami, Oriol Vinyals, and Felix Hill. Multimodal few-shot learning with frozen language models. Advances in Neural Information Processing Systems, 34:200-212, 2021.  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.  
+Martina G Vilas, Timothy Schaumlöffel, and Gemma Roig. Analyzing vision transformers for image classification in class embedding space. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.  
+Kevin Ro Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt. Interpretability in the wild: a circuit for indirect object identification in gpt-2 small. In *The Eleventh International Conference on Learning Representations*, 2023.  
+Peng Wang, Shuai Bai, Sinan Tan, Shijie Wang, Zhihao Fan, Jinze Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, et al. Qwen2-vl: Enhancing vision-language model's perception of the world at any resolution. arXiv preprint arXiv:2409.12191, 2024.  
+Chris Wendler, Veniamin Veselovsky, Giovanni Monea, and Robert West. Do llamas work in English? on the latent language of multilingual transformers. In Lun-Wei Ku, Andre Martins, and Vivek Srikumar (eds.), Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 15366-15394, Bangkok, Thailand, August 2024. Association for Computational Linguistics. doi: 10.18653/v1/2024.acl-long.820. URL https://aclanthology.org/2024.acl-long.820.  
+Zhaofeng Wu, Xinyan Velocity Yu, Dani Yogatama, Jiasen Lu, and Yoon Kim. The semantic hub hypothesis: Language models share semantic representations across languages and modalities, 2024. URL https://arxiv.org/abs/2411.04986.  
+Yuhui Zhang, Alyssa Unell, Xiaohan Wang, Dhruba Ghosh, Yuchang Su, Ludwig Schmidt, and Serena Yeung-Levy. Why are visually-grounded language models bad at image classification? arXiv preprint arXiv:2405.18415, 2024.
+
+![](images/5ea2e162ae25ec2b4099e8da5244b514048acd70a256f861df0d4468b2839f61.jpg)  
+A FILTERED DATASET EXAMPLES
+
+![](images/4899b1fb2ed6a64ca78cb7b688042bd488f7e3298954029ee2518eaed7c8e290.jpg)
+
+![](images/261ec74d68451e3dafcfba1b2bf553e9bbc2470d0fa5365a6c7ee833c886f2cb.jpg)  
+Figure 4: Example of dataset images for the object identification tasks.
+
+![](images/6bd63076bd059ed82fb53a0905a217e4c1a9557186137f1a72682bd9a2b27266.jpg)
+
+# B CURATED IMAGES AND QUESTIONS EXAMPLES
+
+![](images/eb5b6f9f27f8004d1f4fd510ff514f0d8d0a7db1b9a39a89c110575874be4119.jpg)  
+(a) "What is on the bench? It is a"  $\rightarrow$  book
+
+![](images/2e551b2609818e75fb34f48ab0d4cd1044fb03d3e9e74c02469e8303cd61fdac.jpg)  
+(b) "What is below the street sign? It is a"  $\rightarrow$  bottle
+
+![](images/03e9580f5ba4c85b2c31072badfaf876675ff3de31108670dc07ca4e31ad5e15.jpg)  
+(c) "What is on the truck? It is a"  $\rightarrow$  dog
+
+![](images/b566fa7956d4d0b36d726ebeed8f470646daa713e5df8ad46469403e480b8bb0.jpg)  
+(d) "What is on the seat? It is a"  $\rightarrow$  cup  
+Figure 5: Examples of curated images and their questions. We ask the model the question and prefill its answer with "It is a". The correct answer is underlined.
+
+# C OBJECT CORRESPONDENCES
+
+![](images/c526e66913a0c1455c2acaf26c6aa499671ba16612eaa7b598837c0e5cbf5336.jpg)  
+Figure 6: Token-to-object correspondence in LLaVA-1.5 across transformer layers. Analysis of 170 COCO images shows peak correspondence (23.7%) occurs in later layers (average layer 25.7/33).
+
+![](images/649abe2e0f27e5d60c391c5cd808e2704e96783f686087642d3e35885251586b.jpg)  
+Figure 7: Token-to-object correspondence in Qwen2VL-2B showing increasing alignment in deeper layers, peaking at layer 25.1/29 with  $6.5\%$  correct mapping. Unlike LLaVA, Qwen exhibits non-zero correspondence in early layers.
+
+# D LLAVA-PHI-3 BLOCKING
+
+![](images/00d3abc71a437ee6bd62820f57d5f5b6684145fbb0b47df49c1d9b10c8736984.jpg)
+
+![](images/e1efd0bc4878b55c7dde09ae394f1e7744aaa2b1c33942192ff0de3120823003.jpg)
+
+![](images/4203283f1e67817cd019b318f5e9d2cb897284aaaa6f65a093e86a6fe61b309b.jpg)
+
+![](images/1247937820d1280504c529e1c9bd71a0f8813fa28549bf569c62eb724bf30a93.jpg)
+
+![](images/6d0ac4b34ad4f749b6e4ef20364e0b1b995da540b2fc8423d582cf7c3827f451.jpg)
+
+![](images/287cce761d01a645bfb8270fab8c28b6fa5a14ff87070fc8b580b8bad0ee757b.jpg)  
+Figure 8: Attention blocking results for LLaVA-Phi-3 using a sliding window of 10 layers. We find that similar to LLaVA-1.5, blocking attention from object tokens to the last token position has the strongest effect in middle-to-late layers, while blocking attention from visual tokens to the last visual row shows minimal impact throughout the network.

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2025/Towards Multiple Character Image Animation Through Enhancing Implicit Decoupling/full.md

@@ -0,0 +1,630 @@
+# TOWARDS MULTIPLE CHARACTER IMAGE ANIMATION THROUGH ENHANCING IMPLICIT DECOUPLING
+
+Jingyun Xue $^{1,2}$ , Hongfa Wang $^{2,3}$ , Qi Tian $^{2}$ , Yue Ma $^{2,4}$ , Andong Wang $^{2}$ , Zhiyuan Zhao $^{2}$ , Shaobo Min $^{2}$ , Wenzhe Zhao $^{2}$ , Kaihao Zhang $^{5}$ , Heung-Yeung Shum $^{3,4}$ , Wei Liu $^{2}$ , Mengyang Liu $^{2\dagger}$ , Wenhan Luo $^{4‡}$ $^{1}$ Shenzhen Campus of Sun Yat-sen University,  $^{2}$ Tencent Hunyuan,  $^{3}$ Tsinghua University  
+ $^{4}$ HKUST,  $^{5}$ Harbin Institute of Technology, Shenzhen  
+https://multi-animation.github.io/
+
+![](images/15279a180e548e2ccda15855fcafe9ef064ababb910ec609b8881f2ad71f78e5.jpg)
+
+![](images/9b5fc1f7aec84300996843ad43563f571778ec54f192275825892fdc96ab6c0d.jpg)
+
+![](images/a9b14b9293bea03f8fa5ede77850fed235cf505b8c75bcc8d070c02903d407a6.jpg)
+
+![](images/3b7a1c6650e854fb2b78eb60d1c5bf385730d824329b096e7047ea6e287fa749.jpg)
+
+![](images/699a07b46bfc0b3172e75c793e842cde1a7a33756c848e5f6ca2ec9d2de78481.jpg)  
+Figure 1: Pose-controllable single-character image animation (top) and dual-character image animation (bottom) given the reference image.
+
+![](images/3f0e2159a38cb3e0a41498c843395c4236b0a290dde1fb282f9d213bbd6db281.jpg)
+
+![](images/0dd5d9426384dc48f5230f65f8ad87ea0bca9cbb625b74e84688e0170bcc17ec.jpg)
+
+# ABSTRACT
+
+Controllable character image animation has a wide range of applications. Although existing studies have consistently improved performance, challenges persist in the field of character image animation, particularly concerning stability in complex backgrounds and tasks involving multiple characters. To address these challenges, we propose a novel multi-condition guided framework for character image animation, employing several well-designed input modules to enhance the implicit decoupling capability of the model. First, the optical flow guider calculates the background optical flow map as guidance information, which enables the model to implicitly learn to decouple the background motion into background constants and background momentum during training, and generate a stable background by setting zero background momentum during inference. Second, the depth order guider calculates the order map of the characters, which transforms the depth information into the positional information of multiple characters. This facilitates the implicit learning of decoupling different characters, especially in accurately separating the occluded body parts of multiple characters. Third, the reference pose map is input to enhance the ability to decouple character texture and pose information in the reference image. Furthermore, to fill the gap of fair evaluation of multi-character image animation, we propose a new benchmark comprising about 4,000 frames. Extensive qualitative and quantitative evaluations demonstrate that our method excels in generating high-quality character animations, especially in scenarios of complex backgrounds and multiple characters.
+
+# 1 INTRODUCTION
+
+Character image animation task targets animating a given static character image to a video clip using a sequence of motion signals, such as pose, while preserving the visual appearance. It has attracted much attention and has been extensively explored in research (Zhang et al., 2022). Existing character image animation can be divided into three categories: GAN-based, 3D-based, and
+
+![](images/f297ff9f1b6f3b34834e282ad6df0d38b0366d981264aff55fe3e759cd73c57b.jpg)  
+Figure 2: Challenges faced by existing methods: (a) By overlaying video frames, it is observed that models generate unreasonable backgrounds. (b) Models struggle to accurately identify characters in multi-character image animations, resulting in incorrect character identities. (c) Models have difficulty in accurately generating the occluded body parts of multiple characters.
+
+diffusion-based frameworks. GAN-based methods (Siarohin et al., 2019; Wang et al., 2021) leverage a warping function to transfer the reference image into the target pose, and then the GAN model is employed to generate the missing parts. 3D-based methods (Jiang et al., 2023) reconstruct a character avatar from monocular videos and then render the avatar into a character video based on the pose sequence. Diffusion-based methods generate refined character animation by adding conditional control. For example, Hu et al. (2023) generate a series of coherent action videos by incorporating pose sequences, while Xu et al. (2023) do so by integrating semantic map sequences.
+
+Despite generating visually plausible videos, existing methods still face several challenges. GAN-based methods often struggle to generate realistic animations due to their limited ability to transfer motion, which affects details such as hair movement, and facial expressions. 3D-based methods (Jiang et al., 2023; Yu et al., 2023) are generally limited by the precision of SMPL and struggle to derive hand and facial details. In addition, 3D-based methods cannot fill in the empty background left after character movement. Besides, 3D-based methods perform poorly in generating high-quality texture details and clothing movement. In contrast, diffusion-based methods alleviate these problems due to their powerful generative capability. However, diffusion-based methods still face serious challenges when generating higher-quality and more complex character animations. As shown in Fig. 2, existing methods often have the drawback of generating unreasonable backgrounds. Specifically, due to the camera shake and/or video effects popularly present in the background, the model is affected by the noise, leading to abrupt changes, flickering, and artifacts in the background. Additionally, when generating animated videos with multiple characters, existing methods tend to generate chaotic character identities and erroneous occluded body parts.
+
+Our work is dedicated to addressing these challenges. Through our investigation, we have the following findings. First, the background instability results from the model's low robustness to noisy data with background variations in the training set. The common solution is to construct a high-quality training set without background variation, which is costly and limits the size of the dataset, especially considering that data for multiple characters is scarce. We aim for the model to achieve strong generalization with sufficient and cheap training data with background variations. A feasible solution is to decouple the noise features from the video, improving the utility of the noisy data. Second, the errors in multiple character image animation primarily stem from that the model cannot further decouple multiple character features into individual character features. Perhaps we can enable the model to learn this implicit decoupling process through extensive multiple-character data, but a more economical and expeditious approach is to provide guiding information to direct the model in learning this process. These findings indicate that it is extremely important to introduce the implicit decoupling ability into the model for the task of multiple-character image animation.
+
+In this work, to address the aforementioned challenges, we propose a diffusion-based framework for multiple-character image animation, allowing for the input of various types and amounts of guidance information. Based on the mentioned insights, we carefully design three guiders to enhance the implicit decoupling capability of the model. (1) We design an optical flow guider that calculates the background motion optical flow map as guidance. This enables the model to implicitly learn to decouple the background motion into background constants and background momentum during training, and generate a stable background by inputting zero background momentum during inference. (2) We present the depth order guider calculating the order map of the characters, which transforms
+
+![](images/1ae5bc116b201cd151844d2a47e9567014a4c298111850c57c84a7a66076f4bc.jpg)  
+Figure 3: The overview of the proposed framework. The left half illustrates the data flow of the multiple condition guiders, with green and black arrows denoting training data flow, and red and black arrows indicating inference data flow. The gray box represents different inputs for training and inference. The right half shows the denoising U-Net and ReferenceNet.
+
+the depth information into the positional information of multiple characters in terms of their relative front and back positions. This facilitates the implicit learning of decoupling different characters, especially in accurately separating the occluded body parts of multiple characters, leading to more stable multiple-character animation. (3) We introduce the reference pose map to enhance the ability to decouple character texture and pose information in the reference image. Additionally, we propose a multi-character benchmark including about 4,000 frames, which empowers the community to comprehensively evaluate the generation ability in complex multiple character image animations. To the best of our knowledge, we are the first to collect such a benchmark. We conduct extensive quantitative and qualitative experiments to illustrate the superiority of our approach.
+
+In summary, our contributions are as follows:
+
+- We propose a multi-condition guided framework with multiple guiders for multiple-character image animation, which exhibits strong robustness against noisy data with unstable backgrounds. By leveraging large-scale raw data for training, this framework effectively addresses the multiple-character image animation.  
+- We enhance the implicit decoupling capability of the model. Technically, the optical flow guider decouples the background momentum to ensure background stability, the depth order guider provides multiple character positional information to address the occlusion among body parts, and reference pose guider inputs the source pose to align the character with the target pose.  
+- We address the lack of a benchmark in multiple character image animation. A new benchmark called Multi-Character Bench is introduced, which contains about 4,000 frames for fair and comprehensive evaluation.  
+- Extensive quantitative and qualitative evaluations are conducted using two public datasets and Multi-Character Bench. The results show the superiority of the proposed method.
+
+# 2 RELATED WORK
+
+Diffusion Models for Video Generation. Diffusion models excel in generating high-quality contents (Esser et al., 2023; Ma et al., 2023; 2024c; Chen et al., 2024b; Feng et al., 2024; Tan et al., 2024; Kong et al., 2025). Ho et al. (2020); Song et al. (2020) propose to generate images using Diffusion models. Many studies (Wu et al., 2023) adapt image synthesis pipelines for video generation. Khachatryan et al. (2023) incorporate motion dynamics in generated frames via cross-frame attention in zero-shot style. Guo et al. (2023b) finetune a plug-and-play motion module that can be integrated into any text-to-image models to obtain animations in a personalized style. Zhang et al. (2024); Chen et al. (2024a) achieve both text-to-video and image-to-video generation of high resolution. Besides, Guo et al. (2023a) extract semantics from images to condition T2V generation.
+
+![](images/6c0c3ebf569b9b88a3187d54aab5846821de694cabe1783991de2003647cad7d.jpg)  
+Figure 4: The pipeline for calculating the skeletal dilation map and optical flow map. The optical flow map is obtained by superimposing the skeletal dilation map onto the optical flow vector map.
+
+Pose-Controllable Character Image Animation. Generating realistic character video from the driving signal (e.g., key points, semantic maps) has been extensively studied in recent years. Early approaches (Mirza & Osindero, 2014; Wang et al., 2021) employ GAN-based models for conditional video synthesis. Wang et al. (2019) use conditional GANs with optical flow, temporal consistency, and multiple discriminators for diverse pose videos. Chan et al. (2019) transfer movements from a source person to a target person through keypoints. Siarohin et al. (2021) incorporate consistent regions that describe locations, shapes, and poses. To address the pose gap between objects in the source and driving images, Zhao & Zhang (2022) effectively align object poses using thin-plate spline motion estimation and multi-resolution occlusion mask. With the development of the diffusion model, existing methods use pose sequences to guide the character video generation (Zhang et al., 2023). Karras et al. (2023) propose a finetuning framework to adapt the Stable Diffusion model to a pose-and-image guided video synthesis model. Hu et al. (2023) design ReferenceNet to merge detailed human body features, ensuring character appearance consistency, while employing a pose guider and temporal modeling for controllable and continuous movements. To enhance the performance, Wang et al. (2023) disentangle the control conditions (i.e., character foreground, background, and poses) by introducing multiple ControlNets for different feature embeddings and a human attribute pre-training framework is proposed. Moreover, Ma et al. (2024a) propose a two-stage training scheme to address the lack of comprehensive paired video-pose datasets.
+
+# 3 PRELIMINARIES
+
+Video Latent Diffusion Models. The powerful text-to-image (T2I) model with the addition of temporal motion modules endows it with the ability to generate video (Guo et al., 2023b), i.e., video latent diffusion model (VLDM). Specifically, inserting a temporal motion module after spatial attention enables the T2I model to generate video. The temporal motion modules run across temporal frames to enhance motion smoothness and content consistency. The encoder of VAE (Kingma & Welling, 2013; Razavi et al., 2019) compresses the video clip  $v_{1:n}$  to a latent space feature  $z = \mathcal{E}(v_{1:n})$ . Besides, the initial input tensor  $z \in \mathbb{R}^{c \times h \times w}$  of the T2I model should add a temporal dimension and repeat it  $n$  times, becoming  $z_{1:n} \in \mathbb{R}^{n \times c \times h \times w}$ . VLDM uses U-Net (Ronneberger et al., 2015) or DiT (Peebles & Xie, 2023) to estimate the noise, with the loss function of
+
+$$
+\mathcal {L} = \mathbb {E} _ {\mathcal {E} (v _ {1: n}), c, \epsilon_ {1: n} \sim \mathcal {N} (0, I), t} \left[ \| \epsilon - \epsilon_ {\theta} \left(z _ {1: n} ^ {t}, t, c\right) \| _ {2} ^ {2} \right], \tag {1}
+$$
+
+where  $\epsilon_{\theta}(\cdot)$  is the network for predicting noise.  $c$  denotes the conditional information.  $t$  represents the timestep of denoising process and  $z_{1:n}^{t}$  is the intermediate result of denoising at timestep  $t$ .
+
+# 4 METHODOLOGY
+
+Given a reference image  $I_0$  and pose sequences  $\{P_0, P_1, P_2, \dots, P_N\}$ , we aim to generate character video clip with both plausible motion and faithful visual appearance (foreground and background) regarding the reference image. To address the issues of identity errors and body occlusion in multi-character image animation, or background instability, building large-scale high-quality datasets and conducting extensive training is costly and laborious. In contrast, enhancing the implicit decoupling ability of the model through input guidance is more economical and feasible. To achieve this, we propose a novel framework equipped with multi-condition guiders, as shown in Fig. 3.
+
+![](images/5bee771307ffde58d18bcdfcec56599143d0265e80d30412ad528aaf0d3ab687.jpg)  
+Figure 5: The pipeline for calculating the depth order map, which strictly separates characters and assigns distinct values. The yellow module "Skeletal Dilation" is shown in Fig. 4 (a).
+
+# 4.1 OPTICAL FLOW GUIDER
+
+As shown in Fig. 2 (a), when directly training on noisy data, the model is unable to achieve background alignment due to the interference caused by noise. As a result, there are abrupt changes, flickering, and artifacts in the generated background. We propose using background optical flow maps as guidance to direct the model in learning the implicit decoupling process, which decouples the background into a constant and a momentum. Here, background momentum represents the relative motion of the current frame's background compared to the previous frame, which is equivalent to the optical flow. After decoupling the background motion, the model can learn to generate stable backgrounds even trained on noisy data. During inference, we input zero background motion to achieve stable background. Unlike the work (Liang et al., 2024) using optical flow to enhance temporal consistency, we use the motion vector information in optical flow to decouple noise features.
+
+Fig. 4 presents the pipeline for calculating the background optical flow map in the training stage (Yang et al., 2023; Güler et al., 2018; Sun et al., 2018). The optical flow estimator  $\mathcal{E}_{flow}$  (OpenMMLab, 2021) predicts optical flow maps from adjacent frames  $\{v_{1}, v_{2}, \ldots, v_{N}\}$ . The dilation operation is then applied on pose skeletons  $\{p_{1}, p_{2}, \ldots, p_{N}\}$  to obtain binary mask sequences  $\{\mathcal{M}_1, \mathcal{M}_2, \mathcal{M}_3, \ldots, \mathcal{M}_N\}$ , which are used for defining the control region. We blend optical flows with the binary mask and encode them using 8 layers of inflated 3D convolution. The region in blended optical flows is set to 1.0. In inference, we directly set the background optical flow, i.e., the background momentum, to zero. Formally, the optical flow guider is implemented as:
+
+$$
+\mathcal {M} _ {i} = \mathcal {D} \left(p _ {i}\right), \tag {2}
+$$
+
+$$
+c _ {f l o w} ^ {i} = g _ {\mathrm {o p}} \left(\left(1 - \mathcal {M} _ {i}\right) \odot \mathcal {E} _ {f l o w} \left(v _ {i}, v _ {i - 1}\right)\right), \tag {3}
+$$
+
+where  $\mathcal{D}$  represents the dilation operation,  $\odot$  is Hadamard product, and  $g_{\mathrm{op}}$  is the optical flow guider.
+
+# 4.2 DEPTH ORDER GUIDER
+
+Concurrent methods struggle to accurately identify different characters and generate correct identities. They lack spatial order information about different characters in front of the camera, which is critical for generating occluded body parts. To address this issue, we design a depth order guider that can simultaneously enhance the implicit decoupling of characters into individual ones and provide characters' spatial order information. Specifically, it calculates the depth order map of characters, decoupling multiple characters into individuals while providing their sequential positional/depth relationships in front of the camera. The regions of different characters are separated by assigning distinct values, with occluded body parts being strictly assigned to characters positioned in the front.
+
+Fig. 5 illustrates calculating the depth order map during training. For each frame, we extract individual character poses and calculate their skeletal dilation maps. Subsequently, we merge the character-wise skeleton dilation maps to get the intersection and subtract it to derive the non-intersection area. Next, split by characters, as shown by the green arrow in Fig. 5, we have the character-wise non-intersection area maps. Then, we align them with the depth maps, compute the average depth of the non-intersection region for each character, and sort them by the average depth. We assign values to each character's region based on their position ranking. For example, as shown in Fig. 5, the front-most character region gets "yellow value" while the second front character region gets "red value".
+
+Table 1: Quantitative comparison on Tiktok dataset. The best and second-best results are indicated in red and blue respectively. AnimateAnyone† is trained on our noisy dataset. Ours* is trained on the Tiktok training set.  
+
+<table><tr><td>Method</td><td>FID↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>L1↓</td><td>FID-VID↓</td><td>FVD↓</td></tr><tr><td>MRAA (Siarohin et al., 2021)</td><td>54.47</td><td>0.672</td><td>29.39</td><td>0.296</td><td>3.21E-04</td><td>66.36</td><td>284.82</td></tr><tr><td>TPSMM (Zhao &amp; Zhang, 2022)</td><td>53.78</td><td>0.673</td><td>29.18</td><td>0.299</td><td>3.23E-04</td><td>72.55</td><td>306.17</td></tr><tr><td>DreamPose (Karras et al., 2023)</td><td>79.46</td><td>0.509</td><td>28.04</td><td>0.450</td><td>6.91E-04</td><td>80.51</td><td>551.56</td></tr><tr><td>DisCo (Wang et al., 2023)</td><td>51.29</td><td>0.699</td><td>28.70</td><td>0.333</td><td>1.10E-04</td><td>61.41</td><td>379.56</td></tr><tr><td>DisCo+ (Wang et al., 2023)</td><td>48.29</td><td>0.713</td><td>28.78</td><td>0.320</td><td>1.03E-04</td><td>52.56</td><td>334.67</td></tr><tr><td>MagicAnimate (Xu et al., 2023)</td><td>32.09</td><td>0.714</td><td>29.16</td><td>0.239</td><td>3.13E-04</td><td>21.75</td><td>179.07</td></tr><tr><td>MagicPose (Chang et al., 2023)</td><td>25.50</td><td>0.752</td><td>29.53</td><td>0.292</td><td>0.81E-04</td><td>46.30</td><td>216.01</td></tr><tr><td>AnimateAnyone (Hu et al., 2023)</td><td>-</td><td>0.718</td><td>29.56</td><td>0.285</td><td>-</td><td>-</td><td>171.90</td></tr><tr><td>AnimateAnyone†</td><td>54.42</td><td>0.685</td><td>29.01</td><td>0.316</td><td>1.06E-04</td><td>47.93</td><td>236.28</td></tr><tr><td>Ours*</td><td>29.15</td><td>0.735</td><td>29.61</td><td>0.287</td><td>0.79E-04</td><td>35.28</td><td>153.47</td></tr><tr><td>Ours</td><td>27.70</td><td>0.760</td><td>29.70</td><td>0.272</td><td>0.73E-04</td><td>14.30</td><td>117.81</td></tr></table>
+
+Table 2: Quantitative comparison on TED-talks dataset. The best and the second-best results are indicated in red and blue respectively.AnimateAnyone† is trained on our noisy dataset.  
+
+<table><tr><td>Method</td><td>FID↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>L1↓</td><td>FID-VID↓</td><td>FVD↓</td></tr><tr><td>MRAA (Siarohin et al., 2021)</td><td>50.36</td><td>0.762</td><td>31.90</td><td>0.266</td><td>0.50E-04</td><td>82.79</td><td>493.02</td></tr><tr><td>TPSMM (Zhao &amp; Zhang, 2022)</td><td>23.71</td><td>0.771</td><td>32.30</td><td>0.252</td><td>0.49E-04</td><td>32.12</td><td>260.67</td></tr><tr><td>DisCo (Wang et al., 2023)</td><td>75.48</td><td>0.575</td><td>27.99</td><td>0.309</td><td>1.21E-04</td><td>66.18</td><td>393.04</td></tr><tr><td>DisCo+ (Wang et al., 2023)</td><td>63.28</td><td>0.596</td><td>28.12</td><td>0.300</td><td>1.11E-04</td><td>55.81</td><td>343.20</td></tr><tr><td>MagicAnimate (Xu et al., 2023)</td><td>41.58</td><td>0.529</td><td>28.28</td><td>0.310</td><td>1.73E-04</td><td>33.61</td><td>223.54</td></tr><tr><td>MagicPose (Chang et al., 2023)</td><td>23.39</td><td>0.723</td><td>30.08</td><td>0.236</td><td>0.81E-04</td><td>27.53</td><td>214.23</td></tr><tr><td>Moore-AnimateAnyone</td><td>25.93</td><td>0.710</td><td>30.99</td><td>0.310</td><td>0.46E-04</td><td>41.20</td><td>262.49</td></tr><tr><td>AnimateAnyone†</td><td>47.68</td><td>0.691</td><td>29.59</td><td>0.283</td><td>1.15E-04</td><td>30.07</td><td>241.76</td></tr><tr><td>Ours</td><td>18.21</td><td>0.779</td><td>30.88</td><td>0.198</td><td>0.46E-04</td><td>10.24</td><td>81.73</td></tr></table>
+
+The result is the depth order map. During inference, the calculation of the average depth value of the skeleton dilation areas is skipped. Instead, we directly compute the order values of the characters in the reference image and assign these values to the corresponding areas of the character dilation skeletons. Given a training video  $v$ , where the  $i$ -th frame is denoted as  $v_{i}$ , supposing that there are  $J$  characters with pose skeleton  $\{p_1, \dots, p_J\}$  in  $v_{i}$ . The training processes can be illustrated as
+
+$$
+a _ {i, 1}, \dots a _ {i, J} = \mathcal {D} \left(p _ {1}, \dots , p _ {J}\right), \tag {4}
+$$
+
+$$
+m _ {i, j} = a _ {i, j} - \left(1 - \bigcap_ {k \in \{1, \dots , J \}} \left(a _ {i, k}\right)\right), j \in \{1.. J \}, \tag {5}
+$$
+
+$$
+\operatorname {r a n k} _ {j} = f _ {\text {s o r t}} \left(m _ {i, 1} \odot f _ {\mathrm {d}} \left(v _ {1}\right), \dots , m _ {i, j} \odot f _ {\mathrm {d}} \left(v _ {i}\right)\right), \tag {6}
+$$
+
+$$
+c _ {\text {d e p t h}, j} = m _ {i, j} \odot L _ {\text {r a n k} _ {j}} + \left(1 - m _ {i, j} \odot c _ {\text {d e p t h}, j + 1}\right), c _ {\text {d e p t h}, J + 1} = 0, \tag {7}
+$$
+
+$$
+c _ {\text {d e p t h}} = g _ {\mathrm {d p}} \left(c _ {\text {d e p t h}, 1}\right), \tag {8}
+$$
+
+where  $\mathcal{D}$  represents the dilation operation,  $a_{i,j}$  denotes the character-wise skeleton dilation map.  $\cap$  is intersection operation and  $m_{i,j}$  is the non-intersection region of each character.  $f_{\mathrm{d}}$  represents the depth calculation network,  $f_{\mathrm{sort}}$  is average depth sorting (descending order), and  $rank_j$  is the depth ranking of character  $j$ .  $L_{rank_j}$  denotes the value assigned to  $j$  based on depth ranking and  $c_{\mathrm{depth},j}$  is the depth order map of the farthest  $j$  characters.  $g_{\mathrm{dp}}$  represents the depth order guider.
+
+# 4.3 REFERENCE POSE GUIDER
+
+The positions of characters in inference images and pose sequences often exhibit inconsistencies. Diffusion-based methods employ spatial cross-attention to facilitate the interaction between the pose frame and reference image, enabling it to learn character texture information from the reference images and accurately map them to the corresponding pose positions. We realize that this process involves the model implicitly decoupling the character's pose and fine-grained texture information from the reference image, and then mapping the texture to the target position by aligning the character's pose with the target pose. Consequently, we can assist the model in this decoupling process by introducing the reference pose as a prior. Since the reference pose map and the pose frame are consistent in format, the model can directly align the pose and map the characters efficiently without
+
+![](images/15e4ba13cf2c8f0d6dab37adec395ba5d57fcb240000cd77fb5e5e577b9e3e4f.jpg)  
+Figure 6: Qualitative comparisons between baselines and our approach on dataset Tiktok (Top three rows) and TED-talks (bottom two rows). MRAA* and TPSMM* present these methods utilizing ground-truth videos as driving signals.
+
+calculating poses from the reference image. In this way, the model can reduce the learning load and focus on mapping character textures. Given the reference image  $x$ , the  $c_{\mathrm{ref\_pose}}$  can be written as
+
+$$
+c _ {\text {r e f}} = g _ {\text {r p}} \left(f _ {\text {s}} (x)\right), \tag {9}
+$$
+
+where  $f_{\mathrm{s}}$  represents the skeleton extraction network and  $g_{\mathrm{rp}}$  denotes the reference pose guider.
+
+# 4.4 SKELETAL DILATION MAP
+
+Skeletal dilation map is used to mask the character and separate it from the background, and Fig. 4 illustrates its calculation. The skeletal dilation map does not perfectly cover the character area compared to segmentation maps, but it achieves better results. Using segmentation map as mask strictly requires highly accurate body segmentation maps as driving signals. However, given only an inference image and pose sequence during inference, it is challenging to generate a segmentation map sequence that perfectly aligns with the characters in the reference image. The discrepancies in segmentation map accuracy between training and inference lead to model misalignment and poor performance. In contrast, skeletal dilation maps can be generated directly from pose sequences and generalize well to characters of varying heights and body types. For both training and inference, we use the same skeletal dilation maps, enabling the model to learn to adapt its mask range and become less sensitive to mask precision. Further details can be found in the appendix (Sec. B.3).
+
+# 4.5 MODEL ARCHITECTURE
+
+Fig. 3 shows our framework. Specifically, the reference image is compressed using VAE and then fed into ReferenceNet (Cao et al., 2023). It interacts with the features in the U-net model through spatial cross-attention. Meanwhile, the CLIP-encoded reference image is used as the prompt to replace the text prompt in the diffusion model. The three proposed conditions and the pose drive signal sequence are encoded by the same structured guider, which consists of convolutional layers. These encoded features are then incorporated into the initial noise latent. The whole training objective can be formulated as
+
+$$
+c _ {\text {m u l t i}} = c _ {\text {p o s e}} + c _ {\text {f l o w}} + c _ {\text {d e p t h}} + c _ {\text {r e f}}, \tag {10}
+$$
+
+$$
+\mathcal {L} = \mathbb {E} _ {\mathcal {E} (v _ {1: n}), c, \epsilon_ {1: n} \sim \mathcal {N} (0, I), t} \left[ | | \epsilon - \epsilon_ {\theta} \left(z _ {1: n} ^ {t}, t, x, c _ {\mathrm {m u l t i}}\right) | | _ {2} ^ {2} \right], \tag {11}
+$$
+
+where  $c_{\mathrm{multi}}$  denotes the overall control condition. The definitions of  $v_{1:n}$ ,  $\epsilon_{1:n}$ , and  $z_{1:n}^{t}$  are analogous to Eq. 1.
+
+Table 3: Quantitative comparison on Multi-Character Bench. The best and the second-best results are indicated in red and blue respectively.AnimateAnyone† is trained on our noisy dataset.  
+
+<table><tr><td>Method</td><td>FID↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>L1↓</td><td>FID-VID↓</td><td>FVD↓</td></tr><tr><td>DisCo (Wang et al., 2023)</td><td>77.61</td><td>0.793</td><td>29.65</td><td>0.239</td><td>7.64E-05</td><td>104.57</td><td>1367.47</td></tr><tr><td>DisCo+ (Wang et al., 2023)</td><td>73.21</td><td>0.799</td><td>29.66</td><td>0.234</td><td>7.33E-05</td><td>92.26</td><td>1303.08</td></tr><tr><td>MagicAnime (Xu et al., 2023)</td><td>40.02</td><td>0.819</td><td>29.01</td><td>0.183</td><td>6.28E-05</td><td>19.42</td><td>223.82</td></tr><tr><td>MagicPose (Chang et al., 2023)</td><td>31.06</td><td>0.806</td><td>31.81</td><td>0.217</td><td>4.41E-05</td><td>30.95</td><td>312.65</td></tr><tr><td>Moore-AnimateAnyone</td><td>33.04</td><td>0.795</td><td>31.44</td><td>0.213</td><td>5.02E-05</td><td>22.98</td><td>272.98</td></tr><tr><td>AnimateAnyone†</td><td>35.59</td><td>0.796</td><td>31.10</td><td>0.208</td><td>4.87E-05</td><td>22.74</td><td>236.48</td></tr><tr><td>Ours</td><td>26.95</td><td>0.830</td><td>31.86</td><td>0.173</td><td>4.01E-05</td><td>14.56</td><td>142.76</td></tr></table>
+
+![](images/057367f07bf2eb662f8a841d8ad7e9b041238110d2e71821bb6b853b57f08953.jpg)  
+Figure 7: Qualitative comparisons on the Multi-Character bench.
+
+# 5 EXPERIMENT
+
+Dataset. The robustness of our model to noisy data enables direct training on unfiltered videos. We collect 4000 character-action videos of 2M frames as our training set. We analyze the level of contamination in the noisy dataset in the appendix (Sec. A.1).
+
+Training strategy. Our model is trained in two stages. In the first stage, we freeze the VAE (Van Den Oord et al., 2017) and CLIP (Radford et al., 2021) image encoder, and remove the temporal motion modules. U-Net, ReferenceNet, and multiple-condition guiders are trained to align their spatial generative capacities. In addition, we utilize the weights of Stable Diffusion v1.5 (Rombach et al., 2022) to initialize this training stage. In the second stage, we incorporate temporal motion modules along with all parameters from stage one, aiming to endow the model with temporal smoothness. Besides, we use the weights ofAnimateDiff v2 (Guo et al., 2023b) to initialize this training stage.
+
+Implementation Details. We sample 16 frames of video, resize and center-crop them to a resolution of  $896 \times 640$ . Experiments are conducted on 8 NVIDIA A800 GPUs. Both stages are optimized using Adam with a learning rate  $1 \times 10^{-5}$ . In the first stage, we train our model for 60K steps with a batch size of 4, and in the second stage, we train for 60K steps with a batch size of 1. At inference, we apply DDIM (Song et al., 2020) sampler for 50 denoising steps, with classifier-free guidance (Ho & Salimans, 2022) scale of 1.5. See the appendix (Sec. C.1) for more details.
+
+# 5.1 COMPARISONS
+
+Dataset and metrics. Following the previous methods (Wang et al., 2023), we evaluate our method on TikTok videos (Wang et al., 2023) and TED-talks (Siarohin et al., 2021). Additionally, due to the lack of benchmarks for multiple-character video generation, we collect 20 multiple-character dancing videos with 3917 frames, named Multi-Character. This dataset serves as a benchmark for evaluating models' capabilities in generating pose-controllable videos with multiple characters. Our evaluation metrics adhere to the existing research literature. Specifically, we employ conventional image metrics to assess the quality of individual frames, including L1 error, PSNR (Hore & Ziou, 2010), SSIM (Wang et al., 2004), LPIPS (Zhang et al., 2018), and FID (Heusel et al., 2017). For
+
+Table 4: Quantitative ablation results on TikTok Dataset.  
+
+<table><tr><td>Method</td><td>FID↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>L1↓</td><td>FID-VID↓</td><td>FVD↓</td></tr><tr><td>w/o. All Conditions</td><td>54.42</td><td>0.685</td><td>29.01</td><td>0.316</td><td>1.06E-04</td><td>47.93</td><td>236.28</td></tr><tr><td>w/o. Optical Flow</td><td>52.94</td><td>0.710</td><td>28.21</td><td>0.318</td><td>0.96E-04</td><td>34.35</td><td>178.48</td></tr><tr><td>w/o. Ref. Pose</td><td>34.74</td><td>0.740</td><td>29.13</td><td>0.285</td><td>0.82E-04</td><td>19.58</td><td>139.32</td></tr><tr><td>w/o. Depth Order</td><td>27.43</td><td>0.754</td><td>29.98</td><td>0.270</td><td>0.79E-04</td><td>14.50</td><td>119.26</td></tr><tr><td>Ours</td><td>27.70</td><td>0.760</td><td>29.70</td><td>0.272</td><td>0.73E-04</td><td>14.30</td><td>117.81</td></tr></table>
+
+![](images/514b6cf278b0057a88f0c99ba10878d25d995b2b972806f860c61849178a89f2.jpg)  
+w/o. Optical Flow Condition
+
+![](images/401a39a62c2c0226290730d4462ee05586f23badef3a6dab8bdc48c443535705.jpg)
+
+![](images/f69623f95bf41a09e9fa814819f8bf716defa743fe31aabcea80d34a29453356.jpg)  
+w/. Optical Flow Condition  
+Figure 8: Qualitative comparison results of ablation variants without optical flow condition. Transparent and overlay the images to clearly see the changes in the background.
+
+![](images/42e77fa027df9de6c3096493ab5be6d549ac8757ebb8f7eadc2ac1a23518db14.jpg)
+
+![](images/995b5ddb14b72237a8137d2a4b5b1cab85b4103427f42cd56a9fd6ef969c9d57.jpg)
+
+video evaluation metrics FID-VID (Balaji et al., 2019) and FVD (Unterthiner et al., 2018), we form a sample by concatenating each consecutive 16 frames.
+
+Counterparts. We compare with several state-of-the-art methods for character image animation. (1) MRAA (Siarohin et al., 2021) and TPSMM (Zhao & Zhang, 2022) are GAN-based methods. (2) DreamPose (Karras et al., 2023), Disco (Wang et al., 2023), MagicAnimate (Xu et al., 2023), MagicPose (Chang et al., 2023) andAnimateAnyone (Hu et al., 2023) are VLDM-based methods. Note that we evaluate theAnimateAnyone reproduced by MooreThreads (MooreThreads, 2024) on TED-talks and Multi-Character. (3) We reproduceAnimateAnyone trained on the noisy dataset.
+
+Unified evaluation standards. We notice that not all approaches adhere to a uniform generation size. As methods with different generation sizes yield different metric results, potentially leading to unfair comparisons, we standardize the generation sizes by center-cropping and resizing to  $512 \times 512$ . Under this unified standard, we reevaluate methods that do not conform to this generation size and directly refer to the original results of methods that comply.
+
+Evaluation on TikTok dataset. Table 1 presents the quantitative comparison of the TikTok dataset. Our model performs best on five metrics and second best on the FID and LPIPS metrics. Notably, our model excels particularly in video metrics FID-VID and FVD. Compared with the second-best method in FID-VID and FVD, our model shows a significant improvement of  $39\%$  and  $35\%$  respectively. TheAnimateAnyone† we reproduce is trained on the noisy dataset and performs poorly, showing that noisy data greatly weakens the model. In contrast, the module we design enables our model to exhibit strong robustness to noisy data and perform excellently when trained on noisy data. Additionally, Ours* trained on Tiktok training set shows slight improvement over baselines, indicating our method excels when trained on clean data. The visual comparison is shown in the top three rows of Fig. 6. Our approach performs better in pose following and visual quality. Specifically, the first and third rows show that our method not only accurately follows the pose but also generates hand details. The second row suggests the powerful pose-following capability of our method, which is the sole approach capable of accurately generating the pose with the arm raised in reverse.
+
+Evaluation on TED-talks dataset. As reported in Table 2, our model achieves SOTA performance across all metrics except PSNR. Compared with MagicPose achieving the second-best of FID-VID and FVD, our model demonstrates a substantial enhancement of  $49\%$  and  $48\%$  respectively. Again, it is observed that the base model performs poorly for the influence of noisy data. The bottom two rows in Fig. 6 illustrate the visual comparison. In the fourth row, only our method successfully maintains the consistency of the character holding white notes in the inference image. In the last row, our method exhibits the best performance in the pose following, with the fewest artifacts.
+
+Evaluation on Multi-Character bench. Our results in Table 3 significantly outperform others across all metrics on Multi-Character bench. In terms of the video metrics FID-VID and FVD, we outperform the second-best method by  $25\%$  and  $36\%$ , respectively. The qualitative comparisons in Fig. 7 highlight the significant advantages of our approach in maintaining identity consistency (top row), pose following (second & third rows), and preserving spatial relationships (bottom row).
+
+![](images/d5bfbdc791f1f461276b4ecb1924f81f062ad0d73ed2b457d39b04e6cc63a38a.jpg)  
+w/o. Depth Order Condition
+
+![](images/858e7d40ffddb1d6ccbea276a1acc9b5c4053bb2cd327f837444e2ecbd816ce1.jpg)  
+w/. Depth Order Condition
+
+![](images/dd3eb99161a8a1b040d0e510d37360fe800ceff0b42dc359c96e81137e2c2896.jpg)  
+GroundTruth
+
+![](images/82e082307ca4d5c002275bca22f3932303846a6e1dcb0b7539ba4bd5c50cb4fd.jpg)  
+Figure 9: Qualitative comparison results of ablation variants without depth order condition.  
+Reference image
+
+![](images/2ad001c4e3f5d86455b5fe7c8747167e81f7b3b088e85fc00b6f2bf6294333fe.jpg)  
+w/o. reference pose
+
+![](images/a486719d181eccc59ceddb24433019d3d2ed52acbaa16699691dd5fe69f27d08.jpg)
+
+pose
+
+![](images/e21513e07d03b2d83ab300cb71109ba02ba0d0c54769d123b4ea4c48fafd41ee.jpg)
+
+![](images/f2a72e116ef8a29ef7821e7525af0fb538192bdc063151a12b7438436db953c0.jpg)
+
+![](images/428da340d8afd234874e86e65414410649ffeb117679c0da6ca9857e089416ab.jpg)
+
+![](images/bb4874b5efa9f8210be0426a98d5967f0995cf84fdb0e7cd922f7c323fade2e3.jpg)
+
+![](images/05fd7048532feee5d5278cd58c8941103e5318d94e1589370744be1b404bb2b7.jpg)  
+Reference image  
+Figure 10: Qualitative comparison results of ablation variants without reference pose condition.
+
+![](images/3c30833ad964998f3edf6dc17a3598dfe4d7a4057fb97dc50ae58add01be3b7b.jpg)  
+w/. reference pose
+
+![](images/7b2fe62de1d719c3138cb3c182f2ae68480fbcba35c9a1916ab39e1de62d3426.jpg)
+
+pose
+
+![](images/581a2f7d5ec4656d0451eb6145d6c1b9c27788d82dc60458b43c5cac549ac76a.jpg)
+
+![](images/4485c92fdb9965940d1066c8059dbce6d18cc14c78e64585d11456e7da48c0fb.jpg)
+
+![](images/dcce6dcd78876cb5721276a7e918304fc76231163d3999e7c30b7bf180ead5ca.jpg)
+
+![](images/65b8aee7a9b1e5ca37fd8ad1f4f300a596c4f74b28a5ae65800b777a8675e1b5.jpg)
+
+# 5.2 ABLATION STUDY
+
+To investigate the roles of the proposed conditions, we examine three variants without Optical Flow, Depth Order, and Reference Pose, respectively. In Table 4, the proposed full method ("Ours") with three conditions outperforms other variants, demonstrating the effectiveness of the three conditions.
+
+Optical flow. In Table 4, the variant without optical flow exhibits the most significant decline, suggesting that optical flow has the most pronounced positive impact on the model. The comparison results shown in Fig. 8 illustrate where its primary effects are background stability. We overlay the transparent six frames to form the right image, enabling a clear depiction of the moving region. As indicated by the red boxes, the background in the results of the variant without optical flow shows noticeable shaking, whereas the background in the results of the full method remains consistently stable. This validates that noisy data with unstable backgrounds leads to the generation of unstable backgrounds, whereas the incorporation of the optical flow condition can solve this problem.
+
+Depth order. Fig. 9 validates the positive impact of the depth order condition. The red boxes highlight the overlapping areas of multiple characters. The variant without depth order condition fails to generate the hand of the character on the left placed behind the right character, instead generating a malformed hand. Conversely, the full method with the depth order condition generates overlapping areas of multiple characters, i.e., the hand of the left character is positioned behind the right one, and the hand of the right character is in front of the head of the left one.
+
+Reference pose. Fig. 10 illustrates the comparison when the character position of the inference image and pose sequence are not aligned. The full method with reference pose performs better in terms of image quality and character consistency. The variant without reference pose maintains the background consistency, while failing to maintain character consistency (see the changes of clothing of the character). In contrast, the full method with reference pose effectively achieves alignment between character and pose sequence, thereby maintaining character and background consistency.
+
+# 6 CONCLUSION
+
+In this paper, we design three guiders to enhance the implicit decoupling ability of a pose-controllable character image animation framework that integrates multiple conditions, addressing the unstable background and poor handling of body occlusions in multiple character scenes. The optical flow guider decouples the background to facilitate the learning of stable background generation. The depth order guider decouples multiple character features into individuals to solve the problem of multiple character generation. The reference pose guider enhances the learning of characters' appearance. Moreover, we have curated and released a benchmark dataset of pose-controllable videos with multiple characters. Experiment studies show the effectiveness of our method.
+
+# ACKNOWLEDGMENT
+
+This work was supported in part by the National Natural Science Foundation of China (Grant No. 62372480), in part by the Guangdong Basic and Applied Basic Research Foundation (No. 2023A1515012839), in part by Huawei Gift Fund (No. HUAWEI25IS02), and in part by HKUST-MetaX Joint Lab Fund (No. METAX24EG01-D).
+
+# REFERENCES
+
+Yogesh Balaji, Martin Renqiang Min, Bing Bai, Rama Chellappa, and Hans Peter Graf. Conditional gan with discriminative filter generation for text-to-video synthesis. In International Joint Conference on Artificial Intelligence, pp. 2, 2019.  
+Mingdeng Cao, Xintao Wang, Zhongang Qi, Ying Shan, Xiaohu Qie, and Yinqiang Zheng. Mas-actr: Tuning-free mutual self-attention control for consistent image synthesis and editing. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 22560-22570, 2023.  
+Caroline Chan, Shiry Ginosar, Tinghui Zhou, and Alexei A Efros. Everybody dance now. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 5933-5942, 2019.  
+Di Chang, Yichun Shi, Quankai Gao, Jessica Fu, Hongyi Xu, Guoxian Song, Qing Yan, Xiao Yang, and Mohammad Soleymani. Magicdance: Realistic human dance video generation with motions & facial expressions transfer. arXiv preprint arXiv:2311.12052, 2023.  
+Haoxin Chen, Yong Zhang, Xiaodong Cun, Menghan Xia, Xintao Wang, Chao Weng, and Ying Shan. Videocaster2: Overcoming data limitations for high-quality video diffusion models. arXiv preprint arXiv:2401.09047, 2024a.  
+Qihua Chen, Yue Ma, Hongfa Wang, Junkun Yuan, Wenzhe Zhao, Qi Tian, Hongmei Wang, Shaobo Min, Qifeng Chen, and Wei Liu. Follow-your-canvas: Higher-resolution video outpainting with extensive content generation. arXiv preprint arXiv:2409.01055, 2024b.  
+Patrick Esser, Johnathan Chiu, Parmida Atighehchian, Jonathan Granskog, and Anastasis Germanidis. Structure and content-guided video synthesis with diffusion models. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 7346-7356, 2023.  
+Kunyu Feng, Yue Ma, Bingyuan Wang, Chenyang Qi, Haozhe Chen, Qifeng Chen, and Zeyu Wang. Dit4edit: Diffusion transformer for image editing. arXiv preprint arXiv:2411.03286, 2024.  
+Riza Alp Güler, Natalia Neverova, and Iasonas Kokkinos. Denseseposé: Dense human pose estimation in the wild. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7297-7306, 2018.  
+Xun Guo, Mingwu Zheng, Liang Hou, Yuan Gao, Yufan Deng, Chongyang Ma, Weiming Hu, Zhengjun Zha, Haibin Huang, Pengfei Wan, et al. I2v-adapter: A general image-to-video adapter for video diffusion models. arXiv preprint arXiv:2312.16693, 2023a.  
+Yuwei Guo, Ceyuan Yang, Anyi Rao, Yaohui Wang, Yu Qiao, Dahua Lin, and Bo Dai. Animatediff: animate your personalized text-to-image diffusion models without specific tuning. arXiv preprint arXiv:2307.04725, 2023b.  
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in Neural Information Processing Systems, 2017.  
+Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598, 2022.  
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840-6851, 2020.
+
+Alain Hore and Djemel Ziou. Image quality metrics: Psnr vs. sSIM. In IEEE International Conference on Pattern Recognition, pp. 2366-2369, 2010.  
+Li Hu, Xin Gao, Peng Zhang, Ke Sun, Bang Zhang, and Liefeng Bo. Animate anyone: Consistent and controllable image-to-video synthesis for character animation. arXiv preprint arXiv:2311.17117, 2023.  
+Tianjian Jiang, Xu Chen, Jie Song, and Otmar Hilliges. Instantavator: Learning avatars from monocular video in 60 seconds. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 16922-16932, 2023.  
+Johanna Karras, Aleksander Holynski, Ting-Chun Wang, and Ira Kemelmacher-Shlizerman. Dreampose: Fashion image-to-video synthesis via stable diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 22623-22633. IEEE, 2023.  
+Levon Khachatryan, Andranik Movsisyan, Vahram Tadevosyan, Roberto Henschel, Zhangyang Wang, Shant Navasardyan, and Humphrey Shi. Text2video-zero: Text-to-image diffusion models are zero-shot video generators. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 15954-15964, 2023.  
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.  
+Zhe Kong, Yong Zhang, Tianyu Yang, Tao Wang, Kaihao Zhang, Bizhu Wu, Guanying Chen, Wei Liu, and Wenhan Luo. Omg: Occlusion-friendly personalized multi-concept generation in diffusion models. In European Conference on Computer Vision, pp. 253-270. Springer, 2025.  
+Feng Liang, Bichen Wu, Jialiang Wang, Licheng Yu, Kunpeng Li, Yinan Zhao, Ishan Misra, Jia-Bin Huang, Peizhao Zhang, Peter Vajda, et al. Flowvid: Taming imperfect optical flows for consistent video-to-video synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8207-8216, 2024.  
+Yue Ma, Xiaodong Cun, Yingqing He, Chenyang Qi, Xintao Wang, Ying Shan, Xiu Li, and Qifeng Chen. Magicstick: Controllable video editing via control handle transformations. arXiv preprint arXiv:2312.03047, 2023.  
+Yue Ma, Yingqing He, Xiaodong Cun, Xintao Wang, Siran Chen, Xiu Li, and Qifeng Chen. Follow your pose: Pose-guided text-to-video generation using pose-free videos. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 4117-4125, 2024a.  
+Yue Ma, Yingqing He, Hongfa Wang, Andong Wang, Chenyang Qi, Chengfei Cai, Xiu Li, Zhifeng Li, Heung-Yeung Shum, Wei Liu, et al. Follow-your-click: Open-domain regional image animation via short prompts. arXiv preprint arXiv:2403.08268, 2024b.  
+Yue Ma, Hongyu Liu, Hongfa Wang, Heng Pan, Yingqing He, Junkun Yuan, Ailing Zeng, Chengfei Cai, Heung-Yeung Shum, Wei Liu, et al. Follow-your-emoji: Fine-controllable and expressive freestyle portrait animation. arXiv preprint arXiv:2406.01900, 2024c.  
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.  
+MooreThreads. Moore-animateanyone. https://github.com/MooreThreads/Moore-AnimateAnyone, 2024.  
+OpenMMLab. MMFlow: Openmmlab optical flow toolbox and benchmark. https://github.com/open-mmlab/mmflow, 2021.  
+William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4195-4205, 2023.  
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, pp. 8748-8763, 2021.
+
+Ali Razavi, Aaron Van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. Advances in Neural Information Processing Systems, 32, 2019.  
+Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10684-10695, 2022.  
+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-assisted intervention-MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pp. 234-241. Springer, 2015.  
+Aliaksandr Siarohin, Stéphane Lathuilière, Sergey Tulyakov, Elisa Ricci, and Nicu Sebe. First order motion model for image animation. Advances in Neural Information Processing Systems, 32, 2019.  
+Aliaksandr Siarohin, Oliver J Woodford, Jian Ren, Mengei Chai, and Sergey Tulyakov. Motion representations for articulated animation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13653-13662, 2021.  
+Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020.  
+Deqing Sun, Xiaodong Yang, Ming-Yu Liu, and Jan Kautz. Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8934-8943, 2018.  
+Jingfan Tan, Hyunhee Park, Ying Zhang, Tao Wang, Kaihao Zhang, Xiangyu Kong, Pengwen Dai, Zikun Liu, and Wenhan Luo. Blind face video restoration with temporal consistent generative prior and degradation-aware prompt. In Proceedings of the 32nd ACM International Conference on Multimedia, pp. 1417-1426, 2024.  
+Thomas Unterthiner, Sjoerd Van Steenkiste, Karol Kurach, Raphael Marinier, Marcin Michalski, and Sylvain Gelly. Towards accurate generative models of video: A new metric & challenges. arXiv preprint arXiv:1812.01717, 2018.  
+Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. Advances in Neural Information Processing Systems, 30, 2017.  
+Tan Wang, Linjie Li, Kevin Lin, Chung-Ching Lin, Zhengyuan Yang, Hanwang Zhang, Zicheng Liu, and Lijuan Wang. Disco: Disentangled control for referring human dance generation in real world. arXiv e-prints, pp. arXiv-2307, 2023.  
+Ting-Chun Wang, Ming-Yu Liu, Andrew Tao, Guilin Liu, Jan Kautz, and Bryan Catanzaro. Few-shot video-to-video synthesis. arXiv preprint arXiv:1910.12713, 2019.  
+Ting-Chun Wang, Arun Mallya, and Ming-Yu Liu. One-shot free-view neural talking-head synthesis for video conferencing. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10039-10049, 2021.  
+Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing, pp. 600-612, 2004.  
+Jay Zhangjie Wu, Yixiao Ge, Xintao Wang, Stan Weixian Lei, Yuchao Gu, Yufei Shi, Wynne Hsu, Ying Shan, Xiaohu Qie, and Mike Zheng Shou. Tune-a-video: One-shot tuning of image diffusion models for text-to-video generation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 7623-7633, 2023.  
+Zhongcong Xu, Jianfeng Zhang, Jun Hao Liew, Hanshu Yan, Jia-Wei Liu, Chenxu Zhang, Jiashi Feng, and Mike Zheng Shou. Magicanimate: Temporally consistent human image animation using diffusion model. arXiv preprint arXiv:2311.16498, 2023.
+
+Lihe Yang, Bingyi Kang, Zilong Huang, Xiaogang Xu, Jiashi Feng, and Hengshuang Zhao. Depth anything: Unleashing the power of large-scale unlabeled data. arXiv preprint arXiv:2401.10891, 2024.  
+Zhendong Yang, Ailing Zeng, Chun Yuan, and Yu Li. Effective whole-body pose estimation with two-stages distillation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4210-4220, 2023.  
+Zhengming Yu, Wei Cheng, Xian Liu, Wayne Wu, and Kwan-Yee Lin. Monohuman: Animatable human neural field from monocular video. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 16943-16953, 2023.  
+Beiyuan Zhang, Yue Ma, Chunlei Fu, Xinyang Song, Zhenan Sun, and Ziqiang Li. Follow-your-multipose: Tuning-free multi-character text-to-video generation via pose guidance. arXiv preprint arXiv:2412.16495, 2024.  
+Jianfeng Zhang, Hanshu Yan, Zhongcong Xu, Jiashi Feng, and Jun Hao Liew. Magicavatar: Multi-modal avatar generation and animation. arXiv preprint arXiv:2308.14748, 2023.  
+Pengze Zhang, Lingxiao Yang, Jian-Huang Lai, and Xiaohua Xie. Exploring dual-task correlation for pose guided person image generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7713-7722, 2022.  
+Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 586-595, 2018.  
+Jian Zhao and Hui Zhang. Thin-plate spline motion model for image animation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3657-3666, 2022.
+
+# APPENDIX
+
+In this supplementary material, we present:
+
+- Dataset Supplement (Sec. A)
+
+- Analysis of noisy training dataset. (Sec. A.1)  
+- Details of the training dataset. (Sec. A.2)  
+- Details of the Multi-character benchmark. (Sec. A.3)
+
+- Methodology Supplement (Sec. B)
+
+- Pseudo code of depth order guider. (Sec. B.1)  
+- Long video inference. (Sec. B.2)  
+- Further details of skeletal dilation map. (Sec. B.3)
+
+- Experimental Supplement (Sec. C)
+
+- More implementations. (Sec. C.1)  
+- Unified standard for comparative experiments. (Sec. C.2)
+
+- Limitation (Sec. D)  
+- More Qualitative Results (Sec. E)
+
+- Additional Comparison. (Sec. E.1)  
+- Additional Ablation. (Sec. E.2)  
+- Additional Visualizations. (Sec. E.3)
+
+# A DATASET SUPPLEMENT
+
+# A.1 ANALYSIS OF NOISY TRAINING DATASET
+
+![](images/c8edf36fc8f703ab68eba6c4d769f83972af6f8de6873b4b7514c0ae7fbbc011.jpg)  
+bof-mean=1.0
+
+![](images/ad7bbcfbaf0836bedd008fe1753aea47b7702b3c9f41d53ea7115f755b8ab181.jpg)  
+Figure 11: Distribution histogram and image examples of background optical flow mean.
+
+![](images/d20161c73410a66bfdf0d9930f327becf75cc35ff9ac454812b5580c2763084b.jpg)  
+Background Optical Flow Mean (bof-mean)  
+bof-mean=1.5
+
+![](images/5a5c0977f6fdc54eae0dc1923eb2a5b9f03c57bc3a02242d41042de351211ec9.jpg)  
+bof-mean=2.0
+
+![](images/966c1b36a82c468da02ba11549510a4ffcfb585b197c35dc73fb6b839a1f757d.jpg)  
+bof-mean2.0
+
+We analyze the level of contamination in the noise dataset. Specifically, we calculate the background optical flow using a skeletal dilation mask to exclude character regions, then average it across frames to derive the background optical flow mean for each video. Fig. 11 (right) shows videos with different background optical flow means. It is observed that only when the background optical flow mean is at least less than 1, the motion of the background of the video is imperceptible to human eyes. Fig. 11 (left) illustrates the distribution of the background optical flow mean in the training set. It indicates that only about  $12\%$  of the training set videos have a background optical flow mean of less than 1. A noise data proportion of  $88\%$  indicates severe contamination of the dataset. Therefore, it is necessary to incorporate background optical flow maps into the network for training.
+
+We randomly sample 400 videos (approximately  $10\%$  of the dataset) from our dataset and analyze the distribution of videos based on the count of characters they contain. Fig. 12 (left) illustrates the proportion of videos regarding different numbers of characters in the sampling set. Specifically,
+
+![](images/53c42f9c6b87b9809a4c8f470ab7bf737696e411f60bd9bd5a972e6ee47b1df3.jpg)  
+Figure 12: Sampling distribution of character counts in videos and body occlusion rates in double-character videos.
+
+![](images/f5f3e629d085777e846f6f4ecfc4a73e147f21be524011844754404ebc2e8e5d.jpg)
+
+single-character videos account for  $66.25\%$ , double-character videos account for  $29.5\%$ , and videos of triple characters and above account for a total of  $4.25\%$ . It can be concluded that the majority of videos in our dataset are single-character or dual-character, with very few videos containing three or more characters.
+
+Subsequently, we compute the body occlusion rates among characters across all sampled multi-character videos. Specifically, we calculate the intersection area of the skeletal dilation maps for multiple characters in each frame of the video, and then divide it by the union area of these skeletal dilation maps for the same frame. This yields the body occlusion rate for that frame. Subsequently, we compute the average body occlusion rate across all frames of a video and analyze the distribution across all videos. Fig. 12 (right) illustrates the distribution of body occlusion rates in multi-character videos. We can find that  $25\%$  of the videos have a body occlusion rate within 0.05,  $25\%$  fall between 0.05 and 0.13,  $25\%$  are between 0.13 and 0.21, and the remaining  $25\%$  have body occlusion rates exceeding 0.21. It indicates that, on average, the body occlusion rates of most multiple-character videos fall between 0 and 0.21, with videos exhibiting high occlusion rates above 0.21 being relatively rare. Meanwhile, we directly observe that there are many multi-character videos with body occlusion rates around 0.
+
+# A.2 DETAILS OF THE TRAINING DATASET
+
+Table 5: The detailed composition of the training dataset.  
+
+<table><tr><td>Source</td><td>Videos</td><td>Frames</td><td>Proportion</td></tr><tr><td>Tiktok</td><td>2,493</td><td>1,379,449</td><td>68.5%</td></tr><tr><td>YouTube</td><td>938</td><td>435,293</td><td>21.6%</td></tr><tr><td>Kuaishou</td><td>424</td><td>115,101</td><td>5.7%</td></tr><tr><td>Bilibili</td><td>162</td><td>83,785</td><td>4.2%</td></tr></table>
+
+We collect 4,017 character videos with the amount of 2,013,628 frames as our training set. The data come from public videos on TikTok, YouTube, and other websites. The detailed composition of the training dataset is shown in Table 5.
+
+# A.3 DETAILS OF THE MULTI-CHARACTER BENCHMARK
+
+We collect 20 multiple-character dancing videos of 3917 frames in total, from social media, named Multi-Character. Table 6 shows the detailed sources of Multi-Character.
+
+Table 6: The source of Multi-character benchmark.  
+
+<table><tr><td>Video Name</td><td>Url</td><td>Timestamp</td></tr><tr><td>Daovm348PQQ_0</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>00:10-00:14</td></tr><tr><td>Daovm348PQQ_1</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>00:16-00:25</td></tr><tr><td>Daovm348PQQ_2</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>00:47-00:53</td></tr><tr><td>Daovm348PQQ_3</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>01:11-01:15</td></tr><tr><td>Daovm348PQQ_4</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>01:16-01:21</td></tr><tr><td>Daovm348PQQ_5</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>01:22-01:25</td></tr><tr><td>Daovm348PQQ_6</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>02:02-02:09</td></tr><tr><td>Daovm348PQQ_7</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>02:11-02:15</td></tr><tr><td>Daovm348PQQ_8</td><td>https://www.youtube.com/watch?v=Daovm348PQQ</td><td>02:16-02:20</td></tr><tr><td>HpFDXGAo25c_0</td><td>https://www.youtube.com/watch?v=HpFDXGAo25c</td><td>00:20-00:33</td></tr><tr><td>HpFDXGAo25c_1</td><td>https://www.youtube.com/watch?v=HpFDXGAo25c</td><td>00:38-00:43</td></tr><tr><td>HpFDXGAo25c_2</td><td>https://www.youtube.com/watch?v=HpFDXGAo25c</td><td>00:44-00:52</td></tr><tr><td>jx_VseYOi5A_0</td><td>https://www.youtube.com/watch?v=xx_VseYOi5A</td><td>00:32-00:37</td></tr><tr><td>jx_VseYOi5A_1</td><td>https://www.youtube.com/watch?v=xx_VseYOi5A</td><td>00:40-00:53</td></tr><tr><td>jx_VseYOi5A_2</td><td>https://www.youtube.com/watch?v=xx_VseYOi5A</td><td>01:02-01:07</td></tr><tr><td>jx_VseYOi5A_3</td><td>https://www.youtube.com/watch?v=xx_VseYOi5A</td><td>01:08-01:12</td></tr><tr><td>ka3BfUsvRqE_0</td><td>https://www.youtube.com/watch?v=ka3BfUsvRqE</td><td>00:21-00:27</td></tr><tr><td>ka3BfUsvRqE_1</td><td>https://www.youtube.com/watch?v=ka3BfUsvRqE</td><td>00:28-00:33</td></tr><tr><td>ka3BfUsvRqE_2</td><td>https://www.youtube.com/watch?v=ka3BfUsvRqE</td><td>02:56-03:05</td></tr><tr><td>ycInNCB8rbA_0</td><td>https://www.youtube.com/watch?v=ycInNCB8rbA</td><td>00:10-00:15</td></tr></table>
+
+# B METHODOLOGY SUPPLEMENT
+
+# B.1 PSEUDOCODE OF DEPTH ORDER GUIDER
+
+Algorithm 1: Pseudocode for depth order map mask extraction  
+```latex
+Input: Given a training video  $v$  of length  $N$ , where the  $i$ -th frame is denoted as  $v_{i}$ , and suppose that there are  $J$  characters on  $v_{i}$ . The skeleton extraction network is denoted as  $f_{s}$ . The expansion network is denoted as  $f_{e}$ . The depth extraction network is denoted as  $f_{d}$ . The average depth sorting (ascending order) operator is denoted as  $f_{sort}$ . The value assigned to  $r_{j}$  based on depth ranking is denoted as  $L_{r_{j}}$ . The depth guider is denoted as  $g_{dp}$ .  
+1 Initialize  $c_{depth,1,0}, \ldots, c_{depth,N,0} = 0$ .  
+2 Initialize an array  $C_R$ .  
+3 for  $i = 1$  to  $N$  do  
+4  $\begin{array}{rl} & {a_{i,1}, \ldots, a_{i,J} = f_{e}(f_{s}(v_{i}))}\\ & {\textbf{for } j = 1 \textbf{to } J \textbf{do}}\\ & {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ & {\quad r_1, \ldots, r_J = f_{sort}(m_{i,1} \odot f_d(v_i), \ldots, m_{i,J} \odot f_d(v_i))}\\ & {\textbf{for } r_j = r_1 \textbf{to } r_J \textbf{do}}\\ & {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ & {\qquad c_{depth,i,r_j} = m_{i,r_j} \odot L_{r_j} + ((1 - m_{i,r_j}) \odot c_{depth,i,(r_j - 1)})} \\ & {\qquad C_R[i] \leftarrow c_{depth,i,r_j}} \\ & {\qquad c_{depth} = g_{dp}(C_R)} \\ & {\qquad return c_{depth}} \\ & {\qquad c_{depth} = g_{dp}(C_R)} \\ & {\qquad return c_{depth}} \\ & {\qquad c_{depth} = g_{dp}(C_R)} \\ & {\qquad return c_{depth}} \\ & {\qquad c_{depth} = g_{dp}(C_R)} \\ & {\qquad return c_{depth}} \\ & {\qquad c_{depth} = g_{dp}(C_R)} \end{array}$
+```
+
+# B.2 LONGVIDEO INFERENCE
+
+We employ the overlap method for long video inference to maintain the consistency of long video. As illustrated in Fig. 13, we divide the pose sequence into multiple shorter segments for inference, with overlapping parts between adjacent segments. For the overlapping parts of two segments, we
+
+![](images/8e8229c13f39513c6f6f688fbadced83e8e6965a48a99b5196a672991f28dab1.jpg)  
+Figure 13: The pipeline for inferring long videos.
+
+perform addition and averaging to generate temporal smoothing between the two segments. In this work, we perform inference on every 16 frames with a stride of 8 frames, and then stitch them together using an overlap of 8 frames.
+
+# B.3 FURTHER DETAILS OF SKELETAL DILATION MAP
+
+![](images/5e01fbd083ccf4be6a40314dd39f3e6c4d79d8480f801c202aa201e5d034de6f.jpg)  
+Figure 14: The pipeline for calculating skeletal dilation map.
+
+We use skeletal dilation map as mask to cover the character region in the "optical flow maps" and "depth order maps". The purpose of masking off the character regions in "Optical Flow Maps" is to prevent character motion from affecting the decoupling of the background. Additionally, the skeletal dilation map is used to refer to different character areas in "Depth Order Maps", facilitating the decoupling of characters into individual ones.
+
+Fig. 14 shows the pipeline for calculating the skeletal dilation map, which is directly generated from the pose frame. Maybe one concern is that skeletal dilation map often does not cover the character region comprehensively. However, in our attempts, the skeleton dilation map achieves better results compared to the segmentation map that fully covers the human body. We analyze the reasons as follows: (1) The consistency between training and inference processes enables the model to learn to ignore the deviations of skeleton dilation maps. Specifically, skeletal dilation map represents only rough but not precise character regions, during both training and inference. When using the skeletal dilation map as mask during training, the model learns to map rough character regions to precise character regions and applies this capability during inference. Similarly, research on the "regional image animation" task and "modify region animation" task (Ma et al., 2024b) have confirmed that models animate the objects represented by the mask region, rather than animating the mask region itself. Fig. 15 (a) shows that when the skeleton dilation map does not perfectly cover the character, the character still separates perfectly from the background, and the character animation remains outstanding. Fig. 15 (b) illustrates that the parts of the character that exceed the coverage range of the skeletal dilation map can also be animated perfectly. In contrast, Fig. 15 (c) shows a bad case where the portion of the clothing extending beyond the coverage of the skeletal expansion map is too large, resulting in poor animation effects for the clothing. (2) The use of the skeleton dilation map exhibits high tolerance and insensitivity to small amounts of noise. Specifically, the skeleton dilation map requires the model to learn to adapt its mask range, resulting in less sensitivity to mask precision. This enables the same skeletal dilation map to perform well across various scenarios involving different body types and clothing, demonstrating strong generalization capability. In contrast, the use of segmented images imposes strict requirements for high accuracy, and segmentation maps with different body types or clothing, or even slightly different fingers can lead to significant degradation. (3) The discrepancies in segmentation map accuracy between training and inference lead to model misalignment and poor performance. During training, using the segmentation map generated from the original video will make the model highly sensitive to errors. However, given only the
+
+![](images/71d8d067d4860aeb56319e4fe35aa9153436f038e6cd049c20e870a8a7554422.jpg)  
+(a)
+
+![](images/2038a6fa5e2dd3d2ee7aef5ad32b541fa72738f83875617623f3c19051a7693a.jpg)  
+(b)
+
+![](images/326347ecb024c378eab8cc5b4af0973464617d9f6492215f41bee1f0659f5fa9.jpg)  
+(c)  
+Figure 15: The results using skeletal dilation map as mask.
+
+inference image and pose sequence during inference, it is challenging to generate a segmentation map sequence that perfectly aligns with the characters in the reference image. Therefore, minor misalignments in the inference segmentation map can lead to significant degradation of the results.
+
+# C EXPERIMENTAL SUPPLEMENT
+
+# C.1 MORE IMPLEMENTATIONS
+
+In the data processing, we utilize the DWPose (Yang et al., 2023) to extract pose sequence from videos, and PWC-Net (Sun et al., 2018) from the open-source toolbox MMFlow (OpenMMLab,
+
+Table 7: Inconsistent inference standards across all methods.  
+
+<table><tr><td>Method</td><td>Inference Size</td><td>Center-Crop</td><td>Uninterrupted Frames</td></tr><tr><td>MRAA (Siarohin et al., 2021)</td><td>384×384</td><td>×</td><td>✓</td></tr><tr><td>TPSMM (Zhao &amp; Zhang, 2022)</td><td>384×384</td><td>×</td><td>✓</td></tr><tr><td>DreamPose (Karras et al., 2023)</td><td>512×640</td><td>×</td><td>✓</td></tr><tr><td>DisCo (Wang et al., 2023)</td><td>256×256</td><td>✓</td><td>×</td></tr><tr><td>MagicAnimate (Xu et al., 2023)</td><td>512×512</td><td>✓</td><td>✓</td></tr><tr><td>MagicPose (Chang et al., 2023)</td><td>512×512</td><td>×</td><td>✓</td></tr></table>
+
+Table 8: The ablation experiment on the inference standards of Disco+.  
+
+<table><tr><td>Inconsistent Standards</td><td>FID↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>FID-VID↓</td><td>FVD↓</td></tr><tr><td>origin size, w/o center-crop</td><td>30.75</td><td>0.668</td><td>29.03</td><td>0.292</td><td>59.90</td><td>292.80</td></tr><tr><td>256×256, w/o uninterrupted frames</td><td>28.31</td><td>0.674</td><td>29.15</td><td>0.285</td><td>55.17</td><td>267.75</td></tr><tr><td>512×512, w/o uninterrupted frames</td><td>48.29</td><td>0.713</td><td>28.78</td><td>0.320</td><td>52.56</td><td>334.67</td></tr><tr><td>512×512, w/ uninterrupted frames</td><td>48.29</td><td>0.713</td><td>28.78</td><td>0.320</td><td>47.73</td><td>312.49</td></tr></table>
+
+2021) to calculate optical flow vectors. Additionally, we use the Depth Anything (Yang et al., 2024) to extract depth maps from videos.
+
+When conducting long video inference, we perform inference on every 16 frames with a stride of 8 frames, and then stitch them together using an overlap of 8 frames. Besides, we resize and center-crop the "Reference Image" and "Pose Sequence" to a uniform resolution of  $896 \times 640$  pixels ( $512 \times 512$  pixels in comparative experiments). We apply the DDIM sampler for 50 denoising steps, with classifier-free guidance.
+
+Regarding the inference resources, our method requires 24GB of VRAM to generate a  $640 \times 896$  video, 16GB of VRAM for a 480P ( $480 \times 854$ ) video, and 12GB of VRAM for a 360P ( $360 \times 640$ ) video. Thus, our model can run on most commodity-level GPUs with the necessary adjustment to an appropriate video generation resolution.
+
+# C.2 UNIFIED STANDARD FOR COMPARATIVE EXPERIMENTS
+
+We notice in the comparative experiment that not all approaches adhere to a uniform inference size and other inference details. As depicted in Table 7, there are primarily three inconsistent standards that may affect fair comparisons. In the following, the method  $\mathrm{Disco + }$  will be used as an example to illustrate how inconsistent standards affect the fair comparison of metrics, as shown in Table 8. First, Table 7 shows some works resize without center-crop, which can result in significant differences in the test set, as illustrated in Table 8. Second, as shown in the comparison between the second and third rows of Table 8, different inference sizes result in different metric values. Table 7 shows the inconsistent inference sizes of each method. Third, compared to other methods, Disco has a bug in measurement, resulting in fewer video segments being sampled when calculating FID-VID and FVD. This will lead to a decrease in FID-VID and FVD of Disco, as shown in Table 8.
+
+Because methods with different standards result in different metrics, potentially leading to unfair comparisons, we standardize the inference sizes by center-cropping and resizing to  $512 \times 512$ , and we fix the bug in the measurement of disco. Under this unified standard, we reevaluate methods that do not conform to this inference size and directly refer to the data from the original literature of methods that do comply. Most previous works directly refer to the inconsistent statistical data from other works for comparison, resulting in unfair comparisons. We are the first to conduct comparative work under a unified standard.
+
+# D LIMITATION
+
+In this work, we are dedicated to addressing the challenges in pose-controllable multiple-character animation. However, there are still several problems we have not resolved. First, similar to most diffusion-based approaches, our model struggles to generate highly refined facial and hand details. Second, our model also struggles to generate substantial swaying of long skirts, Hanfu, or other large-area clothing very well, as shown in Fig. 15 (c). Third, our model also faces challenges in handling complex multiple-character scenarios, such as those involving four or more characters, or extensive swapping of positions among characters.
+
+# E MORE QUALITATIVE EXPERIMENTS
+
+# E.1 ADDITIONAL COMPARISON
+
+![](images/b0ed50ed1fafd027f4e80638ab2f645b86fc9a929061393d970677e0129771b0.jpg)  
+Figure 16: Qualitative comparison between ours andAnimateAnyone†.
+
+![](images/a29639cf47e694ae32ab9c2a2b6df17244ac767c98d207065f4976dc4072012a.jpg)
+
+![](images/2e579346a8a2fe7e211f5046a6ef4bdde57c8cdda9e3f8bdaa2e15a1bcd66067.jpg)
+
+![](images/5971c393d823e4d258250b4e093384af1fd269a7c87238b2ab9ac8cfbb3d8e28.jpg)
+
+We reproduceAnimateAnyone† on our noisy dataset, but achieve worse results compared to the originalAnimateAnyone, as shown in Tables 1, 2, and 3. This indicates that our noisy dataset's poor video quality significantly weakens the model. Fig. 16 illustrates the qualitative comparison between ours andAnimateAnyone†. In the top three rows, it can be observed that the backgrounds generated byAnimateAnyone† exhibit numerous artifacts. In the bottom two rows,AnimateAnyone† loses arm when generating occluded body parts of multiple characters. In contrast, ours addresses these issues, demonstrating that the proposed conditions have a significant positive impact on the model.
+
+Fig. 17 illustrates a comparison of facial regions between ours and baselines on the TikTok dataset. It can be observed that our method generates the optimal character face, particularly achieving consistency with the reference image in terms of expressions and hairstyles. Note that we use the reference image, rather than the groundtruth, as the correct sample.
+
+![](images/a440a122906e392dd77370f2e09c69b559b020ff2f46d5f7616a9e2bb4fc49c7.jpg)  
+Figure 17: Qualitative comparison of facial regions.
+
+# E.2 ADDITIONAL ABLATION
+
+![](images/fc39f9536eaaa351d1d96b4c161e4c333ec11edf6468bc112b9204a6e374999c.jpg)
+
+![](images/300d5b537538f22efc642f156d24fe176075e616cd3d42c45c69341f6ec0c231.jpg)
+
+![](images/83152afdf75b870e3b330a685c95acfb5b1f0305d71a552cb6cb1cadf52673d7.jpg)
+
+![](images/5a61f1f2b952166a80de218f9e555ed8971f690f8c00d1f5125c8fddf9b758b4.jpg)
+
+![](images/66a250e3f725d6a5965f0227ddf0d5202691b4295d729f521360a620a06fada0.jpg)  
+Reference Image
+
+![](images/cdacaec2bb2b2dd5d732c74da97f91689c24fef129edd545a5cf7401fef65315.jpg)
+
+![](images/1cc724fcfb2d835565b3f5ef4f0fe66114e5204f041ee3f008e5299820e440f6.jpg)
+
+![](images/28b8257cb64a986cbfe710c2fb02578a90e80dd5c787ed79ef208ad58d8a2a84.jpg)
+
+![](images/b7d29a7248ac5836115ccb1a96de16bdcfd14e2db73dd8b8aa43f7eb1885dc5c.jpg)
+
+![](images/8fb18489de6a1c50106ee57c3b6fcf4c86d4650a19630610d06d8bb2954faff1.jpg)  
+w/o. Depth Order Condition
+
+![](images/4a3934d8526bd218fcffd4cc3976445e6911fd3120e2b02fc25a8589add8f4fb.jpg)
+
+![](images/2189ab5876efec5127a20317dd313ba9de46447b51b667f9f527897448ce0ba9.jpg)
+
+![](images/b2e851743649589cc5c96550a8d97d8ee1317128d1b1321ce54610081d5d39a5.jpg)
+
+![](images/f39151d48394e65b00f3891be10c0da939c8538035ba4bcd08c7d34fd5ea799b.jpg)
+
+![](images/765585398eecefda4d309796786aded959f6af5377309a81f4ef8ce761ded16a.jpg)  
+w/. Depth Order Condition
+
+![](images/04777c7e5af9a90e33b4a0445c4d6da12791a012a994ad13d954b863d610c610.jpg)
+
+![](images/beacf8d5c1e2bd51483ba06ce078bbbf9ef5e87bdaffd202a934a9d502fd4aee.jpg)
+
+![](images/7580a977486ea4551d0df9391695b8ed261dbe47fdfb25342e1cb4f31c3a899f.jpg)
+
+![](images/8f45d45c95852290d23a03622168ffe083c3c214a22e19b7ee6fdbb8be10277c.jpg)
+
+![](images/31a5ffe1f2eef22f4505b2f1c3448f3f6c60db18ae2d69bd5afe7b38cee9cbb9.jpg)  
+GroundTruth  
+Figure 18: Additional visualizations of ablation variants without depth order condition.
+
+![](images/17beea158770661f0534cedb419e0e8d74ff0a2d9358defb7e4ac4562c75d590.jpg)  
+Reference image
+
+![](images/85c44d9693a9e8598c5315559b93fa88535429c0ba26960d66b5257148bbd984.jpg)  
+w/o. reference pose
+
+![](images/77d49365da1a325820b03e0e5c291eb6b270a117f63253f5e1f891ccd947b8f3.jpg)
+
+pose
+
+![](images/d84c5b564140f3c56602695b4ab15d56e448b43a022227a69042f5ad4150e0b7.jpg)
+
+![](images/32ef86433bd5db8944ce62537032cb6462030c272ca50d736d803855cea16403.jpg)
+
+![](images/d6b9b4c53444b6e8a54c18fb17ce1d6bdeddf2967dd9c660c5a3594a43c5f61c.jpg)
+
+![](images/3442afd3d589757da873188289c984bdc7c92f9c973a345b2905d91fb599ca6a.jpg)
+
+![](images/01c0a18be5d4f4c2199f53a5e37b97054493e3a3db1197e2784be6bcf977c9af.jpg)  
+Reference image  
+Figure 19: Additional visualizations of ablation variants without reference pose condition.
+
+![](images/f05c4426cf8367246d61a31fb80cbca484e5f553c6b1648851695c6b730a7677.jpg)  
+w/. reference pose
+
+![](images/9dd4dbc5a93b06dcd7ce3ee88854d2748262f154cf990d08781519753f564191.jpg)
+
+ose
+
+![](images/f4874d8ba9405f88b19f191dae695e0e48e106162d887be2d8970f2e5a6bbc27.jpg)
+
+![](images/571b8f23069209444cf8706daae31eb7fb02c03da278af3def20867b3339d218.jpg)
+
+![](images/c63c88b648309224b596ea19af3b09de999ea30dd3a0315cf62edfbd78cd9ea0.jpg)
+
+![](images/e18ce6bcf23f8c505997a90827e0cf6dda57a2f92e4c5a0980d7d7e2f67b43f4.jpg)
+
+# E.3 ADDITIONAL VISUALIZATIONS
+
+![](images/57c67117b0b9892029585adfb5e2c1c61154bfa8a364747e5cae5f513dd83605.jpg)  
+Reference
+
+![](images/47634fbe9292b44397a83ce3fe2d20a09aa7d6fd6684c09ae969a625d9b3be3b.jpg)
+
+![](images/f7def5e16f15f51b297d968a1c60ca7c1f449f69277260e5683953cd77df9f3d.jpg)  
+Animation
+
+![](images/16d8b3add1499c1dd3c4c510893a6530faf99c79fc116236eeac44f351a66725.jpg)
+
+![](images/0f517c9ab825fcf8703c0d1c0d59c7398321c5cc141ffe2279672d9439380e10.jpg)
+
+![](images/115ac0f92229d35e9dda14b1415427714598aa30031047c76bf2220ad7b6e554.jpg)  
+Figure 20: Additional visualizations, example 1, different poses for different characters.
+
+![](images/aea9e150c188b3109e3ff4b1bcbc6345515349026554e169bb185488e2823f77.jpg)  
+Figure 21: Additional visualizations, example 2, character rotation.
+
+![](images/adcfedc0bf1a51feb0d8d3ade4896c7fc78207ff28ae74b1b04ae2c3cff92568.jpg)  
+Figure 22: Additional visualizations, example 3, three or four characters animations.
+
+![](images/7810b97119921801f9c27ea00abd00f22ce58609a86462aa9eaaaef547a8b35a.jpg)  
+Figure 23: Additional visualizations, example 4, single character dance - Just Because You're So Beautiful.
+
+![](images/ebf11cb35c305fcd8b82a8bf9326297b785a301ce55e5a3170551f478911728d.jpg)  
+Figure 24: Additional visualizations, example 5, single character dance - the viral kemusan.
+
+![](images/1f0a3edf6f84319f9e8cf2210a741e42b824fe58dfdb2641ac6e7d4353db5477.jpg)  
+Figure 25: Additional visualizations, example 6, single character dance - the rabbit dance.
+
+![](images/5b594737ffd33c58110a4a660ca5ca310d738b57cffb133dbd5b9e8339265ccd.jpg)  
+Figure 26: Additional visualizations, example 7, multiple character dance - the viral kemusan.
+
+![](images/eaa179e2d03f0025663dcf1132cfeeb6e60d26ef44a7cee7f81bf70669ee2b59.jpg)  
+Figure 27: Additional visualizations, example 8, multiple character dance - the rabbit dance.

2025/Towards Multiple Character Image Animation Through Enhancing Implicit Decoupling/images.zip

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2025/Towards Neural Scaling Laws for Time Series Foundation Models/c03e4c74-4f7c-4ad8-ab2a-01887da879c7_origin.pdf

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2025/Towards Neural Scaling Laws for Time Series Foundation Models/full.md

@@ -0,0 +1,546 @@
+# TOWARDS NEURAL SCALING LAWS FOR TIME SERIES FOUNDATION MODELS
+
+Qingren Yao $^{1,2}$ , Chao-Han Huck Yang $^{3}$ , Renhe Jiang $^{4}$ , Yuxuan Liang $^{2*}$ , Ming Jin $^{1*}$ , Shirui Pan $^{1}$
+
+<sup>1</sup>Griffith University <sup>2</sup>The Hong Kong University of Science and Technology (Guangzhou)
+
+$^{3}$ NVIDIA Research  $^{4}$ The University of Tokyo
+
+# ABSTRACT
+
+Scaling laws offer valuable insights into the design of time series foundation models (TSFMs). However, previous research has largely focused on the scaling laws of TSFMs for in-distribution (ID) data, leaving their out-of-distribution (OOD) scaling behavior and the influence of model architectures less explored. In this work, we examine two common TSFM architectures—encoder-only and decoder-only Transformers—and investigate their scaling behavior on both ID and OOD data. These models are trained and evaluated across varying parameter counts, compute budgets, and dataset sizes. Our experiments reveal that the negative log-likelihood of TSFMs exhibits similar scaling behavior in both OOD and ID settings. We further compare the scaling properties across different architectures, incorporating two state-of-the-art TSFMs as case studies, showing that model architecture plays a significant role in scaling. The encoder-only Transformers demonstrate better scalability than the decoder-only Transformers in ID data, while the architectural enhancements in the two advanced TSFMs primarily improve ID performance but reduce OOD scalability. While scaling up TSFMs is expected to drive performance breakthroughs, the lack of a comprehensive understanding of TSFM scaling laws has hindered the development of a robust framework to guide model scaling. We fill this gap in this work by synthesizing our findings and providing practical guidelines for designing and scaling larger TSFMs with enhanced model capabilities.
+
+# 1 INTRODUCTION
+
+Time series analysis is an important piece of data mining, facilitating decision-making and scientific inference across various domains (Zhang et al., 2023). As an important analysis task, time series forecasting has long been studied and drives a wide range of practical applications, from energy, climate and quantitative finance to urban computing and system management (Jin et al., 2023; Nie et al., 2024; Wen et al., 2024). Various methods have been proposed for this task, ranging from classical statistic models (Hyndman & Athanasopoulos, 2013), bespoke dynamical models (Prado, 2020), to the more recent deep-learning based approaches (Wen et al., 2022). Despite their competitive performance, the methods are typically designed for specific tasks, poor to generalize to other domains (Fan et al., 2023; Rasul et al., 2023). Concurrently, we are witnessing a paradigm shift in time series forecasting from task-specific models to universal models, with the emergence of time series foundation models (TSFMs). Timer (Liu et al., 2024), Moirai (Woo et al., 2024), and more recently proposed Time-MoE (Shi et al., 2024b) show trends of scaling in both data volume and model size, aiming to achieve performance breakthroughs through more resource investment.
+
+The neural scaling law (Kaplan et al., 2020) quantitatively describes how model performance grows with the scaling of three basic training factors: model parameters, computational resources and training dataset size. Establishing such scaling laws is crucial for developing TSFMs, as it provides a framework for predicting expected performance gains, enabling the community to rationally allocate efforts toward key designs. The exploration on scaling laws for TSFMs is still in an initial stage; recent research has primarily focused on studying ID scaling behavior (Edwards et al.,
+
+2024; Shi et al., 2024a). In practical applications, TSFMs primarily face challenges from unseen scenarios (Wang et al., 2024), where OOD forecasting capability is most critical. This raises an unresolved question: do neural scaling laws also apply to predict out-of-distribution forecasting performance? Moreover, various architectures of TSFMs have been arising, but they typically focus on performance improvement at specific scales. No studies have investigated the scaling behaviors across different architectures, leaving a key question unanswered: how do model architectures affect scalability? Although we are seeing an increasing investment in training resources for TSFMs, the bottlenecks and potential driving factors for developing larger TSFMs remain unclear. This raises another practical question: how to design TSFMs from the perspective of scalability?
+
+In this paper, we aim to provide empirical answers to the above research questions. To investigate the scaling laws in OOD scenarios, we trained a family of encoder-only Transformer-based TSFMs, varying three basic training factors: model sizes, compute budgets, and training set sizes. We evaluated their performance on both ID and OOD test sets and established scaling laws for three training factors in each scenario. To examine the impact of model architecture on scaling behavior, we trained decoder-only Transformer based TSFMs and compared them with the encoder-only versions. Additionally, we included two state-of-the-art TSFMs, Moirai (Woo et al., 2024) and Chronos (Ansari et al., 2024), as case studies for detailed analysis. Our experiment results suggest that the negative log-likelihood loss of TSFMs exhibits similar scaling behavior in both OOD and ID scenarios; encoder-only Transformers have a similar scalability with decoder-only Transformers, with a slight advantage in ID data; the architectural modifications introduced by two advanced TSFMs mainly improve ID performance but compromise OOD scalability. Based on the findings and comparative analysis, we finally provided design principles for TSFMs from a scaling perspective.
+
+Our contributions are summarized as follows:
+
+- Scaling laws across data distributions. We extend the scaling laws for TSFMs from ID scenarios to OOD scenarios across three core training factors: model size, computational resources, and dataset size, establishing a foundation for predicting expected OOD performance gains of TSFMs.  
+- Scaling laws across model architectures. We investigate the scaling patterns of different TSFM architectures, showing the scalability varies depending on the model architecture and the design of specific modules.  
+- Scaling laws-guided design principles. We provide practical design principles for TSFMs from the perspective of data, model and compute scaling, via analyzing the commonalities and differences in scaling behaviors across data distributions and model architectures
+
+# 2 PRELIMINARY
+
+To investigate the scaling laws of TSFMs, we curated a large, diverse, and balanced dataset for pre-training. Leveraging this dataset, we trained both encoder-only and decoder-only transformers within two state-of-the-art TSFMs: Moirai and Chronos. For comparative analysis, we evaluated these models on (i) in-distribution and (ii) out-of-distribution test sets, focusing on key performance metrics to examine the scaling behavior across architectures.
+
+# 2.1 DATASETS
+
+A large scale, diverse, balanced and high quality pre-training dataset is the foundation to build FMs. To this end, we constructed our time series corpus for TSFM pre-training from the large-scale open time series archive, LOTSA (Woo et al., 2024). The corpus comprises approximately 17B time points from 39 datasets spanning seven distinct domains. To ensure that the model performs fairly across all domains, we maintained a balanced ratio of data from different domains. Furthermore, we performed quality filtering on the corpus by constraining the signal-to-noise ratio of a time series to be greater than  $20\mathrm{dB}$ , ensuring that the pre-training corpus exhibits strong predictability. A detailed breakdown of the data sources is provided in Appendix A, with a summary in Table 1.
+
+To assess the impact of pre-training data scale on model performance, we partitioned the corpus into subsets containing 10M, 100M, and 1B time points, ensuring that each subset maintained similar diversity. For each subset,  $95\%$  of the data was allocated for model training, with the remaining  $5\%$  reserved as a validation set to evaluate in-distribution forecasting performance. Additionally,
+
+Table 1: Dataset summary. M indicates million and B indicates billion.  
+
+<table><tr><td>Domain</td><td>Transport</td><td>Climate</td><td>Energy</td><td>CloudOps</td><td>Health</td><td>Sales</td><td>Web</td><td>Total</td></tr><tr><td>Datasets</td><td>8</td><td>2</td><td>14</td><td>3</td><td>9</td><td>1</td><td>2</td><td>39</td></tr><tr><td>Time points</td><td>4.82B</td><td>4.73B</td><td>4.76B</td><td>2.15B</td><td>232M</td><td>140M</td><td>40M</td><td>16.8B</td></tr><tr><td>Proportion</td><td>28.52%</td><td>28.06%</td><td>28.21%</td><td>12.76%</td><td>1.38%</td><td>0.83%</td><td>0.24%</td><td>100%</td></tr></table>
+
+![](images/88b914c3cc3e05e39379228fa8894a2e0cf736e6bfff58fb1295cb29499c8827.jpg)  
+Figure 1: Architectures of Baseline Time Series Foundation Models. As the most widely used two Transformer architectures, encoder-only Transformer and decoder-only Transformer are selected to as our baseline. A time series is divided into multiple patches, each treated as a token and fed into the Transformer model. The shaded patches represent the future horizon to be predicted.
+
+we used a subset from a widely recognized long-sequence prediction benchmark (Wu et al., 2023) to test the model's out-of-distribution forecasting capabilities. To further enhance the reliability, we also incorporated a subset of the Monash dataset (Godahewa et al., 2021) as additional OOD test data. The details of the dataset composition and properties are provided in Appendix A Table 3.
+
+# 2.2 MODELS
+
+TSFMs are predominantly built upon the Transformer architecture (Wen et al., 2022). For our baseline models, we selected two widely adopted architectures: the encoder-only Transformers (Woo et al., 2024) (Moirai) and the decoder-only Transformers (Ansari et al., 2024) (Chronos). The primary distinction between them lies in the attention mechanisms applied to the inputs, as illustrated in Figure 1. To better adapt them for time series forecasting, we introduce three key modifications in input layer, positional encoding and prediction head. More details are given in Appendix B.
+
+Patch Embedding. There are several approaches for generating inputs for transformer-based TSFMs, including point embedding, patch embedding, and lagged feature embedding. Due to the high computational cost of point embedding for long sequences and the limited robustness of lagged feature embedding, we adopt patch embedding in our models. This method, initially introduced by Vision Transformers (Dosovitskiy et al., 2020) and later adapted by PatchTST (Nie et al., 2023) for time series forecasting, divides the time series into non-overlapping segments, which are then projected into a feature space.
+
+Rotary Position Embedding. This technique (RoPE) has rapidly gained popularity as a positional encoding method in recent large language models (Su et al., 2024). Given the improved pre-training efficiency observed with RoPE (Woo et al., 2023), we adopt RoPE as a replacement for the original Transformer's positional encoding. RoPE encodes absolute positions using a rotation matrix while embedding relative position dependencies directly into the self-attention mechanism.
+
+Mixture of Distributions. Our models are designed to predict the probability distribution of future time series. However, real-world time series often exhibit complex distributions, including outliers, heavy tails, and extreme skew, which pose significant challenges for accurate modeling. To address
+
+these complexities, we incorporate a more flexible output likelihood by utilizing Student-T mixture models (Flunkert et al., 2017). Compared to the commonly used Gaussian mixture models, Student-T mixture models offer greater robustness in handling outliers and heavy-tailed distributions. An empirical comparison between the two mixture distributions is shown in Appendix B Figure 9.
+
+Our models are characterized by several key hyper-parameters: the number of layers ( $n_{\mathrm{layer}}$ ), the input/output dimensions of the residual stream ( $d_{\mathrm{m}}$ ), the dimensions of the intermediate feed-forward layers ( $d_{\mathrm{ff}}$ ), the number of attention heads per layer ( $n_{\mathrm{heads}}$ ), and the dimension of the attention output ( $d_{\mathrm{head}}$ ). The overall model size can be expressed as:
+
+$$
+\begin{array}{l} N \approx n _ {\text {l a y e r}} \left(4 d _ {\mathrm {m}} n _ {\text {h e a d s}} d _ {\text {h e a d}} + 2 d _ {\mathrm {m}} * d _ {\mathrm {f f}}\right) \\ = 2 d _ {\mathrm {m}} n _ {\text {l a y e r}} \left(2 n _ {\text {h e a d s}} d _ {\text {h e a d}} + d _ {\mathrm {f f}}\right) \tag {1} \\ = 1 2 n _ {\text {l a y e r}} d _ {\mathrm {m}} ^ {2} \quad \text {w i t h t h e s t a n d a r d} \quad n _ {\text {h e a d}} \cdot d _ {\text {h e a d}} = d _ {\mathrm {m}} = d _ {\mathrm {f f}} / 4, \\ \end{array}
+$$
+
+where the embedding layer, prediction head, biases and other sub-leading terms are excluded for a cleaner scaling laws. The embedding layer uses a patch size of 32 with  $32d_{\mathrm{m}}$  parameters. The mixture distribution prediction head comprises multiple independent linear layers that predict each Student-t mixture distribution parameter for a patch separately, with  $512d_{\mathrm{m}}$  parameters in total. In the study, we explore models with  $\sim 10^{3}$  to  $\sim 10^{8}$  trainable parameters.
+
+# 2.3 TRAINING AND EVALUATION DETAILS
+
+In this study, we focus exclusively on uni-variate time series forecasting to avoid the confounding effects introduced by multivariate time series, such as variable interactions, correlations, and the complexities of modeling multivariate relationships. Future research will address these factors, aiming to establish more comprehensive scaling laws for multivariate time series models.
+
+Training Details. Our training objective is to optimize the mixture distribution log-likelihood. We utilize the AdamW optimizer with a batch size of 128, and a maximum learning rate of  $10^{-3}$  with a linear warm-up of  $10^{4}$  training steps, followed by cosine decay for the remaining  $9 \times 10^{4}$  steps. To facilitate learning data representations across diverse domains with varying series lengths and sample sizes, we visited each sample with probability  $p_i = t_i / T$ , where  $t_i$  is the series' time points and  $T$  is the corpus' total time points. In addition, we follow the approach used in Moirai (Woo et al., 2024) and Timer (Liu et al., 2024) by capping the sampling probability at 0.05 to ensure a more balanced contribution from each dataset. We then randomly selected a segment from each chosen sample.
+
+Evaluation Details. We evaluate the model on a randomly selected  $10\%$  subset of the test data every  $10^{3}$  steps to reduce computational costs. For performance measurement, we observed that non-normalized metrics like MAE and MSE are highly sensitive to the amplitude of time series data, often causing the overall average to be disproportionately influenced by high-amplitude datasets. To mitigate this issue, we primarily use the normalized metric, mean absolute percentage error (MAPE), along with the negative log-likelihood (NLL), to assess forecasting performance. For a more comprehensive understanding of TSFM scaling laws, we also include additional results using symmetric mean absolute percentage error (SMAPE), mean absolute scaled error (MASE), and continuous ranked probability score (CRPS) in Appendix D.4. Detailed descriptions of these metrics are provided in the Appendix C.3.
+
+# 3 SCALING LAWS FOR TIME SERIES FOUNDATION MODELS
+
+In this section, we first present experimental results using the encoder-only Transformer to explore scaling laws across different data distributions. Following this, we conduct a comparative study on the scaling behavior of encoder-only and decoder-only TSFMs, Chronos and Moirai, to investigate how various scaling factors influence the characteristics of time series models.
+
+# 3.1 SCALING LAWS ACROSS DATA DISTRIBUTIONS
+
+Parameter Scaling. In Figure 2, we display the ID and OOD performance of a wide variety of encoder-only Transformers, ranging from small models with 1K parameters through large models
+
+![](images/13cf0b4b2c393f1e8f7e95bdf975269df3e24d176051a152b5ef43674699f8c6.jpg)  
+Figure 2: Parameter Scaling. The scaling effect of total trainable model parameters on the in-distribution (ID) and out-of-distribution (OOD) forecasting performance, which is evaluated using NLL and MAPE metrics. When evaluated with NLL, both ID and OOD results follow an approximate power law scaling with parameter count, exhibiting consistent trends across different data distributions. The blue and red horizontal dashed lines represent the baselines of the exponential smoothing (ETS) forecasting method.
+
+![](images/60c4b33f13e635b2b58c09ede057a4833439a325718369bd5a0d1fa35d760e9c.jpg)
+
+with 100M parameters. We trained models on the full pre-training corpus to convergence and report the minimum NLL and MAPE. We can see that both ID and OOD performance roughly follow power-law behavior over five orders of magnitude in model sizes. Formally, the power law can be expressed as:
+
+$$
+L (N) \approx \left(\frac {N _ {c}}{N}\right) ^ {\alpha_ {N}}, \tag {2}
+$$
+
+where  $L$  is the performance metric function (i.e., MAPE, or NLL),  $N$  is a given parameter count,  $N_{c}$  is the normalization coefficient, and  $\alpha_{N}$  is the exponent value that indicates the degree of performance improvement expected as we scale up  $N$ .
+
+Observing the NLL metric, the lines fitting the scaling laws for both ID and OOD data exhibits a roughly constant shift and close slopes. This implies that while models incur a consistent performance bias when transferred to OOD data, their scaling patterns correlate well with their performance on the ID data. When evaluated using MAPE, the power-law for OOD scenario shows a bigger exponent value than ID scenario. This indicates that increasing model size yields greater improvements in OOD performance than ID performance. In other words, for models with weak OOD generalization capabilities, increasing model size may enable them to perform equally well on both ID and OOD data.
+
+To evaluate whether the benefits of large-scale pre-training are warranted, we compare the pretrained models with the classical exponential smoothing (ETS) forecasting method. The results indicate that the pre-trained models consistently outperform ETS on ID data and progressively excel on OOD data as the model size increases. This suggests that pre-trained models must reach a certain scale, at least 3M parameters in this case, to demonstrate a level of superiority on OOD data that justifies their high pre-training cost.
+
+Compute Scaling. Following the similar method in (Kaplan et al., 2020), we estimate the compute budget using the formula  $C = 6NBS$ , where  $B$  is the batch size,  $S$  is the number of parameter updates, i.e. the input sequence length, and 6 is the factor to account for the forward and backward passes. The ID and OOD test loss for compute budget varying over six orders of magnitude are shown in Figure 3. We see that the optimal results for each compute budget are achieved by different model sizes  $N$ , but the lowest loss decreases according to a approximate power law with respect to the amount of training compute. The lowest losses appear as the heavy lines, which can be fit with
+
+$$
+L (C) \approx \left(\frac {C _ {c}}{C}\right) ^ {\alpha C}. \tag {3}
+$$
+
+![](images/20dd71f7c6fdeadd707ac0dac38ee186e1ab04a57a6c42c692a98f614a59f844.jpg)  
+Figure 3: Compute Scaling. The computation scaling results indicate that model performance scales approximately according to a power law with increasing compute, consistent across both ID and OOD scenarios. The ID and OOD results illustrate that there is an lower bound for loss and MAPE on both test data under a given computational budget.
+
+![](images/19a98cd47fa872dc6807f70bbdfc0d1b5c4e26028671eb0460c676b932e1cd15.jpg)
+
+![](images/d82eda6a08d61e1dfd2d37146f30271a01a1b48503aa3b6c18f4a53c18a51306.jpg)  
+Figure 4: Data Scaling. The blue and red plots illustrate how data volume affects the ID and OOD forecasting performance of encoder-only Transformers, evaluated using NLL and MAPE metrics. The results indicate that in both scenarios, model performance scales approximately as a power law with data volume.
+
+![](images/126b22f04eae942118d01d06856957b24246664b95ac62f61121b442b6ea8366.jpg)
+
+We observed significant noise in NLL and MAPE during training, which may be caused by the learning rate scheduler and the random sampling evaluation strategy. Similar to parameter scaling, the ID and OOD NLL show similar scaling patterns; however, when evaluated using MAPE, the OOD power law shows larger exponent values than the ID power law.
+
+Data Scaling. We display empirical trends for the performance as function of dataset size  $D$  in Figure 4. For the trend, we trained multiple 1B encoder-only Transformers on a series of subsets of the pre-training dataset and report the averaged evaluation results during training. We see that the NLL and MAPE can be fit well with simple power-law
+
+$$
+L (D) \approx \left(\frac {D _ {c}}{D}\right) ^ {\alpha_ {D}}. \tag {4}
+$$
+
+Different from the parameter scaling and computational scaling, we found that when evaluated using NLL, ID and OOD performance do not exhibit the same scaling behavior. Instead, ID performance is more sensitive to the scaling of dataset size compared to OOD performance. However, similar to the observations on the other two factors, the impact of data scaling on MAPE in OOD data is greater than its impact on MAPE in ID data. This suggests that the scaling of various factors yields greater improvements in OOD performance than ID performance.
+
+Cross-distribution Scaling Effects. We summarize the key findings on how model ID and OOD performance scales with the model parameters, data volume, and compute. (1) In both ID and OOD uni-variate time series forecasting, model performance follows a simple power law as a function of model parameters, data volume, and compute. (2) For NLL, the model exhibits similar scaling
+
+![](images/84674262e82244bd9957f1baf246e85f15af6812e8ed0882d4aaa90015caa3df.jpg)  
+Figure 5: Scaling Laws of Encoder-only vs. Decoder-only Transformer. This figure presents a comparison of scaling behaviors on NLL between encoder-only and decoder-only Transformer across three different axes: number of parameters, compute, and dataset size. Overall, both models exhibit similar scalability patterns with respect to model parameters, computation and dataset sizes across ID and OOD data, but differ in ID performance.
+
+![](images/0e8c5ed0ba1c49f236cef1e813633908d7eeb24081a5b77479253cd23b793fb8.jpg)
+
+![](images/049dd342562c23c9c06d5c120fb91bb86cb8387dd5a31603007035346cd8f3ac.jpg)
+
+![](images/a12757f4171979bc50687ef28e9ca1f73217555a1e50b9fae850b1846d37da5a.jpg)  
+Figure 6: Scaling Laws of Encoder-only Transformer vs. Moirai. This figure compares parameter scaling between the encoder-only Transformer and Moirai. While Moirai demonstrates notable improvements on ID test data compared to the baseline, certain design choices may hinder its scalability on OOD data.
+
+![](images/976aeca4fde38110e1def88e80331f70b4c0332625fe99de1347b32d3a158a24.jpg)
+
+patterns in both ID and OOD scenarios, in terms of model size or compute resources. (3) When using MAPE as the metric, scaling of all three factors results in greater improvements in OOD performance compared to ID performance.
+
+# 3.2 SCALING LAWS ACROSS MODEL ARCHITECTURES
+
+The above results suggest that the power-law captures the scaling behavior of encoder-only Transformers in both ID and OOD scenarios. Similarly, we analyze the scaling properties of decoder-only Transformers, along with two other state-of-the-art TSFMs, Chronos and Moirai, to assess the impact of model architectures on scaling behavior. These models are trained on the dataset we built with the same training setup.
+
+Encoder-only vs. Decoder-only Transformer. Figure 5 presents a comparison of the scaling patterns between encoder-only and decoder-only Transformers. In parameter scaling, the encoder-only Transformer shows a slight advantage, with a marginally higher power-law exponent on ID data and lower NLL values. For compute and data scaling, the two architectures demonstrate nearly identical scalability across ID and OOD settings, with some performance differences observed on ID data. Overall, the two architectures exhibit similar scalability, aside from a performance difference observed in ID data.
+
+Encoder-only Transformer vs. Moirai. Moirai is a TSFM based on an encoder-only Transformer architecture. It introduces "any-variate" attention to capture relationships between multiple variables. Additionally, Moirai incorporates a multi-scale patch embedding to handle different frequency patterns, and a more diverse mixture distribution to model real-world probability distributions. Figure 6 show a comparison between the scaling behavior of encoder-only Transformer and
+
+![](images/7501de54bfac9c53ecdc7a043e8930c8c09be2e26a24685bfff055ce67057e87.jpg)  
+Figure 7: Scaling Laws of Decoder-only Transformer vs. Chronos. This figure presents a comparison of parameter scaling between decoder-only Transformer and Chronos. The design introduced by Chronos enhances ID time series forecasting, but fails to improve the generalization of OOD.
+
+![](images/4daecd5e02c510c45b17886730569e2ea50912358ecd400a90e21229b0ff6df9.jpg)
+
+Moirai. On ID data, Moirai demonstrates better performance. However, for OOD data, as the number of model parameters increases, Moirai is gradually surpassed by the encoder-only Transformer. Comparing the power-law lines of the two models, we see that Moirai shows a smaller slope, indicating relatively weaker scalability. Collectively, Moirai shows significant improvements on ID time series forecast than our baseline, but some designs may limit its scalability on OOD data.
+
+Decoder-only Transformer vs. Chronos. Chronos-T5 is an encoder-decoder Transformer-based TSFM that, like decoder-only Transformers, follows an auto-regressive prediction approach but uses a separate encoder to extract contextual information. It adopts point-wise prediction and transform numerical regression into discrete probability prediction. Figure 7 compares the scaling patterns of decoder-only Transformers and Chronos-T5. When evaluated with NLL, Chronos-T5 exhibits power laws with very small exponents, indicating limited scalability. We attribute this to the discrete probability prediction, as NLL on a discrete distribution is not distance-sensitive, meaning the loss remains high unless the predicted value exactly matches the label. Compared to NLL, symmetric mean absolute percentage error (SMAPE, definition is shown in Appendix C.3) is a more appropriate metric for evaluating the two models. Chronos-T5 shows a slight advantage in performance and scalability on ID data. However, on OOD data, the decoder-only Transformer performs better. Despite some performance differences, both models exhibit similar scaling patterns on OOD data. Overall, the design improvements in Chronos-T5 enhance ID time series forecasting, but they do not effectively improve OOD generalization.
+
+Scaling Effects Summary. Our analysis of the scaling properties across architectures reveals several key findings: (1) Model architecture and design play a crucial role in determining scalability. (2) Encoder-only models and decoder-only models show a similar scalability on OOD data; while encoder-only models have a slightly higher parameter scalability and a certain performance advantage on ID data. (3) While Chronos and Moirai improve ID forecasting, these gains do not extend effectively to OOD scenarios.
+
+# 4 DESIGN PRINCIPLES FOR TIME SERIES FOUNDATION MODELS
+
+Building on our findings regarding the scaling laws in TSFMs, we elaborate design principles to guide the development of scalable models. These principles are framed around three key dimensions: training data, model parameters and architecture, as well as computational resources.
+
+Training Data. Our experiments show that increasing the size of the training dataset leads to a greater performance improvement on OOD data compared to ID data. Enlarging the pre-training dataset is crucial for achieving better generalization. Based on Equation 4, we can expect that doubling the size of the pre-training dataset will result in an OOD MAPE reduction to approximately  $90\%$  of its previous value, and an ID MAPE reduction to approximately  $97\%$ . But maintaining diversity within the dataset is equally important while increasing the data volume.
+
+![](images/912bccc988a1c12213b8231571a2caa6693d4e6ca519970092a54f788c7ba239.jpg)  
+Figure 8: Case studies on the "Emergent Abilities" of scaling time series models. We present three examples of zero-shot (unseen) out-of-distribution (OOD) time series prediction. In the us births, nn5 daily, and australian electricity datasets, we observed that model behavior deviates from expected power law patterns, instead exhibiting characteristics more akin to emergent phenomena.
+
+Model Parameters and Architecture. Our study highlights that model size is the most critical factor for improving TSFM performance. Compared to collecting more data, increasing model size yields the greater benefit for both ID and OOD forecast. However, as model size scales up, data bottlenecks inevitably emerge. From the ratio of exponents in Equations 2 and 4, we infer that scaling up model size should be accompanied by a sublinear increase in dataset size. Specifically, this relationship can be approximated as  $D \propto N^{\frac{\alpha_N}{\alpha_D}} \sim N^{0.8}$ , meaning that doubling the model size requires roughly 1.7 times more data. In terms of model architecture, we found some modifications facilitate ID performance but can not generalize well to OOD scenarios. Model design can impact scalability, and a good design should promote performance, generalization, and scalability.
+
+**Compute.** The power law we established indicates that there is an unbreakable lower bound for NLL loss and MAPE under a given computational budget. This means that, with other factors remaining constant, as the model size increases, more compute resources must be invested to achieve better performance. Our findings suggest that compute resources should prioritize scaling up model size rather than extending training time for fixed models as larger models are more sample-efficient than smaller models. Moreover, different training objectives or model architectures can significantly affect this performance bound, exploring a compute-efficient training paradigm will be promising.
+
+# 5 RELATED WORKS
+
+Neural Scaling Laws. Neural scaling laws seek to provide a predictive framework for optimizing the allocation of computational resources to maximize model performance. In language domains, Kaplan et al. (2020) demonstrated that performance follows a power-law relationship, improving as more computational resources, parameters, and data are utilized. Subsequent research has expanded this to predict other factors, such as downstream task performance (Isik et al., 2024) and inference time (Sardana et al., 2024). In vision domains, scaling laws have been explored in areas like discriminative modeling (Hestness et al., 2017) and visual auto-regressive modeling (Henighan et al., 2020). Recently, Edwards et al. (2024) introduced scaling laws for large time series models, showing that performance scales according to a power law with model size, compute, and dataset size. Shi et al. (2024a) examined the effect of time-series forecasting horizon on model scaling behavior, offering a theoretical framework to explain its influence. However, both studies have focused on in-distribution scenarios, leaving the investigation of scaling laws in out-of-distribution contexts largely unexplored.
+
+Time Series Foundation Models. Foundation models (Das et al., 2024; Goswami et al., 2024) represent a new paradigm aimed at generalizing across diverse domains and tasks by leveraging knowledge from large-scale data in the pre-training phase. They have significantly advanced time series forecasting, particularly in zero-shot scenarios, where predictions are made on data from previously unseen domains. For instance, Woo et al. (2024) introduced Moirai, an encoder-only transformer architecture that employs an "any-variate" attention mechanism to capture dependencies in multivariate time series. Ansari et al. (2024) proposed a method that tokenizes time series values through scaling and quantization into a fixed vocabulary, training a series of transformer-based
+
+models known as Chronos. Liu et al. (2024) developed Timer, a simple decoder-only transformer architecture designed for univariate time series forecasting, while Rasul et al. (2023) introduced Lag-Llama, a decoder-only transformer that integrates lags as covariates to improve forecasting accuracy. These models incorporate various modifications to the standard Transformer architecture for time series data. However, the impact of these changes on model scaling properties has not been systematically studied. As model size increases, it remains an open question whether these modifications will continue to enhance performance.
+
+# 6 DISCUSSION
+
+# 6.1 ADDITIONAL STUDIES
+
+We also conduct the following analysis to better understand the scaling behaviors of TSFMs. Due to the space limit, see their details in Appendix D.2 to D.4.
+
+Emergent Behaviors. Figure 8 shows some examples of zero-shot OOD time series prediction, where the model's performance remains low until the model size reaches a critical threshold, after which performance improves significantly. This scaling behavior deviates from the previously observed power law and is more akin to emergent phenomena (Wei et al., 2022).
+
+Sample Efficiency. Appendix D.2, Figure 14 shows the evaluation results during training. Large models are more sample-efficient than small models, reaching better performance with the same optimization steps and using fewer time points.
+
+Scaling Pattern Depends on Data Distributions. Appendix D.3, Figure 15 - 19 show the scaling behavior of TSFMs on the Monash subset, suggesting that OOD scaling behaviors varies depending on the relationship between the unseen data distribution and the pre-training data distribution.
+
+Scaling Pattern Depends on Performance Metrics. Appendix D.4, Figure 15 - 19 provides a comprehensive analysis of scaling behavior across five performance metrics: NLL, MAPE, SMAPE, MASE, and CRPS. All metrics exhibit a consistent decreasing trend, approximately following a power law.
+
+# 6.2 CONCLUSION
+
+We have observed consistent scaling of encoder-only Transformer NLL with parameter count, training computation and dataset size on both ID and OOD test data, as encapsulated in a power law. The experimental results show that as the number of parameters, computational resources, and training data increase, both models' ID and OOD performance will continue to improve. Furthermore, our established laws suggest that for ID and OOD forecasting performance, larger model may be more important than more data.
+
+We have also study the scaling properties of two common types of TSFMs, encoder-only Transformer and decoder-only Transformer. The two architectures show similar scalability in OOD data across model size, computational resources, and training data. While encoder-only models have a slightly better parameter scalability and performance in ID data based our evaluation setting. Moreover, we further include two advanced TSFMs, Chronos and Moirai, as specific cases to study the impact of module design on scaling behavior. We find that both Chronos and Moirai improve ID forecasting performance, but these gains can not translate effectively to OOD scenarios. Overall, model architecture and module design is important influence factor for scalability.
+
+Our experiments independently investigated the scaling behavior of TSFMs in terms of parameter count, computational resources, and training data size, assuming other factors were unlimited. In future work, it would be valuable to develop a unified model to explore the relationships between these factors. This could help guide the community in optimizing resource allocation when one resource is limited. Additionally, multiple studies have shown that model performance varies across different context window lengths and forecast horizons. In future research, we plan to incorporate the impact of context length and forecast horizon on scalability.
+
+# Acknowledgments and Disclosure of Funding
+
+This research was partly supported by Australian Research Council (ARC) under grants FT210100097 and DP240101547 and the CSIRO - National Science Foundation (US) AI Research Collaboration Program. This work has been also supported by NVIDIA Academic Grant from the NVIDIA Developer Program 2025.
+
+# Ethics Statement
+
+Our research is dedicated exclusively to addressing scientific challenges and does not involve human participants, animals, or materials that pose environmental concerns. We anticipate no ethical risks or conflicts of interest.
+
+# Reproducibility Statement
+
+We provide the implementation details in Appendices B and C, including model configurations and training & evaluation details. The source code and related source of this work are available at https://github.com/Qingrenn/TSFM-ScalingLaws for reproducibility.
+
+# REFERENCES
+
+Alexander Alexandrov, Konstantinos Benidis, Michael Bohlke-Schneider, Valentin Flunkert, Jan Gasthaus, Tim Januschowski, Danielle C. Maddix, Syama Sundar Rangapuram, David Salinas, Jasper Schulz, Lorenzo Stella, Ali Caner Türkmen, and Bernie Wang. Gluonts: Probabilistic and neural time series modeling in python. J. Mach. Learn. Res., 21:116:1-116:6, 2020. URL https://api-semanticscholar.org/CorpusID:226717462.  
+Abdul Fatir Ansari, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, Jasper Zschiegner, Danielle C. Maddix, Michael W. Mahoney, Kari Torkkola, Andrew Gordon Wilson, Michael Bohlke-Schneider, and Yuyang Wang. Chronos: Learning the language of time series. ArXiv, abs/2403.07815, 2024. URL https://api-semanticscholar.org/CorpusID:268363551.  
+Blake Bordelon, Alexander Atanasov, and Cengiz Pehlevan. A dynamical model of neural scaling laws. ArXiv, abs/2402.01092, 2024. URL https://api(semanticscholar.org/ CorpusID:267406160.  
+Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou. A decoder-only foundation model for time-series forecasting. In *Forty-first International Conference on Machine Learning*, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024. URL https://openreview.net/forum?id=ijn2iTJas6h.  
+Hoang Anh Dau, A. Bagnall, Kaveh Kamgar, Chin-Chia Michael Yeh, Yan Zhu, Shaghayegh Gharghabi, Chotirat Ann Ratanamahatana, and Eamonn J. Keogh. The ucr time series archive. IEEE/CAA Journal of Automatica Sinica, 6:1293-1305, 2018. URL https://apisemantic scholar.org/CorpusID:53019704.  
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. ArXiv, abs/2010.11929, 2020. URL https://api_semanticscholar.org/ CorpusID:225039882.  
+Thomas D. P. Edwards, James Alvey, Justin Alsing, Nam H. Nguyen, and Benjamin D. Wandelt. Scaling-laws for large time-series models. ArXiv, abs/2405.13867, 2024. URL https://api.sementicscholar.org/CorpusID:269982332.  
+Patrick Emami, Abhijeet Sahu, and Peter Graf. Buildingsbench: A large-scale dataset of 900k buildings and benchmark for short-term load forecasting. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023.
+
+Wei Fan, Pengyang Wang, Dongkun Wang, Dongjie Wang, Yuanchun Zhou, and Yanjie Fu. Dish-ts: a general paradigm for alleviating distribution shift in time series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 37, pp. 7522-7529, 2023.  
+Valentin Flunkert, David Salinas, and Jan Gasthaus. Deepar: Probabilistic forecasting with autoregressive recurrent networks. ArXiv, abs/1704.04110, 2017. URL https://api.sementicscholar.org/CorpusID:12199225.  
+B. Ghorbani, Orhan First, Markus Freitag, Ankur Bapna, Maxim Krikun, Xavier Garcia, Ciprian Chelba, and Colin Cherry. Scaling laws for neural machine translation. ArXiv, abs/2109.07740, 2021. URL https://api_semanticscholar.org/CorpusID:237532682.  
+Rakshitha Godahewa, C. Bergmeir, Geoffrey I. Webb, Rob J Hyndman, and Pablo Montero-Manso. Monash time series forecasting archive. ArXiv, abs/2105.06643, 2021. URL https://api.sementicscholar.org/CorpusID:234681550.  
+Mononito Goswami, Konrad Szafer, Arjun Choudhry, Yifu Cai, Shuo Li, and Artur Dubrawski. Moment: A family of open time-series foundation models. *ArXiv*, abs/2402.03885, 2024. URL https://api-semanticscholar.org/CorpusID:267500205.  
+Albert Gu, Karan Goel, and Christopher R'e. Efficiently modeling long sequences with structured state spaces. ArXiv, abs/2111.00396, 2021. URL https://api_semanticscholar.org/ CorpusID:240354066.  
+Tatsunori Hashimoto. Model performance scaling with multiple data sources. In International Conference on Machine Learning, pp. 4107-4116. PMLR, 2021.  
+Tom Henighan, Jared Kaplan, Mor Katz, Mark Chen, Christopher Hesse, Jacob Jackson, Heewoo Jun, Tom B. Brown, Prafulla Dhariwal, Scott Gray, Chris Hallacy, Benjamin Mann, Alec Radford, Aditya Ramesh, Nick Ryder, Daniel M. Ziegler, John Schulman, Dario Amodei, and Sam McCandlish. Scaling laws for autoregressive generative modeling. ArXiv, abs/2010.14701, 2020. URL https://api.sementicscholar.org/CorpusID:225094178.  
+Joel Hestness, Sharan Narang, Newsha Ardalani, Gregory Frederick Diamos, Heewoo Jun, Hassan Kianinejad, Md. Mostofa Ali Patwary, Yang Yang, and Yanqi Zhou. Deep learning scaling is predictable, empirically. ArXiv, abs/1712.00409, 2017. URL https://api.sementicscholar.org/CorpusID:2222076.  
+Rob J Hyndman and George Athanasopoulos. Forecasting: principles and practice. 2013. URL https://api.sementicscholar.org/CorpusID:215822294.  
+Berivan Isik, Natalia Ponomareva, Hussein Hazimeh, Dimitris Paparas, Sergei Vassilvitskii, and Sanmi Koyejo. Scaling laws for downstream task performance of large language models. ArXiv, abs/2402.04177, 2024. URL https://api_semanticscholar.org/CorpusID:267499809.  
+Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems, 31, 2018.  
+Jiawei Jiang, Chengkai Han, Wenjun Jiang, Wayne Xin Zhao, and Jingyuan Wang. Libcity: A unified library towards efficient and comprehensive urban spatial-temporal prediction. 2023. URL https://api-semanticscholar.org/CorpusID:258352728.  
+Ming Jin, Qingsong Wen, Yuxuan Liang, Chaoli Zhang, Siqiao Xue, Xue Wang, James Zhang, Yi Wang, Haifeng Chen, Xiaoli Li, et al. Large models for time series and spatio-temporal data: A survey and outlook. arXiv preprint arXiv:2310.10196, 2023.  
+Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeff Wu, and Dario Amodei. Scaling laws for neural language models. ArXiv, abs/2001.08361, 2020. URL https://api_semanticscholar.org/ CorpusID:210861095.
+
+Opher Lieber, Barak Lenz, Hofit Bata, Gal Cohen, Jhonathan Osin, Itay Dalmedigos, Erez Safahi, Shaked Haim Meirom, Yonatan Belinkov, Shai Shalev-Shwartz, Omri Abend, Raz Alon, Tomer Asida, Amir Bergman, Roman Glozman, Michael Gokhman, Avshalom Manevich, Nir Ratner, Noam Rozen, Erez Shwartz, Mor Zusman, and Yoav Shoham. Jamba: A hybrid transformer-mamba language model. ArXiv, abs/2403.19887, 2024. URL https://api.sementicscholar.org/CorpusID:268793596.  
+Xu Liu, Yutong Xia, Yuxuan Liang, Junfeng Hu, Yiwei Wang, Lei Bai, Chao Huang, Zhenguang Liu, Bryan Hooi, and Roger Zimmermann. Largest: A benchmark dataset for large-scale traffic forecasting. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023. URL http://papers.nips.cc/paper_files/paper/2023/hash/ee57cd73a76bd927ffca3dda1dc3b9d4-Abstract-Datasets_and_Benchmarks.html.  
+Yong Liu, Haoran Zhang, Chenyu Li, Xiangdong Huang, Jianmin Wang, and Mingsheng Long. Timer: Generative pre-trained transformers are large time series models. In International Conference on Machine Learning, 2024. URL https://api-semanticscholar.org/CorpusID:267412273.  
+James E Matheson and Robert L Winkler. Scoring rules for continuous probability distributions. Management science, 22(10):1087-1096, 1976.  
+Tung Nguyen, Jason Jewik, Hritik Bansal, Prakhar Sharma, and Aditya Grover. Climatelearn: Benchmarking machine learning for weather and climate modeling. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023. URL http://papers.nips.cc/paper_files/paper/2023/ hash/ed73c36e771881b232ef35fa3a1dec14-Abstract-Datasets_and_Benchmarks.html.  
+Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL https://openreview.net/forum?id=Jbdc0vT0col.  
+Yuqi Nie, Yaxuan Kong, Xiaowen Dong, John M Mulvey, H Vincent Poor, Qingsong Wen, and Stefan Zohren. A survey of large language models for financial applications: Progress, prospects and challenges. arXiv preprint arXiv:2406.11903, 2024.  
+Youngsuk Park, Danielle Maddix, François-Xavier Aubet, Kelvin Kan, Jan Gasthaus, and Yuyang Wang. Learning quantile functions without quantile crossing for distribution-free time series forecasting. In International Conference on Artificial Intelligence and Statistics, pp. 8127-8150. PMLR, 2022.  
+Raquel Prado. Bayesian forecasting and dynamic models. 2020. URL https://api.sementicscholar.org/CorpusID:121876157.  
+Xiangfei Qiu, Jilin Hu, Lekui Zhou, Xingjian Wu, Junyang Du, Buang Zhang, Chenjuan Guo, Aoying Zhou, Christian S. Jensen, Zhenli Sheng, and Bin Yang. Tfb: Towards comprehensive and fair benchmarking of time series forecasting methods. Proc. VLDB Endow., 17:2363-2377, 2024. URL https://api-semanticscholar.org/CorpusID:268793935.  
+Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of machine learning research, 21(140):1-67, 2020.  
+Kashif Rasul, Arjun Ashok, Andrew Robert Williams, Arian Khorasani, George Adamopoulos, Rishika Bhagwatkar, Marin Bilos, Hena Ghonia, Nadhir Vincent Hassen, Anderson Schneider, Sahil Garg, Alexandre Drouin, Nicolas Chapados, Yuriy Nevmyvaka, and Irina Rish. Lag-llama:
+
+Towards foundation models for time series forecasting. ArXiv, abs/2310.08278, 2023. URL https://api.sementicscholar.org/CorpusID:269766909.  
+Nikhil Sardana, Jacob Portes, Sasha Doubov, and Jonathan Frankle. Beyond chinchilla-optimal: Accounting for inference in language model scaling laws. In *Forty-first International Conference on Machine Learning*, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024. URL https://openreview.net/forum?id=0bmXrtTDUu.  
+Jingzhe Shi, Qinwei Ma, Huan Ma, and Lei Li. Scaling law for time series forecasting. ArXiv, abs/2405.15124, 2024a. URL https://api_semanticscholar.org/CorpusID: 270045141.  
+Xiaoming Shi, Shiyu Wang, Yuqi Nie, Dianqi Li, Zhou Ye, Qingsong Wen, and Ming Jin. Time-moe: Billion-scale time series foundation models with mixture of experts. arXiv preprint arXiv:2409.16040, 2024b.  
+Jianlin Su, Murtadha H. M. Ahmed, Yu Lu, Shengfeng Pan, Wen Bo, and Yunfeng Liu. Roformer: Enhanced transformer with rotary position embedding. Neurocomputing, 568:127063, 2024. doi: 10.1016/J.NEUCOM.2023.127063. URL https://doi.org/10.1016/j.neucom.2023.127063.  
+Chang Wei Tan, Christoph Bergmeir, François Petitjean, and Geoffrey I Webb. Time series extrinsic regression: Predicting numeric values from time series data. Data Mining and Knowledge Discovery, 35(3):1032-1060, 2021.  
+Yihe Wang, Yu Han, Haishuai Wang, and Xiang Zhang. Contrast everything: A hierarchical contrastive framework for medical time-series. In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023, 2023a. URL http://papers.nips.cc/paper_files/paper/2023/bitstream/ae7d9c77b5ff9e3b7833a68523b880f2-Abstract-Conference.html.  
+Yuxuan Wang, Haixu Wu, Jiaxiang Dong, Yong Liu, Mingsheng Long, and Jianmin Wang. Deep time series models: A comprehensive survey and benchmark. ArXiv, abs/2407.13278, 2024. URL https://api.sementicscholar.org/CorpusID:271270683.  
+Zhixian Wang, Qingsong Wen, Chaoli Zhang, Liang Sun, Leandro Von Krannichfeldt, and Yi Wang. Benchmarks and custom package for electrical load forecasting. ArXiv, abs/2307.07191, 2023b. URL https://api-semanticscholar.org/CorpusID:259924876.  
+Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed Huai hsin Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, and William Fedus. Emergent abilities of large language models. ArXiv, abs/2206.07682, 2022. URL https://api_semanticscholar.org/CorpusID:249674500.  
+Haomin Wen, Youfang Lin, Lixia Wu, Xiaowei Mao, Tianyue Cai, Yunfeng Hou, Shengnan Guo, Yuxuan Liang, Guangyin Jin, Yiji Zhao, et al. A survey on service route and time prediction in instant delivery: Taxonomy, progress, and prospects. IEEE Transactions on Knowledge and Data Engineering, 2024.  
+Qingsong Wen, Tian Zhou, Chao Zhang, Weiqiu Chen, Ziqing Ma, Junchi Yan, and Liang Sun. Transformers in time series: A survey. In International Joint Conference on Artificial Intelligence, 2022. URL https://api_semanticscholar.org/CorpusID:246863823.  
+Gerald Woo, Chenghao Liu, Akshit Kumar, and Doyen Sahoo. Pushing the limits of pre-training for time series forecasting in the cloudops domain. ArXiv, abs/2310.05063, 2023. URL https://api-semanticscholar.org/CorpusID:263831258.  
+Gerald Woo, Chenghao Liu, Akshit Kumar, Caiming Xiong, Silvio Savarese, and Doyen Sahoo. Unified training of universal time series forecasting transformers. In *Forty-first International Conference on Machine Learning*, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024. URL https://openreview.net/forum?id=Yd8eHMY1wz.
+
+Haixu Wu, Teng Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. Timesnet: Temporal 2d-variation modeling for general time series analysis. ArXiv, abs/2210.02186, 2022. URL https://api.sementicscholar.org/CorpusID:252715491.  
+Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. Timesnet: Temporal 2d-variation modeling for general time series analysis. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL https://openreview.net/forum?id=ju_Uqw384Oq.  
+Chao-Han Huck Yang, Yun-Yun Tsai, and Pin-Yu Chen. Voice2series: Reprogramming acoustic models for time series classification. In International conference on machine learning, pp. 11808-11819. PMLR, 2021.  
+Jiasheng Ye, Peiju Liu, Tianxiang Sun, Yunhua Zhou, Jun Zhan, and Xipeng Qiu. Data mixing laws: Optimizing data mixtures by predicting language modeling performance. arXiv preprint arXiv:2403.16952, 2024.  
+Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, and Lucas Beyer. Scaling vision transformers. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12104-12113, 2022.  
+Kexin Zhang, Qingsong Wen, Chaoli Zhang, Rongyao Cai, Ming Jin, Yong Liu, James Zhang, Y. Liang, Guansong Pang, Dongjin Song, and Shirui Pan. Self-supervised learning for time series analysis: Taxonomy, progress, and prospects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46:6775-6794, 2023. URL https://api_semanticscholar.org/ CorpusID:259203853.
+
+# A PRE-TRAINING DATASETS
+
+Previous work has proposed several public large-scale time series datasets, e.g. LOTSA (Woo et al., 2024) and UTSD (Liu et al., 2024). Instead of using these datasets directly, we built a pre-training corpus through data filtering. There are three main reasons for this: First, these datasets contain numerous periodic time series with substantial pattern redundancy. Second, these datasets show a heavily long-tailed distribution, with multiple domains accounting for less than  $5\%$  of the total. Third, data quality is variable, with some time series missing many observations or having low signal-to-noise ratios. To address these issues, we developed a data filtering pipeline:
+
+Deduplication. In these public datasets, we observed significant redundancy across many subsets. This redundancy typically manifests as numerous repeated patterns within individual samples or high similarity between multiple samples. Such redundancy can negatively impact training efficiency. To address this issue, we applied a down-sampling strategy for datasets with redundant samples based on the data sampling period.
+
+Quality Filtering. We focused on selecting time series data that have no missing values and a signal-to-noise ratio (SNR) greater than 20 dB to ensure better predictability. To calculate the SNR, we first used Fourier transform to identify the low-frequency components of the time series. We then applied a low-pass Butterworth filter to extract these low-frequency components as the signal, with the remaining residuals considered as noise.
+
+Domain Balancing. In the LOTSA dataset, the climate domain accounts for nearly  $90\%$  of the data, while the remaining seven domains collectively represent only  $10\%$ . To create a more balanced pre-training dataset, we selected the transport domain, which contains 4.8 billion time points, as the reference. We then scaled the data from the other domains to a similar magnitude.
+
+Following the aforementioned processing steps, we compiled a high-quality time series dataset containing a total of 16.8 billion time points across 7 domains. The data volume is sufficient to support training models with parameters ranging from  $10^{3}$  to  $10^{8}$  parameters. Furthermore, most domains include at least 100 million time points, ensuring that even when the dataset is divided into subsets of various sizes, domain diversity is preserved. Below, we introduce the datasets for each domain and outline the key properties of the datasets after processing, including domain type, sampling frequency, number of time series, total number of observations, and data source.
+
+Climate. The climate data are sourced primarily from the ERA5 and CMIP6 datasets, which provide time series for various climate-related variables, including temperature, humidity, and pressure levels. During data curation, we observed a high degree of similarity across yearly data. To reduce redundancy, only two years of data from ERA5 and CMIP6 were included.
+
+Energy. The energy data primarily come from the BuildingsBench dataset, which provides time series data on residential and commercial building energy consumption. After applying quality filtering based on SNR, we selected samples from the Buildings-900K, BDG-2, and Sceaux datasets. Additionally, high-quality data from the ProEnFo library, including the GEF, ELF, PDB, Spanish, and Covid19_Energy datasets, were incorporated.
+
+Transport. The primary source for transport data is the LargeST traffic dataset, covering traffic flow in California from 2017 to 2021. We also integrated datasets from LibCity, PEMS, Loop Seattle, and Q-Traffic into our corpus.
+
+Cloud Operations. The cloud operations data is from the large-scale CloudOps time series datasets, which measures various variables such as CPU and memory utilization.
+
+Healthcare. We include a diverse sets of healthcare data from the ucr time series archive.
+
+Web. The web data comprises the Kaggle Web Traffic Weekly dataset and the Wiki-Rolling dataset.
+
+Sales. Here we use the Favorite Sales dataset.
+
+Table 2: Pre-training datasets and key properties.  
+
+<table><tr><td>Dataset</td><td>Domain</td><td>Frequency</td><td># Time Series</td><td># Obs.</td><td>Source</td></tr><tr><td>CMIP6</td><td>Climate</td><td>6H</td><td>196,608</td><td>2,870,476,800</td><td>Nguyen et al. (2023)</td></tr><tr><td>ERA5</td><td>Climate</td><td>H</td><td>212,992</td><td>1,860,698,112</td><td>Nguyen et al. (2023)</td></tr><tr><td>Buildings900K</td><td>Energy</td><td>H</td><td>538,577</td><td>4,718,473,097</td><td>Emami et al. (2023)</td></tr><tr><td>Australian Electricity</td><td>Energy</td><td>30T</td><td>5</td><td>1,153,584</td><td>Godahewa et al. (2021)</td></tr><tr><td>BDG-2 Bear</td><td>Energy</td><td>H</td><td>91</td><td>1,482,312</td><td>Emami et al. (2023)</td></tr><tr><td>BDG-2 Fox</td><td>Energy</td><td>H</td><td>135</td><td>2,324,568</td><td>Emami et al. (2023)</td></tr><tr><td>BDG-2 Panther</td><td>Energy</td><td>H</td><td>105</td><td>919,800</td><td>Emami et al. (2023)</td></tr><tr><td>Sceaux</td><td>Energy</td><td>H</td><td>1</td><td>34,223</td><td>Emami et al. (2023)</td></tr><tr><td>Solar Power</td><td>Energy</td><td>4S</td><td>1</td><td>7,397,222</td><td>Godahewa et al. (2021)</td></tr><tr><td>Covid19 Energy</td><td>Energy</td><td>H</td><td>1</td><td>31,912</td><td>Wang et al. (2023b)</td></tr><tr><td>Spanish</td><td>Energy</td><td>H</td><td>1</td><td>35,064</td><td>Wang et al. (2023b)</td></tr><tr><td>ElecDemand</td><td>Energy</td><td>30T</td><td>1</td><td>17,520</td><td>Godahewa et al. (2021)</td></tr><tr><td>PDB</td><td>Energy</td><td>H</td><td>1</td><td>17,520</td><td>Wang et al. (2023b)</td></tr><tr><td>GEF17</td><td>Energy</td><td>H</td><td>8</td><td>140,352</td><td>Wang et al. (2023b)</td></tr><tr><td>GEF14</td><td>Energy</td><td>H</td><td>1</td><td>17,520</td><td>Wang et al. (2023b)</td></tr><tr><td>ELF</td><td>Energy</td><td>H</td><td>1</td><td>21,792</td><td>Wang et al. (2023b)</td></tr><tr><td>Traffic Weekly</td><td>Transport</td><td>W</td><td>862</td><td>82,752</td><td>Godahewa et al. (2021)</td></tr><tr><td>Q-Traffic</td><td>Transport</td><td>15T</td><td>45,148</td><td>264,386,688</td><td>Jiang et al. (2023)</td></tr><tr><td>PEMS04</td><td>Transport</td><td>5T</td><td>921</td><td>15,649,632</td><td>Jiang et al. (2023)</td></tr><tr><td>PEMS07</td><td>Transport</td><td>5T</td><td>883</td><td>24,921,792</td><td>Jiang et al. (2023)</td></tr><tr><td>PEMS08</td><td>Transport</td><td>5T</td><td>510</td><td>9,106,560</td><td>Jiang et al. (2023)</td></tr><tr><td>PEMS Bay</td><td>Transport</td><td>5T</td><td>325</td><td>16,941,600</td><td>Jiang et al. (2023)</td></tr><tr><td>Loop Seattle</td><td>Transport</td><td>5T</td><td>1,809</td><td>33,953,760</td><td>Jiang et al. (2023)</td></tr><tr><td>LargeST</td><td>Transport</td><td>5T</td><td>42,333</td><td>4,452,510,528</td><td>Liu et al. (2023)</td></tr><tr><td>Azure VM Traces 2017</td><td>CloudOTS</td><td>5T</td><td>159,472</td><td>885,522,908</td><td>Woo et al. (2023)</td></tr><tr><td>Borg Cluster Data 2011</td><td>CloudOTS</td><td>5T</td><td>286,772</td><td>1,075,105,708</td><td>Woo et al. (2023)</td></tr><tr><td>Alibaba Cluster Trace 2018</td><td>CloudOTS</td><td>5T</td><td>116,818</td><td>190,385,060</td><td>Woo et al. (2023)</td></tr><tr><td>Wiki-Rolling</td><td>Web</td><td>D</td><td>47,675</td><td>40,619,100</td><td>Alexandrov et al. (2020)</td></tr><tr><td>Kaggle Web Traffic Weekly</td><td>Web</td><td>W</td><td>145,063</td><td>16,537,182</td><td>Godahewa et al. (2021)</td></tr><tr><td>Favorite Sales</td><td>Sales</td><td>D</td><td>111,840</td><td>139,111,860</td><td>Woo et al. (2024)</td></tr><tr><td>PigArtPressure</td><td>Health</td><td>-</td><td>312</td><td>624,000</td><td>Dau et al. (2018)</td></tr><tr><td>SelfRegulationSCP1</td><td>Health</td><td>0.004 SEC</td><td>3,366</td><td>3,015,936</td><td>Dau et al. (2018)</td></tr><tr><td>SelfRegulationSCP2</td><td>Health</td><td>0.004 SEC</td><td>2,660</td><td>3,064,320</td><td>Dau et al. (2018)</td></tr><tr><td>TDBrain</td><td>Health</td><td>0.002 SEC</td><td>28644</td><td>73,299,996</td><td>Wang et al. (2023a)</td></tr><tr><td>MotorImagery</td><td>Health</td><td>0.001 SEC</td><td>24,192</td><td>72,576,000</td><td>Dau et al. (2018)</td></tr><tr><td>PigCVP</td><td>Health</td><td>-</td><td>312</td><td>624,000</td><td>Dau et al. (2018)</td></tr><tr><td>AtrialFibrillation</td><td>Health</td><td>0.008 SEC</td><td>60</td><td>38,400</td><td>Dau et al. (2018)</td></tr><tr><td>IEEEPPG</td><td>Health</td><td>0.008 SEC</td><td>15,480</td><td>15,480,000</td><td>Tan et al. (2021)</td></tr><tr><td>BIDMC32HR</td><td>Health</td><td>-</td><td>15,898</td><td>63,592,000</td><td>Tan et al. (2021)</td></tr></table>
+
+# B TIME SERIES MODELS
+
+We define time series forecasting as the following problem: given a collection of multivariate time series samples with look back window  $L:(\mathbf{x}_1,\ldots ,\mathbf{x}_L)$  where each  $\mathbf{x}_t$  at time step  $t$  is a vector of dimension of  $M$ , our goal is to forecast  $T$  future values  $(\mathbf{x}_{L + 1},\dots ,\mathbf{x}_{L + T})$ .
+
+Patch Embedding. We split the input  $(\mathbf{x}_1, \ldots, \mathbf{x}_L)$  into  $M$  uni-variate time series  $\mathbf{x}^{(i)} \in \mathbb{R}^{1 \times L}$ , independently forecasting future time series for each variate. Each uni-variate time series  $\mathbf{x}^{(i)}$  is first divided into non-overlapped patches. Specifically, given the patch length as  $P$ , the patching process will generate the sequence of patches  $\mathbf{x}_p^{(i)} \in \mathbb{R}^{P \times N}$  where  $N = \left[\frac{L}{P}\right]$  is the number of patches. Then the patches are mapped to the latent space of  $d_m$  via a learnable linear projection  $\mathbf{W}_p \in \mathbb{R}^{d_m \times P}$ . In our baseline models, the patch size  $P$  is set to 32.
+
+Rotary Position Embedding. Rotary Positional Embedding (RoPE) is a type of position encoding that encodes absolute positional information with a rotation matrix and naturally incorporates explicit relative position dependency in self-attention formulation. In detail, RoPE incorporates absolute position information to the embedding and transform them into queries, keys through function:
+
+$$
+f _ {q, k} (\mathbf {x} _ {m}, m) = \mathbf {R} _ {\Theta , m} ^ {d} \mathbf {W} _ {q, k} \mathbf {x} _ {m}
+$$
+
+where
+
+$$
+\mathbf {R} _ {\Theta , m} ^ {d} = \left[ \begin{array}{c c c c} \mathbf {M} _ {1} & & & \\ & \mathbf {M} _ {2} & & \\ & & \ddots & \\ & & & \mathbf {M} _ {d / 2} \end{array} \right], \mathbf {M} _ {j} = \left( \begin{array}{c c} \cos m \theta_ {j} & - \sin m \theta_ {j} \\ \sin m \theta_ {j} & \cos m \theta_ {j} \end{array} \right)
+$$
+
+is the rotary matrix with pre-defined parameters  $\Theta = \{\theta_{i} = 10000^{-2(i - 1) / d}, i \in [1,2,\dots,d / 2]\}$ ,  $\mathbf{W}_{q,k}$  is the learned query or key projection weights, and  $\mathbf{x}_m \in \mathbb{R}^d$  is the embedding of the  $m$  token. Next, applying RoPE to the dot product of query and key, we can obtain:
+
+$$
+\mathbf {q} _ {m} ^ {\top} \mathbf {k} _ {n} = \left(\mathbf {R} _ {\Theta , m} ^ {d} \mathbf {W} _ {q} \mathbf {x} _ {m}\right) ^ {\top} \left(\mathbf {R} _ {\Theta , n} ^ {d} \mathbf {W} _ {k} \mathbf {x} _ {n}\right) = \mathbf {x} _ {m} ^ {\top} \mathbf {W} _ {q} \mathbf {R} _ {\Theta , n - m} ^ {d} \mathbf {W} _ {k} \mathbf {x} _ {n}
+$$
+
+where  $\mathbf{R}_{\Theta ,n - m}^{d} = \left(\mathbf{R}_{\Theta ,m}^{d}\right)^{\top}\mathbf{R}_{\Theta ,n}^{d}$ . In this way, RoPE naturally incorporates relative position information through rotation matrix product. In general, the self-attention enhanced with RoPE can be written as
+
+$$
+\text {A t t e n t i o n} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) _ {m} = \frac {\sum_ {n = 1} ^ {N} \left(\mathbf {R} _ {\Theta , m} ^ {d} \phi \left(\mathbf {q} _ {m}\right)\right) ^ {\top} \left(\mathbf {R} _ {\Theta , n} ^ {d} \varphi \left(\mathbf {k} _ {n}\right)\right) \mathbf {v} _ {n}}{\sum_ {n = 1} ^ {N} \phi \left(\mathbf {q} _ {m}\right) ^ {\top} \varphi \left(\mathbf {k} _ {n}\right)}.
+$$
+
+where  $\phi(\cdot), \varphi(\cdot)$  are usually non-negative functions, e.g.  $\mathrm{elu}(\cdot) + 1$ , or  $\exp(\cdot)$ . Overall, unlike the traditional approach of adding positional information before the query and key projection, RoPE utilizes a rotary matrix to transform query and key embeddings, effectively leveraging the geometric properties of vectors.
+
+Mixture of Distributions. As described in (Flunkert et al., 2017), our model predicts the parameters of a probability distribution, specifically a mixture of Student-t distributions. The probability density function for a random variable  $x$  following a Student-t distribution is given by:
+
+$$
+p (x; \nu , \mu , \tau) = \frac {\Gamma \left(\frac {\nu + 1}{2}\right)}{\Gamma \left(\frac {\nu}{2}\right) \sqrt {\pi \nu \tau}} \left(1 + \frac {1}{\nu} \left(\frac {x - \mu}{\tau}\right) ^ {2}\right) ^ {- \frac {\nu + 1}{2}}
+$$
+
+with parameters  $\nu > 0, \mu \in \mathbb{R}, \tau > 0$  represent the degrees of freedom (df), location, and scale coefficient, respectively, and  $\Gamma$  is the gamma function. We modeled the predicted distribution using a mixture of four Student-t distributions. Our baseline model employs independent linear layers
+
+![](images/3a09607b2f766dcc42ea23f25440d589221ba28d6ae9d7397358c44456e77d8a.jpg)
+
+![](images/e7fbd60010f769f4108e6f2a81abada01088a5f49b418416a58d11652b294ae7.jpg)
+
+![](images/a74b700bd357c4aa86fda72523daee9e031be42f8ead7340427866775895a979.jpg)  
+Figure 9: Gaussian mixture (GM) distribution vs. Student-t mixture (STM) distribution. Left: GM generally under-performs compared to STM in both ID and OOD test data, particularly as model size increases. Right: GM exhibits unstable loss convergence during training, while STM achieves smoother convergence. These findings underscore the prevalence of long-tail distributions in real-world time series data and highlights the crucial role of Student-t mixtures in effectively modeling these distributions.
+
+![](images/0fbc5b5091ef953d8f055ab84dec4489776eb4c1514cf80df1845537f90bccf8.jpg)
+
+![](images/62cc5270d0c9f9c56f4c916ce968f92c9536f94ddb901773d8c5468bf1442041.jpg)
+
+to predict the distribution parameters: degrees of freedom, location, scale, and mixture weights. To ensure positive values for the scale and df, we apply a soft-plus transformation. The mixture weights are constrained within the probability simplex using a soft-max function. To avoid undefined variance for low df values, we impose a lower bound of 2 on the df parameter.
+
+Additionally, we conducted an experiment to compare the performance of Gaussian mixture distribution and Student-t mixture distribution when used as prior distributions to approximate real-world time series. Figure 9 shows that Student-t mixtures generally outperform Gaussian mixtures, particularly as model size increases. Furthermore, we observe unstable convergence when using Gaussian mixture distribution as the prediction head. These findings highlight the prevalence of long-tail distributions in real-world time series data and emphasize the critical role of Student-t mixtures in effectively modeling such distributions.
+
+Encoder-only Transformer. The encoder-only Transformer has proven effective for time series tasks (Nie et al., 2023). In input time series, the future horizon is replaced by learnable mask tokens combined with position information. After passing through bidirectional attention blocks, the future representations are derived from these mask tokens and mapped to the parameters of a mixture distribution. Through sampling from the mixture distribution, we can get predictions for future patches.
+
+Decoder-only Transformer. The key difference in the decoder-only Transformer is its unidirectional attention. In the input sequence, no masking is needed. The future horizon is predicted based on the preceding token. In other words, given a sequence of input patches, the model is optimized to predict the next patch as a function of all past patches. Similar to LLMs this can be done in parallel over the entire context window, and automatically enables the model to predict the future after having seen varying number of input patches.
+
+# C IMPLEMENTATION DETAILS
+
+# C.1 TRAINING SETUP
+
+Based on the constructed datasets of different sizes, we trained a series of encoder-only Transformers and decoder-only Transformers with increasing data sizes and model parameters. To improve batch processing efficiency and handle varying sequence lengths, we employ sequence packing (Raffel et al., 2020), which reduces padding requirements. Moreover, we train models with forecasting capabilities over varying context and prediction lengths. Rather than defining a fixed context and prediction length, we sample  $15\% - 50\%$  lengths as forecast horizon and the remaining as context horizon, for a given time series. As most probabilistic forecasting models, we minimize the negative log-likelihood of the predicted patch with respect to the ground truth. The loss convergence
+
+![](images/f7612df76e545a081275bcc935f0235ca66d8b48d3781d31e222037b2c2a09c7.jpg)  
+Figure 10: Convergence Process of Training Loss. This figure shows the training loss convergence for encoder-only (left) and decoder-only (right) models of various sizes, ranging from 1K to 1B parameters. Larger models demonstrate faster convergence and lower final training loss, highlighting their improved capacity for optimization.
+
+![](images/a679edcd4f4dcf965c50f9272212ebc5a656d2758a04f517cf61e20a5788fddd.jpg)
+
+process of encoder-only models and decoder-only models is shown in Figure 10. As the number of parameters increases, both encoder-only and decoder-only models tend to converge faster to a lower training loss.
+
+# C.2 EVALUATION SETUP
+
+We primarily used the LSF (Wu et al., 2023) and Monash (Godahewa et al., 2021) datasets to evaluate out-of-distribution generalization ability. For the LSF dataset, we only used the test set and performed evaluations in a non-overlapping rolling window fashion, with the stride equal to the prediction length. For the Monash dataset, considering its large size, we set only one window per sample for evaluation. Table 3 details the evaluation setup per dataset.
+
+Following the statistical method in (Qiu et al., 2024), we compute several statistical characteristics of the datasets used in this study. Table 4 presents the mean statistical features of the pre-training, LSF, and Monash datasets. SNR refers to the signal-to-noise ratio, which quantifies the proportion of meaningful signal relative to random noise in a time series. Shifting captures changes in the probability distribution of a time series over time, with values closer to 1 indicating more severe distributional shifts. Stationarity describes a property of a time series where the mean remains constant over time, and the variance is finite and stable across observations. Transition reflects the regular and identifiable fixed features within a time series, such as trends, periodicity, or other predictable patterns. Overall, our pre-training data and OOD test data both show a high quality and stationarity.
+
+Table 3: Out-of-distribution Evaluation Datasets.  
+
+<table><tr><td>Dataset</td><td>Domain</td><td>Frequency</td><td># Prediction Length</td><td># Samples.</td></tr><tr><td>ETTh1</td><td>Energy</td><td>H</td><td>192</td><td>175</td></tr><tr><td>ETTh2</td><td>Energy</td><td>H</td><td>192</td><td>819</td></tr><tr><td>ETTm1</td><td>Energy</td><td>H</td><td>192</td><td>819</td></tr><tr><td>ETTm2</td><td>Energy</td><td>H</td><td>192</td><td>819</td></tr><tr><td>Electricity</td><td>Energy</td><td>H</td><td>192</td><td>7062</td></tr><tr><td>Weather</td><td>Climate</td><td>H</td><td>192</td><td>1029</td></tr><tr><td>M4 Hourly</td><td>Finance</td><td>H</td><td>48</td><td>414</td></tr><tr><td>M4 Daily</td><td>Finance</td><td>D</td><td>14</td><td>4227</td></tr><tr><td>M4 Weekly</td><td>Finance</td><td>W</td><td>13</td><td>359</td></tr><tr><td colspan="5">Table 3 continued from previous page</td></tr><tr><td>Dataset</td><td>Domain</td><td>Frequency</td><td># Prediction Length</td><td># Samples</td></tr><tr><td>M4 Monthly</td><td>Finance</td><td>M</td><td>18</td><td>48000</td></tr><tr><td>M4 Quarterly</td><td>Finance</td><td>Q</td><td>8</td><td>24000</td></tr><tr><td>M1 Monthly</td><td>Finance</td><td>M</td><td>18</td><td>617</td></tr><tr><td>M3 Monthly</td><td>Finance</td><td>M</td><td>18</td><td>1428</td></tr><tr><td>M3 Other</td><td>Finance</td><td>Q</td><td>8</td><td>174</td></tr><tr><td>NN5 Daily</td><td>Finance</td><td>D</td><td>56</td><td>111</td></tr><tr><td>NN5 Weekly</td><td>Finance</td><td>W</td><td>8</td><td>111</td></tr><tr><td>Tourism Monthly</td><td>Finance</td><td>M</td><td>24</td><td>366</td></tr><tr><td>Tourism Quarterly</td><td>Finance</td><td>Q</td><td>8</td><td>427</td></tr><tr><td>CIF 2016</td><td>Finance</td><td>M</td><td>6</td><td>15</td></tr><tr><td>Traffic Hourly</td><td>Transport</td><td>H</td><td>168</td><td>862</td></tr><tr><td>Rideshare</td><td>Transport</td><td>H</td><td>168</td><td>2304</td></tr><tr><td>Saugeen</td><td>Climate</td><td>D</td><td>30</td><td>1</td></tr><tr><td>Sunspot</td><td>Climate</td><td>D</td><td>30</td><td>1</td></tr><tr><td>Temperature Rain</td><td>Climate</td><td>D</td><td>30</td><td>32072</td></tr><tr><td>Vehicle Trips</td><td>Transport</td><td>D</td><td>30</td><td>329</td></tr><tr><td>Weather</td><td>Climate</td><td>D</td><td>30</td><td>3010</td></tr><tr><td>Car Parts</td><td>Sales</td><td>M</td><td>12</td><td>2674</td></tr><tr><td>FRED MD</td><td>Finance</td><td>M</td><td>12</td><td>107</td></tr><tr><td>Pedestrian Counts</td><td>Transport</td><td>H</td><td>12</td><td>66</td></tr><tr><td>Hospital</td><td>Health</td><td>M</td><td>12</td><td>767</td></tr><tr><td>Covid Deaths</td><td>Health</td><td>D</td><td>30</td><td>266</td></tr><tr><td>KDD Cup 2018</td><td>Energy</td><td>H</td><td>168</td><td>270</td></tr><tr><td>Bitcoin</td><td>Finance</td><td>D</td><td>30</td><td>18</td></tr><tr><td>Us Births</td><td>Health</td><td>D</td><td>30</td><td>1</td></tr></table>
+
+Table 4: Mean Statistic Characteristics.  
+
+<table><tr><td>Dataset</td><td>SNR</td><td>Shifting</td><td>Stationarity</td><td>Transition</td></tr><tr><td>Pre-training</td><td>25.47</td><td>0.33</td><td>0.47</td><td>0.03</td></tr><tr><td>LSF</td><td>19.78</td><td>0.15</td><td>0.50</td><td>0.01</td></tr><tr><td>Monash</td><td>19.84</td><td>0.34</td><td>0.38</td><td>0.09</td></tr></table>
+
+# C.3 EVALUATION METRICS
+
+Mean Absolute Percentage Error (MAPE). MAPE is a commonly used metric to evaluate the accuracy of a forecasting model by expressing the error in percentage terms, making it unit-free and easily interpretable. It measures the average magnitude of the absolute percentage error between the predicted and actual values, relative to the actual values. For a uni-variate time series, the error is defined as:
+
+$$
+e _ {j} ^ {(i)} = y _ {j} ^ {(i)} - \hat {y} _ {j} ^ {(i)}
+$$
+
+where  $y_{j}^{(i)}$  and  $\hat{y}_{j}^{(i)}$  are the target and predicted values of the  $i$ -th time series and  $j$ -th time step, respectively. The MAPE of the  $i - th$  time series is then calculated as:
+
+$$
+\mathrm {M A P E} = \frac {1 0 0}{H} \sum_ {j = t + 1} ^ {t + H} \frac {\left| e _ {j} ^ {(i)} \right|}{\left| y _ {j} ^ {(i)} \right|}
+$$
+
+This formula calculates the percentage error at each time step, averaging it over the forecast horizon  $H$ . MAPE is intuitive and interpretable in percentage terms. However, MAPE can be highly sensitive to small actual values because it divides by  $y_{j}^{(i)}$ , leading to extremely large errors when actual values approach zero.
+
+Symmetric Mean Absolute Percentage Error (SMAPE). SMAPE addresses the shortcoming of MAPE by using a symmetric formula that normalizes both the actual and predicted values. The SMAPE of the  $i$ -th time series is defined to be
+
+$$
+\mathrm {S M A P E} = \frac {2 0 0}{H} \sum_ {j = t + 1} ^ {t + H} \frac {| e _ {j} ^ {(i)} |}{| y _ {j} ^ {(i)} | + | \hat {y} _ {j} ^ {(i)} |}
+$$
+
+The SMAPE ensures that errors are balanced between over- and under-prediction by dividing by the sum of the absolute actual and predicted values. This normalization helps avoid extreme errors when actual values are small, but it can also produce undefined results when both actual and predicted values are zero.
+
+Mean Absolute Scaled Error (MASE). MASE is a scale-independent metric used to evaluate forecasting accuracy by comparing the absolute error of a forecast to the average error of a naive forecasting method, such as a one-step-ahead random walk. MASE provides a robust way to assess model performance across different datasets with varying scales, addressing the limitations of metrics like MAPE that are sensitive to the magnitude of actual values. For a uni-variate time series, the MASE of the  $i$ -th time series is calculated as:
+
+$$
+\mathrm {M A S E} = \frac {\frac {1}{H} \sum_ {j = t + 1} ^ {t + H} | e _ {j} ^ {(i)} |}{\frac {1}{T - 1} \sum_ {k = 2} ^ {T} | y _ {k} ^ {(i)} - y _ {k - 1} ^ {(i)} |}
+$$
+
+where  $|y_k^{(i)} - y_{k-1}^{(i)}|$  represents the difference between consecutive values in the naive forecast. MASE is particularly useful because it accounts for the variability and scale of the time series, making it comparable across datasets. Unlike MAPE, MASE does not produce extreme values for small or zero actual values, making it more stable and reliable for time series evaluation.
+
+Continuous Ranked Probability Score (CRPS). Before we can introduce the CRPS, we need to introduce the weighted quantile loss Park et al. (2022), which is a metric normalized over the test set. We first define the  $\alpha$ -quantile loss, also known as the pinball loss at quantile level  $\alpha$ , to be:
+
+$$
+\Lambda_ {\alpha} (q, y) = (\alpha - 1 _ {y <   q}) (y - q)
+$$
+
+The weighted quantile loss is then the normalized sum of quantile losses,
+
+$$
+\operatorname {w Q L} [ \alpha ] = 2 \frac {\sum_ {(i , j) \in \Omega} \Lambda_ {\alpha} \left(\hat {q} _ {j} ^ {(i)} (\alpha) , y _ {j} ^ {(i)}\right)}{\sum_ {(i , j) \in \Omega} | y _ {j} ^ {(i)} |}
+$$
+
+where  $\Omega = \{(i,j)\in \mathbb{Z}^2:1\leq i\leq n,\tau_i + 1\leq j\leq T_i\}$
+
+The CRPS is a proper scoring rule Matheson & Winkler (1976), meaning that it is minimized when the predictive distribution is equal to the distribution from which the data is drawn.
+
+$$
+\operatorname {C R P S} = \int_ {0} ^ {1} 2 \Lambda_ {\alpha} (F ^ {- 1} (\alpha), y) d \alpha
+$$
+
+![](images/b81bd0d85b8f069da93ef4d0c6690bbd40329c360d570cba7b6c2936e976a3bb.jpg)
+
+![](images/e2f19ac76f5f1c7aa43d4bf6e30f120ab6b4203596c2f3ceb5e55e1405f2db51.jpg)
+
+![](images/1609c47dd4348aa8277659552343450ae1ddf09812f495a7922e21fc87c66dce.jpg)  
+Figure 11: Emergent Behaviors in us birth Dataset. The sub-figures, arranged from left to right and top to bottom, show prediction results for models of sizes 10K, 1M, 10M, and 100M. Models with sizes 10K and 1M are unable to capture the periodic pattern, whereas models with sizes 10M and 100M accurately reflect it.
+
+![](images/f7fd657e014473a307cabaf11d319be0153c35fe332baa5ca97a73012188fab4.jpg)
+
+However, we are unable to evaluate this quantity since we generally are not able to compute the integral in closed form and only have access to a finite number of quantile predictions. The approximation of the CRPS is an average of the weighted quantile loss over  $K$  quantiles, and thus is also known as the mean weighted quantile loss.
+
+$$
+\mathrm {C R P S} \approx \frac {1}{K} \sum_ {k = 1} ^ {K} \mathrm {w Q L} [ \alpha_ {k} ]
+$$
+
+# D ADDITIONAL RESULTS
+
+# D.1 EMERGENT BEHAVIORS
+
+Figure 8 presents case studies on "emergent abilities" of TSFMs for OOD predictions. As model size increases, one would expect the MAPE to follow a smooth power-law decline, indicating continuous improvement. However, the plots reveal distinct phenomena, where performance changes abruptly rather than gradually. Figures 11 - 13 visually show the transition from the small model prediction results to the large model prediction results. We see that the 10K and 1M models are unable to capture the historical seasonal patterns to make accurate predictions; however, when the model size reaches 10M, it can make accurate predictions, far better than the small model. This behavior is akin to the "emergent behaviors" described in (Wei et al., 2022), where certain capabilities absent in smaller models only appear in larger ones. The emergence of improved performance at certain model sizes could indicate the development of higher-order patterns or representations within the model. These emergent abilities might stem from the model's capacity to recognize and generalize complex temporal patterns that were previously inaccessible at smaller scales. Such abilities is evident particularly in zero-shot OOD prediction, where the model must extrapolate from training data to entirely unseen scenarios.
+
+![](images/864152aa982e9ea6c8c7beaa5399a32cacf93cc606c4d4fc0d2175c4128c845f.jpg)
+
+![](images/6f3071e0eeb67a07d5b13da1be452265997f615c50154994975289e0680db31e.jpg)
+
+![](images/2e4df51771e380e4dcf478d6b118d6469feeda4ec86844d0e8f8d4fdfbf51611.jpg)  
+Figure 12: Emergent Behaviors in nn5 daily Dataset. From left to right and top to bottom, the sub-figures illustrate the prediction outcomes of models with sizes 10K, 1M, 10M, and 100M, respectively. While the smaller models (10K and 1M) fail to recognize the periodic pattern in the data, the larger models (10M and 100M) effectively capture and reflect the periodic structure.
+
+![](images/ea835f3f40c914b485145ef8a42079cb176c730ef5eb6e5808dd8c6ecf9ceef5.jpg)
+
+![](images/d665e1d49d018fed951028ce6d282d052b557b51acc105ff7421af7cb359d847.jpg)
+
+![](images/df067c59427d6c8f3173098e3a8b8f790d79ce09c2e509c492fde4fe9bc0e1f6.jpg)
+
+![](images/48ebbf54ff704849183e63d6c03e5e3ff149bd6c80b269ea9f8e84d54fc07ebe.jpg)  
+Figure 13: Emergent Behaviors in australian electricity Dataset. From left to right and top to bottom, the sub-figures display the prediction results of models with sizes 10K, 1M, 10M, and 100M. The models with sizes 10K and 1M fail to accurately capture the periodic pattern, while those with sizes 10M and 100M successfully do so.
+
+![](images/ec3831ca120c516c7424034745761200c67b090b128de4f72ee8824604747199.jpg)
+
+![](images/3cfbac775adea2bf06f240cc74b6a4e61fd0978bd71703fc41bd26f792d344d7.jpg)  
+Figure 14: The figure illustrates the validation NLL during training for encoder-only models of different sizes. Larger models tend to achieve lower NLL with processing the same time points.
+
+![](images/65562958b75e7a9f1ad31015e1acdbfdd0de7d4f72f1187d608ae5a19e76a108.jpg)  
+Figure 15: Scaling Pattern of NLL. This figure illustrates scaling laws for NLL in relation to model size, compute, and dataset size. The blue lines represent ID performance, while the red and green lines show OOD performance on LSF subset and Monash subset.
+
+![](images/aa2eafe16c9aae5ede26d371403de4a818d8720d2b96c773fb97497c28adbf41.jpg)
+
+![](images/062bd6f387b0562ef01cf0700d1b1313e16917eb36ab3f9766f3e428ed448468.jpg)
+
+# D.2 SAMPLE EFFICIENCY
+
+Figure 14 illustrates the evaluation results on the ID data for encoder-only models of different sizes during training. The key observation is that larger models achieve better performance with fewer training steps compared to smaller models, demonstrating higher sample efficiency. This is important in scenarios where data are limited, as larger models can achieve superior performance without needing to process as much data. An explanatory intuition is that, when the number of parameters is large, even a small change in parameters can cause a large change in outputs. From a theoretical perspective, the Neural Tangent Kernel (Jacot et al., 2018) framework provides insights into why this occurs. As neural networks become infinitely wide, their behavior approximates that of a kernel method, enabling them to effectively minimize the data distribution's generalization error. This allows large models to generalize well with fewer samples, further supporting their observed sample efficiency in empirical settings. Moreover, in NLP (Kaplan et al., 2020) and CV (Zhai et al., 2022) fields, larger models have demonstrated higher sample efficiency than smaller models.
+
+# D.3 SCALING PATTERN DEPENDS ON DATA DISTRIBUTIONS
+
+In Figures 15 - 19, we present the scaling laws evaluated on the Monash subset, along with results from the LSF subset and the ID test data. By comparing the green line, representing the Monash dataset, and the red line, representing the LSF dataset, we observe a relatively consistent offset and similar slopes across various metrics. This suggests that, when transferring the model from the training data distribution to other OOD distributions, there is a predictable decrease in performance that varies by the target dataset. However, the model's gains from scaling—whether by increasing model size, compute, or dataset size—follow a fixed proportional relationship. This indicates that while OOD performance degrades, scaling the model still yields a consistent improvement ratio across different distributions, albeit with varying levels of absolute performance. Moreover, in the future, we can further analyze the reasons why transfer cost arises and establish a law to predict the
+
+![](images/be81f8085821bbb5e9276527b1b386c98f24cc2fd6fb47d7210908208916049e.jpg)  
+Figure 16: Scaling Pattern of MAPE. This figure presents scaling laws for MAPE as functions of model size, compute, and dataset size. ID performance is illustrated by the blue lines, while OOD performance is depicted in red for the LSF subset and green for the Monash subset.
+
+![](images/de7a299b55239501db71c2301c6c6d7eba3e2699340bfb1328c587900aa26a52.jpg)
+
+![](images/81da12111a1f5e900cb1aabf4d6e3d1bf3b07c0ab5762f8a583e71c1bf47d03f.jpg)
+
+![](images/c34af827cfdc356a9a6795ed015bca3da4f563085b09a58f5d6502ac8f54118b.jpg)  
+Figure 17: Scaling Pattern of SMAPE. This figure depicts the scaling laws for SMAPE with respect to model size, computational resources, and dataset size. ID performance is marked by blue lines, while red and green lines correspond to OOD performance on the LSF and Monash subsets, respectively.
+
+![](images/579c0125b39207e08d0c9be697282e751d17522c4de783855c62c67e276f9165.jpg)
+
+![](images/b9f8c41eb7301b4e35e6a2906e8c5edc8fe89416064bec2c3b340e41451b0e56.jpg)
+
+transfer performance loss in population-risk based theoretical analysis (Yang et al., 2021) through Wasserstein measurement.
+
+# D.4 SCALING PATTERN DEPENDS ON PERFORMANCE METRICS.
+
+We investigate the scaling behaviors of five common performance metrics: NLL, MAPE, SMAPE, MASE, and CRPS, (as shown in Figures 15 - 19). All metrics exhibit a decreasing trend following an approximate power-law; however, each metric demonstrates distinct scaling characteristics, reflected in their varying power-law exponents. Previous work by (Ghorbani et al., 2021) establishes a relationship between large language models' log-likelihood loss and the BLEU score in translation tasks. Similarly, future research could explore a transformation between log-likelihood loss and time series forecasting metrics, offering a means to predict forecasting performance from training loss.
+
+# E FURTHER DISCUSSION
+
+Scaling Laws for Multivariate Time Series Forecasting. Extending our findings to multivariate time series forecasting is promising but presents some challenges. First, large and diverse multivariate datasets are limited, unlike the more standardized uni-variate time series datasets. This scarcity makes it difficult to conduct experiments that scale both model size and data volume in a controlled manner. Second, establishing scaling laws for multivariate time series require analyzing the impact of variable count and correlation strength, increasing the experimental complexity. Third, multivariate foundation models lag behind uni-variate ones, facing challenges in designing architectures that accommodate variable counts and capture complex inter-variable dependencies.
+
+Pre-training Data Mixing Strategy. It's significant to investigate the impact of data mixing strategies on model performance, as different strategies can introduce performance bias or even degrade model performance. Time series pre-training datasets, often comprising data from dozens to hundreds of sources, create exponentially large possibilities for combinations. Instead of exhaustive
+
+![](images/9f42c8ff450f7829ad38a470fd0503e15fd617870b5a4659b5cd206b4634fbf9.jpg)  
+Figure 18: Scaling Pattern of MASE. The figure shows how MASE scales with model size, compute, and dataset size. ID performance is indicated by blue lines, whereas OOD performance is highlighted in red and green for the LSF and Monash subsets.
+
+![](images/4d275d0d545f23d7a65bcc7ecc661948cd6cb23be2272ee1f1505b903a7ab897.jpg)  
+Figure 19: Scaling Pattern of CRPS. The figure illustrates the scaling laws for CRPS in relation to model size, compute, and dataset size. Blue lines represent ID performance, while red and green lines depict OOD performance on the LSF and Monash subsets, respectively.
+
+empirical research, a more practical approach is to establish an extrapolable analytical framework that balances experimental complexity (Hashimoto, 2021; Ye et al., 2024). Additionally, deeper analysis of time series dataset characteristics is essential. This would provide insights into the nature of time series data and facilitate the development of better data organization formats, moving beyond simple combination strategies.
+
+Impact of Model Architecture on Scalability. Our research comparing the scalability of encoder-only and decoder-only Transformers can serve as the groundwork for future exploration of other architectures. CNN-based models (Wu et al., 2022), RNN-based models (Gu et al., 2021), and hybrid models (Lieber et al., 2024) each demonstrate unique strengths in sequence modeling. Investigating their scalability within the context of time series modeling is a promising direction for future research, and we hope our findings will inspire further work in this area.
+
+Impact of Specific Module on Scalability. Our case studies on the scalability of Chronos and Moirai demonstrate that certain module designs in these models may hinder their scalability in OOD settings. Key factors, such as the time series embedding approach, attention mechanism, positional encoding, and forecasting strategy, all impact scalability to varying degrees. Among these factors, the embedding approach is particularly crucial, as it directly determines the input patterns fed into the model backbone and influences the operations the model can learn.
+
+Theoretical Framework for Scaling Laws. Our scaling laws, similar to most scaling laws, are empirical findings. We believe a theoretical understanding of the learning dynamics that form the laws can provide a more solid justification. A potential perspective is understanding the optimization process through dynamics modeling (Bordelon et al., 2024). Specifically, based on the dynamical mean field theory, we can derive the infinite limit statistical description of the scaling behavior of training factors. By analyzing the response function in the infinite limit, which gauges the correlation between training factors and test error, we can obtain insights into how the scaling laws manifest.
+
+Correlation Estimation for Metrics. Time series forecasting involves a diverse set of metrics, each measuring prediction error from a unique perspective. However, due to the lack of clear and definitive correlations between NLL loss and these metrics, it's difficult to understand model predic
+
+tion behavior directly from NLL loss. A potential perspective is to develop a prediction framework that bridges these metrics by empirically analyzing their relationships and dependencies (Isik et al., 2024). Such a framework would provide insights into how different metrics reflect model performance under varying conditions, enabling a deeper understanding of prediction behavior.

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+# TOWARDS OPTIMAL MULTI-DRAFT SPECULATIVE DECODING
+
+Zhengmian Hu $^{1,2,*}$ , Tong Zheng $^{1,*}$ , Vignesh Viswanathan $^{2,3}$ , Ziyi Chen $^{1}$ , Ryan A. Rossi $^{2}$ , Yihan Wu $^{1}$ , Dinesh Manocha $^{1,4}$ , Heng Huang $^{1}$
+
+$^{1}$ Department of Computer Science, University of Maryland, College Park, MD, USA  
+$^{2}$ Adobe Research, San Jose, CA, USA  
+$^{3}$ Manning College of Information & Computer Sciences, University of Massachusetts Amherst, MA, USA  
+$^{4}$ Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA
+
+# ABSTRACT
+
+Large Language Models (LLMs) have become an indispensable part of natural language processing tasks. However, autoregressive sampling has become an efficiency bottleneck. Multi-Draft Speculative Decoding (MDSD) is a recent approach where, when generating each token, a small draft model generates multiple drafts, and the target LLM verifies them in parallel, ensuring that the final output conforms to the target model distribution. The two main design choices in MDSD are the draft sampling method and the verification algorithm. For a fixed draft sampling method, the optimal acceptance rate is a solution to an optimal transport problem, but the complexity of this problem makes it difficult to solve for the optimal acceptance rate and measure the gap between existing verification algorithms and the theoretical upper bound. This paper discusses the dual of the optimal transport problem, providing a way to efficiently compute the optimal acceptance rate. For the first time, we measure the theoretical upper bound of MDSD efficiency for vocabulary sizes in the thousands and quantify the gap between existing verification algorithms and this bound. We also compare different draft sampling methods based on their optimal acceptance rates. Our results show that the draft sampling method strongly influences the optimal acceptance rate, with sampling without replacement outperforming sampling with replacement. Additionally, existing verification algorithms do not reach the theoretical upper bound for both without replacement and with replacement sampling. Our findings suggest that carefully designed draft sampling methods can potentially improve the optimal acceptance rate and enable the development of verification algorithms that closely match the theoretical upper bound.
+
+# 1 INTRODUCTION
+
+Autoregressive language models have achieved state-of-the-art results in various language tasks (Brown et al., 2020; Touvron et al., 2023), including chatbots (Luo et al., 2022) and code generation (Chen et al., 2021). These models generate outputs by predicting the next token sequentially. However, this autoregressive decoding process leads to significant computational resource requirements and high latency, posing challenges for user experience and limiting potential applications.
+
+Speculative decoding (Leviathan et al., 2023; Chen et al., 2023a) has been proposed to address the high inference cost issue. The method uses a small, fast draft model to generate candidate results, which are then verified and corrected by a large, accurate target model to maintain the model output distribution. Compared to other acceleration methods, such as knowledge distillation, model quantization, and model pruning, speculative decoding has the advantage of significantly reducing inference latency without sacrificing quality of the generated content.
+
+Multi-Draft Speculative Decoding (MDSD) (Miao et al., 2024; Cai et al., 2024; Li et al., 2024; Spector & Re, 2023) is a recent advancement in speculative decoding. When generating each token, the small draft model generates multiple draft tokens instead of a single one, as in vanilla speculative decoding. The target LLM verifies these tokens in parallel, ensuring that the final output aligns with the target model's distribution while achieving a higher overall acceptance rate than vanilla speculative decoding, as the multiple drafts provide better coverage of the target model's possible outputs.
+
+MDSD algorithms have two main design choices: (1) The draft sampling method. Common approaches include sampling with replacement, where each token is independently sampled from the draft model output distribution, and sampling without replacement, where the probability of selecting a token is updated after each draw to exclude previously selected tokens. (2) The verification algorithm design. Examples include Recursive Rejection Sampling (RRS) (Yang et al., 2024b; Jeon et al., 2024), which sequentially verifies the draft tokens, and K-SEQ (Sun et al., 2024e), which is designed to improve acceptance rate for sampling with replacement.
+
+The acceptance rate, a measure of MDSD algorithm performance, also depends on these two design choices. Any verification algorithm that guarantees the final output aligns with the target model distribution can be viewed as a transport from the draft tokens' distribution to the target model's distribution. For a fixed draft sampling method, the optimal verification algorithm is a solution to an optimal transport problem (Sun et al., 2024e), corresponding to an optimal acceptance rate.
+
+However, the complexity of this optimal transport problem, with the number of variables and constraints growing exponentially with the number of draft tokens, makes it difficult to find efficient solutions. This difficulty has led to two open questions:
+
+(1) For modern LLMs, where the vocabulary size is typically in the thousands, the optimal acceptance rate has never been computed, to the best of our knowledge. Simple linear program (LP) solvers can only compute the optimal transport for small toy models, making it challenging to measure the optimal acceptance rate in practical scenarios.  
+(2) Although it is widely known that existing verification algorithms are only approximate solutions to the optimal transport problem, the gap between their performance and the theoretical upper bound has never been quantified with respect to real text distribution. Without knowing the optimal acceptance rate, it is difficult to assess how suboptimal these algorithms are.
+
+This paper addresses these two open questions. Our contributions include:
+
+- We transform the problem of solving the optimal acceptance rate corresponding to the optimal transport into a subset selection problem by considering the dual of the problem and then applying total unimodularity. This provides a novel perspective for understanding the efficiency of MDSD.  
+- For certain special cases, we propose efficient methods to solve the subset selection problem by noticing convexity-like structures in the set function. This includes sampling with replacement and sampling without replacement. For the first time, we provide a practical method to compute the theoretical acceptance rate upper bound of MDSD for a draft distribution.  
+- For the first time, we measure the theoretical upper bound of MDSD efficiency on real text, and the gap of existing verification algorithms. We compare different draft sampling methods through their optimal acceptance rates and observe that sampling without replacement outperforms sampling with replacement. We evaluate existing verification algorithms, including K-SEQ for with replacement and RRS for without replacement and with replacement sampling, and find that they still have significant gaps from the theoretical upper bound.  
+- We propose a novel draft sampling method that greedily selects high-probability drafts, with only the last draft being random. In some cases, it achieves an even higher optimal acceptance rate than without replacement. We also propose a corresponding verification algorithm that perfectly reaches the theoretical acceptance rate upper bound.
+
+# 2 PRELIMINARIES
+
+# 2.1 SPECULATIVE DECODING FOR ACCELERATING LLM INFERENCE
+
+Let  $\Sigma$  denote the vocabulary set. We have a target model  $P_{\mathrm{target}}(\cdot |x_1,x_2,\dots,x_m)$ , which is a probabilistic model that predicts the probability of the next word. Our goal is to sample from this model as the output.
+
+The process of single step Multi-Draft Speculative Decoding is as follows:
+
+1. For a draft model  $P_{\mathrm{draft}}(\cdot |x_1,x_2,\dots,x_m)$ , sample  $n$  draft tokens  $\widehat{x}^{(1)},\ldots \widehat{x}^{(n)}$  
+2. Compute the probabilities of the target model in parallel:  $P_{\mathrm{target}}(\cdot |x_1,x_2,\dots,x_m)$ ,  $P_{\mathrm{target}}(\cdot |x_1,x_2,\dots,x_m,\widehat{x}^{(1)})$ , ...,  $P_{\mathrm{target}}(\cdot |x_1,x_2,\dots,x_m,\widehat{x}^{(n)})$ . Due to parallel computation, this step is not much slower than computing  $P_{\mathrm{target}}(\cdot |x_1,x_2,\dots,x_m)$  alone.  
+3. Run the verification algorithm  $x_{m + 1}\sim P_{\mathrm{verify}}(\cdot \widehat{x}^{(1)},\dots \widehat{x}^{(n)})$  
+4. If accepted, for some draft  $\widehat{x}^{(i)}$ , we have  $x_{m+1} = \widehat{x}^{(i)}$ . In this case, we can perform another sampling step  $x_{m+2} \sim P_{\mathrm{target}}(\cdot | x_1, x_2, \ldots, x_m, \widehat{x}^{(i)})$ , generating two tokens in one step and achieving acceleration.
+
+Speculative Decoding can generate multiple steps, with multiple drafts at each step and all drafts forming a tree. However, we only consider the single-step case in this paper. For following analysis, we use  $p(\cdot) = P_{\mathrm{target}}(\cdot | x_1, x_2, \ldots, x_m)$  to denote the target distribution and  $p_{\mathrm{draft}}$  for distribution of draft tokens.
+
+# 2.2 SPECULATIVE DECODING WITH A SINGLE DRAFTTOKEN
+
+Informally, the verification algorithm depends on two distributions  $p$  and  $p_{\mathrm{draft}}$ , and one draft token  $j \sim p_{\mathrm{draft}}$ . The goal is to output  $i \sim p$  such that the objective max  $P(i = j)$  is achieved, that is to maximize the probability of random variable  $i$  to be the same as random variable  $j$ .
+
+More formally, given  $p \in \Delta_{\Sigma}$  and  $p_{\mathrm{draft}} \in \Delta_{\Sigma}$  representing two probability distributions over the space  $\Sigma$ , we seek a joint distribution  $\pi \in \Pi(p, p_{\mathrm{draft}})$  such that the marginal distributions are  $p$  and  $p_{\mathrm{draft}}$ , respectively, and the objective  $\max \sum_{i \in \Sigma} \pi(i, i)$  is maximized. This forms a optimal transport problem. The optimal transport is denoted as  $\pi_{p, p_{\mathrm{draft}}}^* \in \Pi(p, p_{\mathrm{draft}})$ , and the optimal objective function value is  $\alpha^*(p, p_{\mathrm{draft}}) = \sum_{i \in \Sigma} \pi_{p, p_{\mathrm{draft}}}^*(i, i)$ .
+
+The problem can be formulated as an LP by representing the joint distribution as a matrix:
+
+$$
+\max  _ {C \in \mathbb {R} ^ {\Sigma \times \Sigma}} \sum_ {i \in \Sigma} C _ {i, i} \text {s . t .} \sum_ {j \in \Sigma} C _ {i, j} = p (i), \sum_ {i \in \Sigma} C _ {i, j} = p _ {\mathrm {d r a f t}} (j), C _ {i, j} \geq 0 \forall i, j \in \Sigma . \tag {1}
+$$
+
+The optimal transport has the following closed-form expression
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {*} (i, j) = C _ {i, j} ^ {*} = \left\{ \begin{array}{l l} \min  (p (i), p _ {\text {d r a f t}} (i)) & i = j \\ \frac {(p (i) - p _ {\text {d r a f t}} (i)) _ {+} (p _ {\text {d r a f t}} (j) - p (j)) _ {+}}{\sum_ {z \in \Sigma} (p _ {\text {d r a f t}} (z) - p (z)) _ {+}} & i \neq j \end{array} , \right. \tag {2}
+$$
+
+and the optimal objective function value is
+
+$$
+\alpha^ {*} (p, p _ {\text {d r a f t}}) = \sum_ {i \in \Sigma} \min  (p (i), p _ {\text {d r a f t}} (i)). \tag {3}
+$$
+
+The conditional distribution of  $i$  given  $j$  when  $(i,j)\sim \pi$  is denoted as  $\pi (\cdot |\cdot j)\in G(\Sigma)$ , where
+
+$$
+\pi (i | j) = \frac {\pi_ {i , j}}{\sum_ {i \in \Sigma} \pi_ {i , j}} = \frac {\pi_ {i , j}}{p _ {\mathrm {d r a f t}} (j)}. \tag {4}
+$$
+
+For the optimal transport, this leads to
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {*} (i | j) = \left\{ \begin{array}{l l} \min  \left(\frac {p (i)}{p _ {\text {d r a f t}} (j)}, 1\right) & i = j \\ \left(1 - \frac {p (j)}{p _ {\text {d r a f t}} (j)} + \frac {(p (i) - p _ {\text {d r a f t}} (i)) _ {+}}{\sum_ {z \in \Sigma} (p _ {\text {d r a f t}} (z) - p (z)) _ {+}} \right. & i \neq j \end{array} \right.. \tag {5}
+$$
+
+The basic single-step, single-draft speculative decoding can be improved in two directions. Multi-step methods generate a draft sequence. Some improvements (Sun et al., 2024d; Hu & Huang, 2024; Sun et al., 2024c) in this scenario are discussed in Appendix B.3. Our paper focuses on the multi-draft direction, where multiple draft tokens are generated at each step.
+
+# 2.3 MULTI-DRAFT SPECULATIVE DECODING
+
+For  $i\in \Sigma$  , define the incidence set  $A_{i}\coloneqq \{\vec{i}\in \Sigma^{n}|\exists j\in [n],\vec{i}_{j} = i\}$
+
+Informally, the verification algorithm depends on two distributions  $p$  and  $p_{\mathrm{draft}}$ , where  $p_{\mathrm{draft}}$  is now a joint distribution of  $n$  tokens. Common constructions of  $p_{\mathrm{draft}}$  include:
+
+- Sampling with replacement: Given a draft model with output distribution  $q(\cdot)$ , independently sample  $n$  times. For  $\bar{i} = (\bar{i}_1, \dots, \bar{i}_n) \in \Sigma^n$ , we have  $p_{\mathrm{draft}}(\bar{i}) = \prod_{j=1}^{n} q(\bar{i}_j)$ .
+
+- Sampling without replacement:  $p_{\mathrm{draft}}(\bar{i}) = \prod_{j=1}^{n} q^{\neg \bar{i}_1, \dots, \bar{i}_{j-1}}(\bar{i}_j)$ , where
+
+$$
+q ^ {- \bar {i} _ {1}, \dots , \bar {i} _ {j - 1}} (x) = \left\{ \begin{array}{l l} \frac {q (x)}{1 - \sum_ {z \in \{\bar {i} _ {1} , \dots , \bar {i} _ {j - 1} \} q (z)}} & x \notin \{\bar {i} _ {1}, \dots , \bar {i} _ {j - 1} \} \\ 0 & x \in \{\bar {i} _ {1}, \dots , \bar {i} _ {j - 1} \} \end{array} . \right. \tag {6}
+$$
+
+- Product of different draft distributions:  $p_{\mathrm{draft}}(\bar{i}) = \prod_{j=1}^{n} q_j(\bar{i}_j)$ .
+
+Given multiple draft tokens  $\bar{i} = (\bar{i}_1, \dots, \bar{i}_n) \sim p_{\mathrm{draft}}$ , the goal is to output  $i \sim p$  such that the objective  $\max P(\exists j \in [n], i = \bar{i}_j)$  or equivalently  $\max P(\bar{i} \in A_i)$  is achieved, that is to maximize the probability of random variable  $i$  to be the same as one of random variables in  $(\bar{i}_1, \dots, \bar{i}_n)$ .
+
+More formally, given  $p \in \Delta_{\Sigma}$  and  $p_{\mathrm{draft}} \in \Delta_{\Sigma^n}$  representing a probability distribution over the space  $\Sigma$  and a probability distribution over the space  $\Sigma^n$ , respectively, we seek a joint distribution  $\pi \in \Pi(p, p_{\mathrm{draft}})$  such that the marginal distributions are  $p$  and  $p_{\mathrm{draft}}$ , respectively, and the objective  $\max \sum_{i \in \Sigma} \sum_{\bar{i} \in A_i} \pi(i, \bar{i})$  is maximized. The optimal transport is denoted as  $\pi_{p,p_{\mathrm{draft}}}^* \in \Pi(p, p_{\mathrm{draft}})$ , and the optimal objective function value is  $\alpha^*(p, p_{\mathrm{draft}}) = \sum_{i \in \Sigma} \sum_{\bar{i} \in A_i} \pi_{p,p_{\mathrm{draft}}}^*(i, \bar{i})$ .
+
+The problem can be formulated as an LP by representing the joint distribution as a tensor:
+
+$$
+\max_{C\in \mathbb{R}^{\Sigma \times \Sigma^{n}}}\sum_{i\in \Sigma}\sum_{\bar{i}\in A_{i}}C_{i,\bar{i}}
+$$
+
+$$
+\text {s . t .} \sum_ {\bar {i} \in \Sigma^ {n}} C _ {i, \bar {i}} = p (i) \quad \forall i \in \Sigma , \quad \sum_ {i \in \Sigma} C _ {i, \bar {i}} = p _ {\mathrm {d r a f t}} (\bar {i}) \quad \forall \bar {i} \in \Sigma^ {n}, \tag {7}
+$$
+
+$$
+C _ {i, \bar {i}} \geq 0 \quad \forall i \in \Sigma , \bar {i} \in \Sigma^ {n}.
+$$
+
+The difficulty lies in the exponential number of variables and constraints.
+
+Several approximation have been proposed for the multi-draft speculative decoding problem, including Recursive Rejection Sampling (RRS) (Yang et al., 2024b; Jeon et al., 2024) and K-SEQ (Sun et al., 2024e). RRS recursively verifies the draft tokens, while K-SEQ improves the acceptance rate for sampling with replacement. Due to space constraints, we move the details of these methods (Appendix A), other related work (Appendix B) and all proofs (Appendix C) to the appendix.
+
+# 3 OPTIMAL ACCEPTANCE RATE AS SUBSET SELECTION PROBLEM
+
+We show that the optimal acceptance rate can be expressed as a subset selection problem:
+
+$$
+\alpha^ {*} (p, p _ {\text {d r a f t}}) = 1 + \min  _ {H \subset \Sigma} \left(\sum_ {i \in H} p (i) - \sum_ {\bar {i} \in H ^ {n}} p _ {\text {d r a f t}} (\bar {i})\right). \tag {8}
+$$
+
+# 3.1 DUAL PROBLEM
+
+We start from the linear programming formulation (7) and derive an equivalent formulation:
+
+$$
+\max_{S\in \mathbb{R}^{\Sigma \times \Sigma^{n}}}\sum_{i\in \Sigma}\sum_{\bar{i}\in \Sigma^{n}}S_{i,\bar{i}}
+$$
+
+$$
+\text {s . t .} \sum_ {\bar {i} \in \Sigma^ {n}} S _ {i, \bar {i}} \leq p (i) \quad \forall i \in \Sigma , \quad \sum_ {i \in \Sigma} S _ {i, \bar {i}} \leq p _ {\text {d r a f t}} (\bar {i}) \quad \forall \bar {i} \in \Sigma^ {n}, \tag {9}
+$$
+
+$$
+S _ {i, \bar {i}} \geq 0 \quad \forall i \in \Sigma , \bar {i} \in \Sigma^ {n}, \quad S _ {i, \bar {i}} = 0 \quad \forall i \in \Sigma , \bar {i} \notin A _ {i}.
+$$
+
+Lemma 1. The two formulations (7) and (9) are equivalent.
+
+This equivalent formulation transforms the transportation problem (Hitchcock, 1941) into a  $b$ -matching problem, whose dual is a  $w$ -vertex cover problem (Schrijver et al., 2003) (with a detailed derivation in Appendix C.1):
+
+$$
+\min  _ {y \in \mathbb {R} ^ {\Sigma}, z \in \mathbb {R} ^ {\Sigma^ {n}}} \sum_ {i \in \Sigma} y _ {i} p (i) + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} p _ {\text {d r a f t}} (\bar {i}) \tag {10}
+$$
+
+$$
+s. t. y _ {i} + z _ {\bar {i}} \geq 1 \quad \forall i \in \Sigma , \bar {i} \in A _ {i}, \quad y _ {i} \geq 0 \quad \forall i \in \Sigma , \quad z _ {\bar {i}} \geq 0 \quad \forall \bar {i} \in \Sigma^ {n}.
+$$
+
+# 3.2 TOTAL UNIMODULARITY
+
+The coefficient matrix of the constraints in (10) is totally unimodular (TUM). The first set of constraints forms an incidence matrix of a bipartite graph, where one side of the nodes corresponds to  $\Sigma$  and the other side corresponds to  $\Sigma^n$ . There is an edge between  $i$  and  $\bar{i}$  if and only if  $\bar{i} \in A_i$ . Therefore, it is a totally unimodular matrix (Biggs, 1993). The second and third sets of constraints have
+
+coefficient matrices that are identity matrices, with  $|\Sigma| + |\Sigma|^n$  variables and  $|\Sigma| + |\Sigma|^n$  constraints. The concatenation of a TUM matrix and an identity matrix is also TUM (Commoner, 1973).
+
+Since the right-hand side of the constraints are integers, the dual problem (10) always has an integer optimal solution (Hoffman & Kruskal, 2010).
+
+# 3.3 SUBSET SELECTION FORMULATION
+
+By restricting the variables in (10) to integers, we obtain:
+
+$$
+\min  _ {y \in \mathbb {Z} ^ {\Sigma}, z \in \mathbb {Z} ^ {\Sigma^ {n}}} \sum_ {i \in \Sigma} y _ {i} p (i) + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} p _ {\text {d r a f t}} (\bar {i}) \tag {11}
+$$
+
+$$
+s. t. y _ {i} + z _ {\bar {i}} \geq 1 \quad \forall i \in \Sigma , \bar {i} \in A _ {i}, \quad y _ {i} \geq 0 \quad \forall i \in \Sigma , \quad z _ {\bar {i}} \geq 0 \quad \forall \bar {i} \in \Sigma^ {n}.
+$$
+
+In the optimal solution,  $y_{i}$  and  $z_{i}$  will not exceed 1, so they can only take values 0 or 1. Therefore, the problem can be further simplified as:
+
+$$
+\min  _ {y \in \{0, 1 \} ^ {\Sigma}} \min  _ {z \in \{0, 1 \} ^ {\Sigma^ {n}}} \sum_ {i \in \Sigma} y _ {i} p (i) + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} p _ {\text {d r a f t}} (\bar {i}) \tag {12}
+$$
+
+$$
+\begin{array}{l} \text {s . t .} y _ {i} + z _ {\bar {i}} \geq 1 \quad \forall i \in \Sigma , \bar {i} \in A _ {i}. \end{array}
+$$
+
+Define  $H = \{i \in \Sigma | y_i = 1\}$ . The problem becomes:
+
+$$
+\min  _ {H \subset \Sigma} \min  _ {z \in \{0, 1 \} ^ {\Sigma^ {n}}} \sum_ {i \in H} p (i) + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} p _ {\text {d r a f t}} (\bar {i}) \tag {13}
+$$
+
+$$
+s. t. z _ {\bar {i}} \geq 1 \quad \forall i \in \Sigma \backslash H, \bar {i} \in A _ {i}.
+$$
+
+The optimal solution for  $z$  is
+
+$$
+z ^ {*} (H) _ {\bar {i}} = \left\{ \begin{array}{l l} 1 & \bar {i} \in \bigcup_ {x \in \Sigma \backslash H} A _ {x} \\ 0 & \bar {i} \notin \bigcup_ {x \in \Sigma \backslash H} A _ {x} \end{array} . \right. \tag {14}
+$$
+
+Substituting this solution, we obtain the subset selection formulation:
+
+$$
+\min  _ {H \subset \Sigma} \sum_ {i \in H} p (i) + \sum_ {\bar {i} \in \bigcup_ {x \in \Sigma \backslash H} A _ {x}} p _ {\text {d r a f t}} (\bar {i}). \tag {15}
+$$
+
+Finally, note that
+
+$$
+\sum_ {\bar {i} \in \bigcup_ {x \in \Sigma \backslash H} A _ {x}} p _ {\text {d r a f t}} (\bar {i}) + \sum_ {\bar {i} \in H ^ {n}} p _ {\text {d r a f t}} (\bar {i}) = 1. \tag {16}
+$$
+
+This completes the derivation of the subset selection formulation (8).
+
+# 4 COMPUTING OPTIMAL ACCEPTANCE RATE IN SPECIAL CASES
+
+In this section, we discuss how to efficiently compute the optimal acceptance rate for certain special cases of the draft distribution  $p_{\mathrm{draft}}$ . For any set function  $f$ , we define the marginal value of an element  $x$  with respect to a set  $H$  as  $f(x|H) = f(H \cup \{x\}) - f(H)$ . We also define the following shorthand notations:  $P(H) = \sum_{i \in H} p(i)$ ,  $Q(H) = \sum_{\bar{i} \in H^n} p_{\mathrm{draft}}(\bar{i})$ ,  $f(H) = P(H) - Q(H)$ .
+
+The optimal acceptance rate can be expressed as  $\alpha^{*}(p,p_{\mathrm{draft}}) = 1 + \min_{H\subset \Sigma}f(H)$ .
+
+# 4.1  $q$ -CONVEX FUNCTIONS
+
+Definition 2 (q-Convex Function). A set function  $Q:2^{\Sigma}\to \mathbb{R}$  is called a q-convex function if there exists a function  $q:\Sigma \rightarrow \mathbb{R}_{\geq 0}$  such that for all  $H\subset \Sigma$  and  $x,y\in \Sigma \setminus H$  with  $x\neq y$ , we have
+
+$$
+\frac {Q (x \mid H)}{q (x)} \leq \frac {Q (y \mid H \cup \{x \})}{q (y)}. \tag {17}
+$$
+
+Intuitively, if we order the elements of  $\Sigma$  arbitrarily and construct a sequence of sets  $H_{i}$  by adding elements one by one, then the curve of  $Q(H_{i})$  against the sum of  $q$  values is always convex.
+
+Theorem 3. For sampling with replacement, the function  $Q$  is a q-convex function.  
+Theorem 4. For sampling without replacement, the function  $Q$  is a  $q$ -convex function.  
+Theorem 5. All  $q$ -convex functions are supermodular functions.
+
+For both sampling with replacement and without replacement, computing  $\alpha^{*}(p,p_{\mathrm{draft}})$  can be formulated as an unconstrained submodular minimization problem, which has polynomial-time algorithms (Iwata, 2008). However, by fully exploiting the properties of  $q$ -convex functions, we can solve the problem even faster, as shown in the next section.
+
+# 4.2 EFFICIENT COMPUTATION
+
+Theorem 6. Suppose that  $Q$  is a  $q$ -convex function,  $Q$  is monotone increasing, and  $p(x) > 0$  for all  $x \in \Sigma$ . For all  $H \subset \Sigma$  and  $x, y \in \Sigma \setminus H$  with  $x \neq y$ , if  $\frac{q(x)}{p(x)} \leq \frac{q(y)}{p(y)}$  and  $f(x|H) \leq 0$ , then  $f(y|H \cup \{x\}) \leq 0$ .
+
+The above theorem requires  $p(x) > 0$ . When  $p(x) = 0$ , there exists an optimal set  $H^{*}$  for (8) that contains  $x$  because  $Q$  is monotone increasing.
+
+# 4.2.1 ALGORITHM
+
+Inspired by Theorem 6, we can compute the optimal acceptance rate efficiently as follows:
+
+1. Find an ordering  $\sigma$  of  $\Sigma$  such that  $\frac{q(\sigma_1)}{p(\sigma_1)} \geq \dots \geq \frac{q(\sigma_{|\Sigma|})}{p(\sigma_{|\Sigma|})}$ .  
+2. Construct a sequence of sets  $H_{i} = \{\sigma_{1},\dots ,\sigma_{i}\}$  
+3. Compute  $\alpha^{*}(p,p_{\mathrm{draft}}) = 1 + \min_{i}f(H_{i})$
+
+Intuitively, we sort the elements by the ratio of  $q$  and  $p$  in non-increasing order and then perform a linear search.
+
+# 4.2.2 COMPLEXITY OF COMPUTING  $Q$  AND  $\alpha^{*}$
+
+For sampling with replacement,  $Q$  has a simple expression  $Q(H) = (\sum_{x\in H}q(x))^n$ . The time complexity for computing  $\alpha^{*}(p,p_{\mathrm{draft}})$  is  $O(|\Sigma |\log |\Sigma |)$  for the sorting step, plus  $O(|\Sigma |)$  for the linear scan.
+
+For sampling without replacement, we can compute  $Q(H) = \frac{W_{n,H}}{W_{n,\Sigma}}$  based on the coefficient of generating function  $W_{n,H} = \mathrm{Coeff}_{t^n} G_H(t) = \mathrm{Coeff}_{t^n} \prod_{i \in H} (1 + q(i)t)$  and apply dynamic programming with recurrence relation  $W_{n,H \cup \{x\}} = W_{n,H} + q(x) W_{n-1,H}$ . The time complexity is  $O(|\Sigma| \log |\Sigma|)$  for the sorting step, plus  $O(n|\Sigma|)$  for computing coefficient of generating function with dynamic programming.
+
+# 5 A GREEDY APPROACH FOR SELECTING DRAFT TAXENS
+
+In this section, we propose a novel method for constructing the draft distribution  $p_{\mathrm{draft}}$  and a corresponding verification algorithm that achieves the optimal acceptance rate for this distribution.
+
+# 5.1 DRAFT CONSTRUCTION
+
+Given a draft model output distribution  $q \in \Delta_{\Sigma}$ , we construct the draft tokens  $\bar{i} = (\bar{i}_1, \dots, \bar{i}_n)$  as follows:
+
+- The first  $n - 1$  tokens are deterministically set to be the top  $n - 1$  tokens according to the probability in  $q$ , i.e.,  $\bar{i}_1, \ldots, \bar{i}_{n-1} = \operatorname{Top}_{n-1}(q)$ , such that  $q(\bar{i}_1) \geq \dots \geq q(\bar{i}_{n-1})$  and  $\max_{i \in \Sigma \setminus \{\bar{i}_1, \ldots, \bar{i}_{n-1}\}} q(i) \leq q(\bar{i}_{n-1})$ .  
+- Only the last token  $\bar{i}_n$  is randomly sampled from  $q$  without replacement (i.e., it is different from the previous  $n - 1$  tokens):  $\bar{i}_n \sim q^{-\operatorname{Top}_{n - 1}(q)}(\cdot) = \frac{q(\cdot)}{1 - \sum_{j = 1}^{n - 1} q(\bar{i}_j)}$ .
+
+The resulting draft distribution is
+
+$$
+p _ {\text {d r a f t}} (\bar {i}) = \left\{ \begin{array}{l l} q ^ {- \operatorname {T o p} _ {n - 1} (q)} (\bar {i} _ {n}) & \bar {i} _ {1}, \dots , \bar {i} _ {n - 1} = \operatorname {T o p} _ {n - 1} (q) \\ 0 & \bar {i} _ {1}, \dots , \bar {i} _ {n - 1} \neq \operatorname {T o p} _ {n - 1} (q) \end{array} . \right. \tag {18}
+$$
+
+# 5.2 VERIFICATION ALGORITHM
+
+The corresponding optimal transport problem for this draft distribution is simple because only one draft token is random. We can design a verification algorithm that strictly achieves the optimal acceptance rate for this draft distribution (with unfolded definition in Appendix D):
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {\text {G r e e d y}} (i | \bar {i}) = \pi_ {p, q} ^ {*} ^ {\text {T o p} _ {n - 1} (q)} (i | \bar {i} _ {n}). \tag {19}
+$$
+
+Theorem 7. The optimal acceptance rate for the greedy draft distribution is
+
+$$
+\alpha^ {*} (p, p _ {\mathrm {d r a f t}}) = \alpha^ {\text {G r e e d y}} (p, p _ {\mathrm {d r a f t}}) = \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} p (i) + \sum_ {i \in \Sigma} \min  (p (i), q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i)). \tag {20}
+$$
+
+Our subset selection formulation (8) provides a convenient way to prove the above theorem.
+
+# 5.3 CONNECTION TO SPECHUB
+
+SpecHub (Sun et al., 2024b) is a recently proposed MDSD method that is only applicable to the case of  $n = 2$ . The draft construction in SpecHub is as follows:
+
+- First, sample the first draft token  $\bar{i}_1$ .  
+- If  $\bar{i}_1 = \mathrm{Top}_1(q)$  is the token with the highest probability in  $q$ , then sample the second draft token  $\bar{i}_2$  without replacement to ensure it is different from  $\bar{i}_1$ .  
+- If  $\bar{i}_1 \neq \operatorname{Top}_1(q)$  is not the token with the highest probability in  $q$ , then deterministically set the second draft token to be the token with the highest probability, i.e.,  $\bar{i}_2 = \operatorname{Top}_1(q)$ .
+
+The resulting draft distribution is:
+
+$$
+p _ {\text {d r a f t}} (\bar {i}) = \left\{ \begin{array}{l l} q \left(\bar {i} _ {1}\right) & \bar {i} _ {2} = \operatorname {T o p} _ {1} (q) \\ \frac {q \left(\bar {i} _ {1}\right)}{1 - q \left(\bar {i} _ {1}\right)} q \left(\bar {i} _ {2}\right) & \bar {i} _ {1} = \operatorname {T o p} _ {1} (q) \\ 0 & \text {o t h e r w i s e} \end{array} . \right. \tag {21}
+$$
+
+We note that the greedy method for  $n = 2$  is essentially equivalent to SpecHub because both methods ensure that at least one draft token is the token with the highest probability in  $q$ . However, the specific draft distributions are different, leading to a simpler verification algorithm for the greedy method.
+
+# 6 EXPERIMENTS
+
+The goal of our experiments is to measure the acceptance rates of various MDSD methods on real text distributions and compare them with the theoretical upper bounds. In the previous sections, we analyzed the theoretical acceptance rate  $\alpha^{*}(p,p_{\mathrm{draft}})$  for three different draft distributions: sampling with replacement, sampling without replacement, and greedy approach (Section 5). We also discussed some existing verification methods (Appendix A), such as RRS and K-SEQ, whose acceptance rates are expected to be lower than the theoretical upper bound. For K-SEQ, its average acceptance rate  $\alpha^{\mathrm{K - SEQ}}$  can be derived theoretically (see Appendix A.2 for details). Our efficient computation methods (Section 4) make it possible, for the first time, to obtain the theoretical upper bound of MDSD for vocabulary sizes of thousands.
+
+To obtain realistic distributions  $p$  and  $p_{\mathrm{draft}}$ , we select real-world datasets for various tasks, including Alpaca (Taori et al., 2023) for instruction-following, WMT'14 De-En (Bojar et al., 2014) for translation, and CNN-DailyMail (Hermann et al., 2015) for summarization. For each task, we use an LLM to generate responses on 1024 data samples, with a maximum length of 128 tokens. We then measure the logits of the target model and the draft model on these generated responses to construct  $p$  and  $p_{\mathrm{draft}}$ .
+
+We evaluated different approaches based on four publicly available large language models, including 1) LLaMA (Touvron et al., 2023), 2) Vicuna (Chiang et al., 2023), the instruction fine-tuned version of LLaMA models, 3) OPT (Zhang et al., 2022), and 4) Qwen2 (Yang et al., 2024a). Specifically, for the LLaMA family, we select LLaMA-7B as the target model and LLaMA-68M as the draft model, which is consistent with previous work (Miao et al., 2024). For the OPT family, we select OPT-6.7B as the target model and OPT-125M as the draft model. Moreover, for the Vicuna family
+
+Table 1: Acceptance rates of different MDSD methods across various models and tasks.  $\Delta \alpha$  means the gap between a verification method and the theoretical upper bound, with statistically significant differences indicated by directional arrows.  
+
+<table><tr><td rowspan="2">Model Pairs</td><td rowspan="2">Draft Sampling</td><td rowspan="2">Method</td><td colspan="2">Alpaca</td><td colspan="2">CNN-DailyMail</td><td colspan="2">WMT&#x27;14</td></tr><tr><td>α</td><td>Δα</td><td>α</td><td>Δα</td><td>α</td><td>Δα</td></tr><tr><td rowspan="8">OPT-125M OPT-6.7B</td><td rowspan="4">With Replacement</td><td>RRS</td><td>85.4 ± 0.1</td><td>-1.5 ↓</td><td>77.3 ± 0.1</td><td>-3.3 ↓</td><td>70.6 ± 0.1</td><td>-1.5 ↓</td></tr><tr><td>K-SEQ</td><td>85.8 ± 0.1</td><td>-1.1 ↓</td><td>78.4 ± 0.1</td><td>-2.2 ↓</td><td>71.1 ± 0.1</td><td>-0.9 ↓</td></tr><tr><td>αK-SEQ</td><td>85.9 ± 0.1</td><td>-0.9 ↓</td><td>78.5 ± 0.1</td><td>-2.2 ↓</td><td>71.0 ± 0.1</td><td>-1.0 ↓</td></tr><tr><td>α*</td><td>86.9 ± 0.1</td><td>-</td><td>80.7 ± 0.1</td><td>-</td><td>72.0 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Without Replacement</td><td>RRS</td><td>88.9 ± 0.1</td><td>-0.9 ↓</td><td>81.5 ± 0.1</td><td>-2.8 ↓</td><td>75.1 ± 0.1</td><td>-1.0 ↓</td></tr><tr><td>α*</td><td>89.9 ± 0.1</td><td>-</td><td>84.3 ± 0.1</td><td>-</td><td>76.0 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Greedy</td><td rowspan="2">Verify α*</td><td>90.7 ± 0.1</td><td>0.0</td><td>84.2 ± 0.1</td><td>-0.1</td><td>77.0 ± 0.1</td><td>-0.0</td></tr><tr><td>90.7 ± 0.1</td><td>-</td><td>84.3 ± 0.1</td><td>-</td><td>77.1 ± 0.1</td><td>-</td></tr><tr><td rowspan="8">LLaMA-68MLLaMA-7B</td><td rowspan="4">With Replacement</td><td>RRS</td><td>71.6 ± 0.1</td><td>-1.5 ↓</td><td>65.3 ± 0.1</td><td>-2.2 ↓</td><td>59.8 ± 0.1</td><td>-1.0 ↓</td></tr><tr><td>K-SEQ</td><td>71.9 ± 0.1</td><td>-1.1 ↓</td><td>66.0 ± 0.1</td><td>-1.5 ↓</td><td>60.0 ± 0.1</td><td>-0.8 ↓</td></tr><tr><td>αK-SEQ</td><td>72.0 ± 0.1</td><td>-1.0 ↓</td><td>66.2 ± 0.1</td><td>-1.3 ↓</td><td>60.2 ± 0.1</td><td>-0.6 ↓</td></tr><tr><td>α*</td><td>73.0 ± 0.1</td><td>-</td><td>67.5 ± 0.1</td><td>-</td><td>60.8 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Without Replacement</td><td>RRS</td><td>75.7 ± 0.1</td><td>-0.8 ↓</td><td>70.5 ± 0.1</td><td>-1.3 ↓</td><td>63.3 ± 0.1</td><td>-0.1</td></tr><tr><td>α*</td><td>76.5 ± 0.1</td><td>-</td><td>71.8 ± 0.1</td><td>-</td><td>63.4 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Greedy</td><td rowspan="2">Verify α*</td><td>78.4 ± 0.1</td><td>-0.1</td><td>73.2 ± 0.1</td><td>0.1</td><td>66.1 ± 0.1</td><td>0.0</td></tr><tr><td>78.4 ± 0.1</td><td>-</td><td>73.1 ± 0.1</td><td>-</td><td>66.1 ± 0.1</td><td>-</td></tr><tr><td rowspan="8">Eagle-0.24B Vicuna-7B</td><td rowspan="4">With Replacement</td><td>RRS</td><td>63.4 ± 0.2</td><td>-1.0 ↓</td><td>56.7 ± 0.1</td><td>-1.1 ↓</td><td>32.9 ± 0.2</td><td>-0.2</td></tr><tr><td>K-SEQ</td><td>63.9 ± 0.2</td><td>-0.5 ↓</td><td>57.0 ± 0.1</td><td>-0.8 ↓</td><td>33.0 ± 0.2</td><td>-0.1</td></tr><tr><td>αK-SEQ</td><td>63.7 ± 0.1</td><td>-0.7 ↓</td><td>57.1 ± 0.1</td><td>-0.7 ↓</td><td>32.9 ± 0.2</td><td>-0.1</td></tr><tr><td>α*</td><td>64.4 ± 0.1</td><td>-</td><td>57.8 ± 0.1</td><td>-</td><td>33.1 ± 0.2</td><td>-</td></tr><tr><td rowspan="2">Without Replacement</td><td>RRS</td><td>70.9 ± 0.2</td><td>-0.6 ↓</td><td>63.4 ± 0.1</td><td>-0.8 ↓</td><td>36.9 ± 0.2</td><td>0.4</td></tr><tr><td>α*</td><td>71.5 ± 0.1</td><td>-</td><td>64.2 ± 0.1</td><td>-</td><td>36.4 ± 0.2</td><td>-</td></tr><tr><td rowspan="2">Greedy</td><td rowspan="2">Verify α*</td><td>72.6 ± 0.2</td><td>-0.2</td><td>65.7 ± 0.1</td><td>-0.1</td><td>39.5 ± 0.2</td><td>0.1</td></tr><tr><td>72.8 ± 0.1</td><td>-</td><td>65.8 ± 0.1</td><td>-</td><td>39.5 ± 0.2</td><td>-</td></tr><tr><td rowspan="8">Eagle-0.26B Qwen2-7B</td><td rowspan="4">With Replacement</td><td>RRS</td><td>59.6 ± 0.2</td><td>-1.0 ↓</td><td>46.7 ± 0.1</td><td>-1.6 ↓</td><td>38.3 ± 0.1</td><td>-0.4 ↓</td></tr><tr><td>K-SEQ</td><td>59.9 ± 0.2</td><td>-0.8 ↓</td><td>47.2 ± 0.1</td><td>-1.1 ↓</td><td>38.3 ± 0.1</td><td>-0.4</td></tr><tr><td>αK-SEQ</td><td>59.9 ± 0.1</td><td>-0.7 ↓</td><td>47.3 ± 0.1</td><td>-1.0 ↓</td><td>38.3 ± 0.1</td><td>-0.3</td></tr><tr><td>α*</td><td>60.7 ± 0.1</td><td>-</td><td>48.3 ± 0.1</td><td>-</td><td>38.7 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Without Replacement</td><td>RRS</td><td>68.3 ± 0.2</td><td>-1.1 ↓</td><td>52.4 ± 0.1</td><td>-1.7 ↓</td><td>43.9 ± 0.1</td><td>-0.1</td></tr><tr><td>α*</td><td>69.4 ± 0.1</td><td>-</td><td>54.1 ± 0.1</td><td>-</td><td>44.0 ± 0.1</td><td>-</td></tr><tr><td rowspan="2">Greedy</td><td rowspan="2">Verify α*</td><td>69.9 ± 0.2</td><td>-0.0</td><td>54.0 ± 0.1</td><td>0.0</td><td>45.5 ± 0.1</td><td>0.1</td></tr><tr><td>70.0 ± 0.1</td><td>-</td><td>53.9 ± 0.1</td><td>-</td><td>45.4 ± 0.1</td><td>-</td></tr></table>
+
+and the Qwen family, we select Vicuna-7B-v1.3 and Qwen2-7B-Instruct as target models, and we use paired draft models provided by EAGEL (Li et al., 2024), with 0.24B parameters and 0.26B parameters, respectively.
+
+Unless otherwise specified, we use a default generation temperature of 0.7 and a draft token number of 3. The total computational cost is less than 50 GPU hours on RTXA6000.
+
+# 6.1 MAIN EXPERIMENT
+
+In the main experiment, we compare the acceptance rates of different MDSD methods across various LLMs and tasks. The results are shown in Table 1. We observe that the existing verify methods, RRS and K-SEQ, still have gaps compared to the theoretical acceptance rate upper bound. Sampling without replacement achieves higher acceptance rates than sampling with replacement, both in terms of the theoretical upper bound and the existing verification algorithms. We can attribute this to the fact that sampling with replacement may lead to duplicate draft tokens, which are less helpful for acceleration. The greedy method obtains the highest acceptance rate, but this is not always the case, as we will see in the ablation study below that the greedy method performs worse when the temperature is 1.
+
+![](images/c970cce3b18e506c437c2c667d52f46610efe65fbcee37d4abb6736012faf929.jpg)  
+(a) WMT'14 De-En  
+(b) CNN-DailyMail
+
+![](images/d5793f343081edb43d4ece870ea4922d9fee39a4679621e014a74b8716610f26.jpg)  
+Figure 1: Comparison of acceptance rate  $\alpha$  for different temperatures across datasets.  
+(a) WMT'14 De-En  
+(b) CNN-DailyMail  
+Figure 2: Comparison of acceptance rate  $\alpha$  for different number of drafts across datasets.
+
+# 6.2 ABLATION STUDY I: IMPACT OF TEMPERATURE
+
+We study the impact of different temperatures on the acceptance rates. The temperature affects the distributions of the target model and the draft model, even if the logits remain unchanged. It also affects the output text during the sampling process, resulting in different responses. Figure 1 shows the results. We use LLaMA-7B as the target model and LLaMA-68M as the draft model for our ablation studies. We can have the following observations:
+
+- The impact of temperature is non-monotonic. Moreover, different methods respond differently to temperature changes.  
+- At low temperatures, all methods fall into two categories. The first includes methods that allow duplicate tokens. When  $T = 0$ , these methods essentially have only one effective draft token, the one with the largest logits on the draft model. The second includes methods that prevent duplicate tokens. When  $T = 0$ , these methods always select the top  $n$  tokens on the draft model.  
+- The gap between the optimal acceptance rate and acceptance rates for previously existing verification methods, RRS and K-SEQ, gradually increases as the temperature rises.  
+- As temperature increases, the gap between methods with replacement sampling and methods without replacement sampling decreases. We can attribute this to the fact that, at high temperatures, the probability distribution is less concentrated, making with replacement sampling strategies have less probability to generate duplicate tokens.
+
+# 6.3 ABLATION STUDY II: IMPACT OF NUMBER OF DRAFTS
+
+We investigate the impact of different numbers of drafts on the acceptance rates. The results are shown in Figure 2. We have the following observations:
+
+- As the number of drafts increases, the coverage of the target model's possible outputs improves, therefore leading to better acceptance rate. This trend holds for all methods.  
+- The draft sampling strategy significantly impacts the benefits derived from an increase in the number of drafts. Sampling without replacement generally benefit more from an increase in drafts compared to sampling with replacement. This is because sampling with replacement can lead to redundant drafts, which do not fully leverage the advantages of increasing the number of drafts.
+
+Table 2: Acceptance rates of different MDSD methods on MT-Bench based on Eagle framework.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2"># Drafts = 2, # Steps = 4</td><td colspan="2"># Drafts = 4, # Steps = 3</td><td colspan="2">EAGLE default sparse tree</td></tr><tr><td>α</td><td>Speed</td><td>α</td><td>Speed</td><td>α</td><td>Speed</td></tr><tr><td colspan="7">T = 0.1</td></tr><tr><td>RRS w/ replacement</td><td>75.3 ± 0.3</td><td>-</td><td>78.4 ± 0.3</td><td>-</td><td>74.7 ± 0.3</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>79.4 ± 0.3</td><td>1.04 (± 0.02) ×</td><td>80.4 ± 0.3</td><td>1.03 (± 0.01) ×</td><td>76.8 ± 0.3</td><td>1.04 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>84.0 ± 0.3</td><td>1.11 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>84.7 ± 0.3</td><td>1.13 (± 0.02) ×</td><td>88.8 ± 0.2</td><td>1.17 (± 0.01) ×</td><td>79.1 ± 0.3</td><td>1.08 (± 0.02) ×</td></tr><tr><td colspan="7">T = 0.6</td></tr><tr><td>RRS w/ replacement</td><td>78.8 ± 0.3</td><td>-</td><td>84.7 ± 0.3</td><td>-</td><td>76.2 ± 0.3</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>82.4 ± 0.3</td><td>1.07 (± 0.02) ×</td><td>88.6 ± 0.2</td><td>1.07 (± 0.01) ×</td><td>77.6 ± 0.3</td><td>1.05 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>82.3 ± 0.3</td><td>1.02 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>82.8 ± 0.3</td><td>1.04 (± 0.02) ×</td><td>90.0 ± 0.2</td><td>1.09 (± 0.01) ×</td><td>78.3 ± 0.3</td><td>1.01 (± 0.02) ×</td></tr><tr><td colspan="7">T = 1.0</td></tr><tr><td>RRS w/ replacement</td><td>76.7 ± 0.3</td><td>-</td><td>83.5 ± 0.3</td><td>-</td><td>72.1 ± 0.3</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>76.4 ± 0.3</td><td>1.00 (± 0.02) ×</td><td>85.3 ± 0.3</td><td>1.05 (± 0.01) ×</td><td>74.1 ± 0.3</td><td>1.03 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>79.5 ± 0.3</td><td>1.01 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>79.2 ± 0.3</td><td>1.02 (± 0.02) ×</td><td>87.8 ± 0.2</td><td>1.08 (± 0.01) ×</td><td>72.9 ± 0.3</td><td>0.97 (± 0.02) ×</td></tr></table>
+
+- The gap between the optimal acceptance rate and acceptance rates for previously existing verification methods, RRS and K-SEQ, gradually increases as the number of drafts rises.
+
+# 6.4 EVALUATING THE GREEDY SAMPLING METHOD ON GENERATION TASKS
+
+In this section, we evaluate the effectiveness and generation efficiency of the proposed Greedy draft sampling method (Section 5) on real-world generation tasks and compare it with other MDSD methods.
+
+We implement the Greedy method within the EAGLE Framework (Li et al., 2024), which supports multi-step MDSD with a draft tree structure. We experiment with three types of tree structures: (1) drafts  $= 2$ , depths  $= 4$ ; (2) drafts  $= 4$ , depths  $= 3$ ; and (3) a sparse tree with up to 4 drafts and 5 steps, which is the default setting in EAGLE. We conduct experiments on the MT-Bench dataset (Zheng et al., 2023) using Vicuna-7B-v1.3 (Chiang et al., 2023) as the target model and its corresponding Eagle model with 0.24B parameters as the draft model.
+
+Table 3 presents the results. As discussed in Section 5.3, the Greedy method and SpecHub have equal acceptance rates when the number of draft tokens is 2. Our experiments confirm this theoretical insight, showing no statistically significant difference between the two methods for any temperature.
+
+The Greedy method demonstrates improved performance at low temperatures. For example, at  $T = 0.1$ , it achieves a higher acceptance rate compared to RRS without replacement, leading to faster generation. However, as the temperature increases, the performance gain of the Greedy method diminishes. This observation is consistent with the ablation study in Figure 1.
+
+# 7 CONCLUSION
+
+In this paper, we studied the acceptance rate of Multi-Draft Speculative Decoding (MDSD).
+
+On the theoretical side, we discovered an equivalence between the optimal acceptance rate and a subset selection problem. We also provided efficient methods to compute the optimal acceptance rate for common draft distributions.
+
+On the practical side, for the first time, we measured the optimal acceptance rate under real text distributions and quantified the gap between existing algorithms and the optimal acceptance rate.
+
+Furthermore, we proposed a practical greedy draft construction method that, in some cases, achieves an even higher acceptance rate than sampling without replacement.
+
+We hope that our work will stimulate further research on improving the efficiency of large language model inference and make these powerful models more accessible and applicable in real-world scenarios.
+
+# ACKNOWLEDGMENT
+
+This work was partially supported by NSF IIS 2347592, 2348169, DBI 2405416, CCF 2348306, CNS 2347617.
+
+# REFERENCES
+
+Norman Biggs. Algebraic graph theory. Number 67. Cambridge university press, 1993.  
+Ondrej Bojar, Christian Buck, Christian Federmann, Barry Haddow, Philipp Koehn, Johannes Leveling, Christof Monz, Pavel Pecina, Matt Post, Herve Saint-Amand, et al. Findings of the 2014 workshop on statistical machine translation. In Proceedings of the ninth workshop on statistical machine translation, pp. 12-58, 2014.  
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 1877-1901. Curran Associates, Inc., 2020.  
+Tianle Cai, Yuhong Li, Zhengyang Geng, Hongwu Peng, Jason D Lee, Deming Chen, and Tri Dao. Medusa: Simple llm inference acceleration framework with multiple decoding heads. arXiv preprint arXiv:2401.10774, 2024.  
+Charlie Chen, Sebastian Borgeaud, Geoffrey Irving, Jean-Baptiste Lespiau, Laurent Sifre, and John Jumper. Accelerating large language model decoding with speculative sampling. arXiv preprint arXiv:2302.01318, 2023a.  
+Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde De Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, et al. Evaluating large language models trained on code. arXiv preprint arXiv:2107.03374, 2021.  
+Zhuoming Chen, Avner May, Ruslan Svirschevski, Yuhsun Huang, Max Ryabinin, Zhihao Jia, and Beidi Chen. Sequoia: Scalable, robust, and hardware-aware speculative decoding. arXiv preprint arXiv:2402.12374, 2024.  
+Ziyi Chen, Xiaocong Yang, Jiacheng Lin, Chenkai Sun, Jie Huang, and Kevin Chen-Chuan Chang. Cascade speculative drafting for even faster llm inference. arXiv preprint arXiv:2312.11462, 2023b.  
+Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E. Gonzalez, Ion Stoica, and Eric P. Xing. Vicuna, March 2023.  
+Frederic G Commoner. A sufficient condition for a matrix to be totally unimodular. Networks, 3(4): 351-365, 1973.  
+Zhenyu He, Zexuan Zhong, Tianle Cai, Jason D Lee, and Di He. Rest: Retrieval-based speculative decoding. arXiv preprint arXiv:2311.08252, 2023.  
+Karl Moritz Hermann, Tomás Kocisky, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. Teaching machines to read and comprehend. In Twenty-eighth Conference on Neural Information Processing Systems, pp. 1693-1701, 2015.  
+Frank L Hitchcock. The distribution of a product from several sources to numerous localities. Journal of mathematics and physics, 20(1-4):224-230, 1941.  
+Alan J Hoffman and Joseph B Kruskal. Integral boundary points of convex polyhedra. 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, pp. 49-76, 2010.
+
+Zhengmian Hu and Heng Huang. Accelerated speculative sampling based on tree monte carlo. In *Forty-first International Conference on Machine Learning*, 2024.  
+Satoru Iwata. Submodular function minimization. Mathematical Programming, 112:45-64, 2008.  
+Wonseok Jeon, Mukul Gagrani, Raghavy Goel, Junyoung Park, Mingu Lee, and Christopher Lott. Recursive speculative decoding: Accelerating llm inference via sampling without replacement. arXiv preprint arXiv:2402.14160, 2024.  
+Ashish J Khisti, Arash Behravesh, Hassan Dbouk, Arash Behboodi, Roland Memisevic, and Christos Louizos. Importance weighted multi-draft speculative sampling. In ICML 2024 Workshop on Theoretical Foundations of Foundation Models, 2024.  
+Yaniv Leviathan, Matan Kalman, and Yossi Matias. Fast inference from transformers via speculative decoding. In International Conference on Machine Learning, pp. 19274-19286. PMLR, 2023.  
+Yuhui Li, Fangyun Wei, Chao Zhang, and Hongyang Zhang. Eagle: Speculative sampling requires rethinking feature uncertainty. arXiv preprint arXiv:2401.15077, 2024.  
+Bei Luo, Raymond YK Lau, Chunping Li, and Yain-Whar Si. A critical review of state-of-the-art chatbot designs and applications. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 12(1):e1434, 2022.  
+Xupeng Miao, Gabriele Oliaro, Zhihao Zhang, Xinhao Cheng, Zeyu Wang, Zhengxin Zhang, Rae Ying Yee Wong, Alan Zhu, Lijie Yang, Xiaoxiang Shi, et al. Specinfer: Accelerating large language model serving with tree-based speculative inference and verification. In Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 3, pp. 932-949, 2024.  
+Giovanni Monea, Armand Joulin, and Edouard Grave. Pass: Parallel speculative sampling. arXiv preprint arXiv:2311.13581, 2023.  
+Jie Ou, Yueming Chen, and Wenhong Tian. Lossless acceleration of large language model via adaptive n-gram parallel decoding. arXiv preprint arXiv:2404.08698, 2024.  
+Alexander Schrijver et al. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003.  
+Benjamin Spector and Chris Re. Accelerating lIm inference with staged speculative decoding. arXiv preprint arXiv:2308.04623, 2023.  
+Hanshi Sun, Zhuoming Chen, Xinyu Yang, Yuandong Tian, and Beidi Chen. Triforce: Lossless acceleration of long sequence generation with hierarchical speculative decoding. arXiv preprint arXiv:2404.11912, 2024a.  
+Ryan Sun, Tianyi Zhou, Xun Chen, and Lichao Sun. SpecHub: Provable acceleration to multi-draft speculative decoding. In Yaser Al-Onaizan, Mohit Bansal, and Yun-Nung Chen (eds.), Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pp. 20620-20641, Miami, Florida, USA, November 2024b. Association for Computational Linguistics. doi: 10.18653/v1/2024.emnlp-main.1148.  
+Ziteng Sun, Uri Mendlovic, Yaniv Leviathan, Asaf Aharoni, Ahmad Beirami, Jae Hun Ro, and Ananda Theertha Suresh. Block verification accelerates speculative decoding. In Workshop on Efficient Systems for Foundation Models II@ ICML2024, 2024c.  
+Ziteng Sun, Jae Hun Ro, Ahmad Beirami, and Ananda Theertha Suresh. Optimal block-level draft verification for accelerating speculative decoding. arXiv preprint arXiv:2403.10444, 2024d.  
+Ziteng Sun, Ananda Theertha Suresh, Jae Hun Ro, Ahmad Beirami, Himanshu Jain, and Felix Yu. Spectr: Fast speculative decoding via optimal transport. Advances in Neural Information Processing Systems, 36, 2024e.  
+Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto. Stanford alpaca: An instruction-following llama model, 2023.
+
+Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothee Lacroix, Baptiste Roziere, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971, 2023.  
+An Yang, Baosong Yang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Zhou, Chengpeng Li, Chengyuan Li, Dayiheng Liu, Fei Huang, et al. Qwen2 technical report. arXiv preprint arXiv:2407.10671, 2024a.  
+Nan Yang, Tao Ge, Liang Wang, Binxing Jiao, Daxin Jiang, Linjun Yang, Rangan Majumder, and Furu Wei. Inference with reference: Lossless acceleration of large language models. arXiv preprint arXiv:2304.04487, 2023.  
+Sen Yang, Shujian Huang, Xinyu Dai, and Jiajun Chen. Multi-candidate speculative decoding. arXiv preprint arXiv:2401.06706, 2024b.  
+Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, et al. Opt: Open pre-trained transformer language models. arXiv preprint arXiv:2205.01068, 2022.  
+Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in Neural Information Processing Systems, 36:46595-46623, 2023.  
+Yongchao Zhou, Kaifeng Lyu, Ankit Singh Rawat, Aditya Krishna Menon, Afshin Rostamizadeh, Sanjiv Kumar, Jean-François Kagy, and Rishabh Agarwal. Distillspec: Improving speculative decoding via knowledge distillation. arXiv preprint arXiv:2310.08461, 2023.
+
+# A APPROXIMATE SOLUTIONS
+
+# A.1 RECURSIVE REJECTION SAMPLING (RRS)
+
+Yang et al. (2024b) and Jeon et al. (2024) use the Recursive Rejection Sampling method. Define the residual distribution  $\mathrm{Res}^{p - q}\in \Delta_{\Sigma}$  for  $p,q\in \Delta_{\Sigma}$  as:
+
+$$
+\operatorname {R e s} ^ {p - q} (i) = \frac {(p (i) - q (i)) _ {+}}{\sum_ {z \in \Sigma} (p (z) - q (z)) _ {+}} \tag {22}
+$$
+
+When  $p = q$ ,  $\operatorname{Res}^{p - q}$  can be defined as an arbitrary distribution.
+
+For  $p_{\mathrm{draft}}$  from sampling with replacement, the RRS algorithm is recursively defined as:
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {\mathrm {R R S , w}} (i | \bar {i}) = \tilde {\pi} _ {p, q} ^ {\mathrm {R R S , w}} (i | \bar {i}) \tag {23}
+$$
+
+where
+
+$$
+\widetilde {\pi} _ {p, q} ^ {\mathrm {R R S}, \mathrm {w}} (i | \bar {i}) = \left\{ \begin{array}{l l} \min  \left(\frac {p (\bar {i} _ {1})}{q (\bar {i} _ {1})}, 1\right) & i = \bar {i} _ {1} \\ \left(1 - \frac {p (\bar {i} _ {1})}{q (\bar {i} _ {1})}\right) _ {+} \widetilde {\pi} _ {\operatorname {R e s} ^ {p - q}, q} ^ {\mathrm {R R S}, \mathrm {w}} (i | \bar {i} _ {2:}) & i \neq \bar {i} _ {1} \end{array} \right. \tag {24}
+$$
+
+and
+
+$$
+\tilde {\pi} _ {p, q} ^ {\mathrm {R R S}, \mathrm {w}} (i | ()) = p (i) \tag {25}
+$$
+
+Here  $\bar{\iota}_{2}$  denotes the sequence  $\bar{i}$  with the first element removed.
+
+For  $p_{\mathrm{draft}}$  from sampling without replacement, the RRS algorithm is defined as:
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {\mathrm {R R S}, \mathrm {w}} (i | \bar {i}) = \widetilde {\pi} _ {p, q} ^ {\mathrm {R R S}, \mathrm {w o}} (i | \bar {i}) \tag {26}
+$$
+
+where
+
+$$
+\widetilde {\pi} _ {p, q} ^ {\mathrm {R R S}, \mathrm {w o}} (i | \bar {i}) = \left\{ \begin{array}{l l} \min  \left(\frac {p \left(\bar {i} _ {1}\right)}{q \left(\bar {i} _ {1}\right)}, 1\right) & i = \bar {i} _ {1} \\ \left(1 - \frac {p \left(\bar {i} _ {1}\right)}{q \left(\bar {i} _ {1}\right)}\right) + \widetilde {\pi} _ {\operatorname {R e s} ^ {p - q}, q} ^ {\mathrm {R R S}, \mathrm {w o}} \left(i | \bar {i} _ {2:}\right) & i \neq \bar {i} _ {1} \end{array} \right. \tag {27}
+$$
+
+and
+
+$$
+\widetilde {\pi} _ {p, q} ^ {\mathrm {R R S}, \mathrm {w o}} (i | ()) = p (i) \tag {28}
+$$
+
+The acceptance rates are denoted as  $\alpha^{\mathrm{RRS,w}}(p,p_{\mathrm{draft}})$  and  $\alpha^{\mathrm{RRS,wo}}(p,p_{\mathrm{draft}})$ , respectively.
+
+# A.2 K-SEQ
+
+Sun et al. (2024e) proposed the K-SEQ method to verify drafts sampled with replacement. Define
+
+$$
+\beta_ {p, q} (\rho) = \sum_ {i \in \Sigma} \min  \left(\frac {p (i)}{\rho}, q (i)\right) \tag {29}
+$$
+
+and let  $\rho$  be the solution to the equation
+
+$$
+1 - \left(1 - \beta_ {p, q} (\rho)\right) ^ {n} = \rho \beta_ {p, q} (\rho) \tag {30}
+$$
+
+The K-SEQ algorithm is defined as:
+
+$$
+\pi_ {p, p _ {\text {d r a f t}}} ^ {\mathrm {K} - \mathrm {S E Q}} (i | \bar {i}) = \widetilde {\pi} _ {p, q, \rho} ^ {\mathrm {K} - \mathrm {S E Q}} (i | \bar {i}) \tag {31}
+$$
+
+where
+
+$$
+\widetilde {\pi} _ {p, q, \rho} ^ {\mathrm {K - S E Q}} (i | \bar {i}) = \left(1 - \frac {p \left(\bar {i} _ {1}\right)}{\rho q \left(\bar {i} _ {1}\right)}\right) _ {+} \widetilde {\pi} _ {p, q, \rho} ^ {\mathrm {K - S E Q}} (i | \bar {i} _ {2:}) + \left\{ \begin{array}{l l} \min  \left(\frac {p \left(\bar {i} _ {1}\right)}{\rho q \left(\bar {i} _ {1}\right)}, 1\right) & i = \bar {i} _ {1} \\ 0 & i \neq \bar {i} _ {1} \end{array} \right. \tag {32}
+$$
+
+and
+
+$$
+\widetilde {\pi} _ {p, q, \rho} ^ {\mathrm {K - S E Q}} (i | ()) = \frac {p (i) - \min  \left\{q (i) , \frac {p (i)}{\rho} \right\} \frac {1 - \left(1 - \beta_ {p , q} (\rho)\right) ^ {n}}{\beta_ {p , q} (\rho)}}{(1 - \beta_ {p , q} (\rho)) ^ {n}} \tag {33}
+$$
+
+The acceptance rate is denoted as  $\alpha^{\mathrm{K - SEQ}}(p,p_{\mathrm{draft}}) = 1 - (1 - \beta_{p,q}(\rho))^n$ , which is theoretically guaranteed to achieve a  $(1 - e^{-1})$ -approximation of the optimal acceptance rate.
+
+# B RELATED WORKS
+
+# B.1 DRAFT MODEL DESIGN
+
+Numerous studies have explored the design of better draft models for speculative decoding. In principle, any autoregressive probabilistic model can serve as a draft model. The simplest approaches include using n-gram models (Ou et al., 2024) or document retrieval as draft models (Yang et al., 2023; He et al., 2023). Small transformer-based language models have also been employed (Leviathan et al., 2023; Chen et al., 2023a), often with distillation techniques to further increase the overlap between the draft and target models (Zhou et al., 2023).
+
+The design of a good draft model involves a trade-off between its similarity to the target model and its computational complexity. More complex draft models lead to higher acceptance rates due to their closer resemblance to the target model, but they also incur higher computational overhead. To achieve a better trade-off, some works have proposed reusing the target model's computational results. For example, Monea et al. (2023) use the original model with "look ahead" tokens, while Cai et al. (2024) add new heads to the last hidden layer of the original model to predict tokens further ahead. Li et al. (2024) reuse the last layer hidden state computation of the large model and introduce a new attention layer to predict the next token. Sun et al. (2024a) employ the target model with a partial key-value cache as the draft model.
+
+# B.2 MULTI-DRAFT SPECULATIVE DECODING
+
+Many related works on Multi-Draft Speculative Decoding (MDSD) have been introduced in other sections. This paper focuses on the single-step Multi-Draft scenario. When MDSD generates multiple steps, with each step involving multiple drafts, it forms a tree structure. Sequoia (Chen et al., 2024) propose a dynamic programming algorithm to search for the optimal tree topology.
+
+As the tree grows deeper, the acceptance probability of certain branches decreases. Cascade Speculative Drafting (Chen et al., 2023b) addresses this issue by assigning the largest draft model to generate draft tokens at shallower levels, which are more likely to be accepted, and gradually using smaller models to generate drafts for less relevant branches.
+
+Khisti et al. (2024) studied the optimal acceptance rate for special case of sampling with replacement for  $n = 2$  drafts, and obtained the following result:
+
+$$
+\alpha^ {*} (p, p _ {\mathrm {d r a f t}}) = \min  _ {H \subset \Sigma} \left\{\sum_ {i \in H} p (i) + \left(\sum_ {i \in \Sigma \backslash H} q (s)\right) ^ {2} + 2 \left(\sum_ {i \in H} q (s)\right) \left(\sum_ {i \in \Sigma \backslash H} q (s)\right) \right\}. \tag {34}
+$$
+
+This is essentially the same as our result (8) under this special case. However, our theory is more general, without any assumption on the draft sampling methods or the number of draft tokens.
+
+# B.3 MULTI-STEP SPECULATIVE DECODING
+
+The basic single-step, single-draft speculative decoding, as introduced in Section 2.1, can be applied to multiple steps, with each step having only one draft and an independent verification process (Leviathan et al., 2023; Chen et al., 2023a). However, such an approach of repeatedly applying single-step verification is not optimal for the multi-step scenario. Some works, such as Sun et al. (2024d); Hu & Huang (2024); Sun et al. (2024c), have designed better verification algorithms specifically for the multi-step setting. These algorithms are tailored for the multi-step scenario while remaining compatible with the single-step case, reducing to the basic speculative sampling algorithm when applied to a draft sequence of length 1.
+
+Multi-step speculative decoding and multi-draft speculative decoding represent different directions for improvement.
+
+As shown in Figure 3, Sun et al. (2024e); Khisti et al. (2024) and our work improve speculative decoding from the multi-draft perspective. When there is only a single draft, it reduces to the case in Leviathan et al. (2023); Chen et al. (2023a). On the other hand, Sun et al. (2024d); Hu & Huang
+
+![](images/fd6639178682eaa6405b75730d541ed88143000be6fefb46bb4f080da8f06eba.jpg)  
+Figure 3: Different directions for improving speculative decoding.
+
+(2024); Sun et al. (2024c) enhance speculative decoding from the multi-step perspective. When there is only a single step, it reduces to the case in Leviathan et al. (2023); Chen et al. (2023a).
+
+Combining both improvements in the multi-draft and multi-step scenario would be ideal, and could be a direction for future research.
+
+# C PROOFS
+
+Proof of Lemma 1. Let  $f_{1}(C)$  and  $f_{2}(S)$  denote the objective function values of (7) and (9), respectively. Let  $v_{1} = f_{1}(C^{*})$  and  $v_{2} = f_{2}(S^{*})$  be the optimal objective function values.
+
+First, we show that the optimal solution of (7) is feasible for (9). Define  $S_{i,\bar{i}} = C_{i,\bar{i}}^{*}$  for  $\bar{i} \in A_i$  and  $S_{i,\bar{i}} = 0$  for  $\bar{i} \notin A_i$ . This solution maintains the objective function value, i.e.,  $f_2(S) = f_1(C^*)$ . Therefore,  $v_2 = f_2(S^*) \geq f_2(S) = f_1(C^*) = v_1$ .
+
+Next, we show that the optimal solution of (9) is feasible for (7). Define  $p^{\mathrm{res}}(i) = p(i) - \sum_{\bar{i} \in \Sigma^n} S_{i,\bar{i}}^* \geq 0$  for  $i \in \Sigma$  and  $p_{\mathrm{draft}}^{\mathrm{res}}(\bar{i}) = p_{\mathrm{draft}}(\bar{i}) - \sum_{i \in \Sigma} S_{i,\bar{i}}^* \geq 0$  for  $\bar{i} \in \Sigma^n$ . We have  $\sum_{\bar{i} \in \Sigma^n} p_{\mathrm{draft}}^{\mathrm{res}}(\bar{i}) = \sum_{i \in \Sigma} p^{\mathrm{res}}(i)$ . Define  $C_{i,\bar{i}} = S_{i,\bar{i}}^* + \frac{p^{\mathrm{res}}(i)p_{\mathrm{draft}}^{\mathrm{res}}(\bar{i})}{\sum_{i \in \Sigma} p^{\mathrm{res}}(i)} \geq S_{i,\bar{i}}^*$ . This solution has a larger objective function value, i.e.,  $f_1(C) \geq f_1(S^*)$ . Therefore,  $v_1 = f_1(C^*) \geq f_1(C) \geq f_2(S^*) = v_2$ .
+
+Combining the two parts, we have  $v_{1} = v_{2}$ , which proves the equivalence of the two formulations.
+
+Proof of Theorem 3. For  $\bar{i} = (\bar{i}_1, \dots, \bar{i}_n) \in \Sigma^n$ , we have  $p_{\mathrm{draft}}(\bar{i}) = \prod_{j=1}^n q(\bar{i}_j)$ . The function  $Q(H) = \sum_{\bar{i} \in H^n} p_{\mathrm{draft}}(\bar{i})$  represents the probability that all  $n$  samples drawn with replacement are in the set  $H$ . Therefore,  $Q(H) = (\sum_{x \in H} q(x))^n$ .
+
+Consider the convex function  $g(x) = x^n$ . To prove the  $q$ -convexity of  $Q$ , it suffices to show that for all  $H \subset \Sigma$  and  $x, y \in \Sigma \setminus H$  with  $x \neq y$ , we have:
+
+$$
+\frac {(Q (H) + q (x)) ^ {n} - Q (x) ^ {n}}{q (x)} \leq \frac {(Q (H) + q (x) + q (y)) ^ {n} - (Q (H) + q (x)) ^ {n}}{q (y)} \tag {35}
+$$
+
+This can be rewritten as:
+
+$$
+\frac {g (Q (H) + q (x)) - g (Q (x))}{q (x)} \leq \frac {g (Q (H) + q (x) + q (y)) - g (Q (H) + q (x))}{q (y)} \tag {36}
+$$
+
+Note that both sides are finite differences of the convex function  $g$ . Define  $a = Q(x)$ ,  $b = Q(x) + q(x)$ , and  $c = Q(x) + q(x) + q(y)$ . It suffices to show that:
+
+$$
+\frac {g (b) - g (a)}{b - a} \leq \frac {g (c) - g (b)}{c - b} \tag {37}
+$$
+
+This follows directly from the convexity of  $g$ .
+
+Proof of Theorem 4. The function  $Q(H) = \sum_{\bar{i} \in H^n} p_{\mathrm{draft}}(\bar{i})$  represents the probability that all  $n$  samples drawn without replacement are in the set  $H$ . To handle the more complex case of sampling without replacement, we use generating functions.
+
+Define the generating function  $G_{H}(t) = \prod_{i\in H}(1 + q(i)t)$  and the coefficient  $W_{n,H} = \mathrm{Coeff}_{t^n}G_H(t)$ . Note that  $Q(H) = \frac{W_{n,H}}{W_{n,\Sigma}}$ .
+
+The coefficients satisfy the following recurrence relation:
+
+$$
+\begin{array}{l} W _ {n, H \cup \{x \}} = \operatorname {C o e f f} _ {t ^ {n}} G _ {H \cup \{x \}} (t) (38) \\ = \operatorname {C o e f f} _ {t ^ {n}} G _ {H} (t) + q (x) t G _ {H} (t) (39) \\ = W _ {n, H} + q (x) W _ {n - 1, H} (40) \\ \end{array}
+$$
+
+To prove the  $q$ -convexity of  $Q$ , it suffices to show that for all  $H \subset \Sigma$  and  $x, y \in \Sigma \setminus H$  with  $x \neq y$ , we have:
+
+$$
+\frac {W _ {n , H \cup \{x \}} - W _ {n , H}}{q (x) W _ {n , \Sigma}} \leq \frac {W _ {n , H \cup \{x , y \}} - W _ {n , H \cup \{x \}}}{q (y) W _ {n , \Sigma}} \tag {41}
+$$
+
+Applying the recurrence relation, it suffices to show that:
+
+$$
+W _ {n - 1, H} \leq W _ {n - 1, H \cup \{x \}} \tag {42}
+$$
+
+Applying the recurrence relation again, it suffices to show that:
+
+$$
+0 \leq q (x) W _ {n - 2, H} \tag {43}
+$$
+
+This holds because the coefficients of  $G$  are always non-negative, i.e.,  $W_{n - 2,H}\geq 0$ .
+
+![](images/c3cae15e3fcf869cc105a8b240f9e63f68688856fa4a261bdb6084603b150a8d.jpg)
+
+Proof of Theorem 5. It suffices to show that for all  $H \subset \Sigma$  and  $x, y \in \Sigma \setminus H$  with  $x \neq y$ , we have:
+
+$$
+Q (x | H) \leq Q (x | H \cup \{y \}) \tag {44}
+$$
+
+By the  $q$ -convexity of  $Q$ , we have:
+
+$$
+\frac {Q (x \mid H)}{q (x)} \leq \frac {Q (y \mid H \cup \{x \})}{q (y)} \tag {45}
+$$
+
+Therefore,
+
+$$
+\frac {Q (x \mid H)}{q (x)} \leq \frac {Q (x \mid H) + Q (y \mid H \cup \{x \})}{q (x) + q (y)} = \frac {Q (H \cup \{x , y \}) - Q (H)}{q (x) + q (y)} \leq \frac {Q (y \mid H \cup \{x \})}{q (y)} \tag {46}
+$$
+
+Similarly, by symmetry, we can reverse  $x$  and  $y$  to obtain:
+
+$$
+\frac {Q (y \mid H)}{q (y)} \leq \frac {Q (H \cup \{x , y \}) - Q (H)}{q (x) + q (y)} \leq \frac {Q (x \mid H \cup \{y \})}{q (x)} \tag {47}
+$$
+
+Therefore,
+
+$$
+\frac {Q (x \mid H)}{q (x)} \leq \frac {Q (H \cup \{x , y \}) - Q (H)}{q (x) + q (y)} \leq \frac {Q (x \mid H \cup \{y \})}{q (x)} \tag {48}
+$$
+
+This implies that:
+
+$$
+Q (x | H) \leq Q (x | H \cup \{y \}) \tag {49}
+$$
+
+![](images/42bc2dc6f651e08c99ba9ecf30c60a6da17fa994138836bdecc977b83599f90e.jpg)
+
+Proof of Theorem 6. We have:
+
+$$
+\begin{array}{l} f (x \mid H) = p (x) - Q (x \mid H) (50) \\ = p (x) \left(1 - \frac {q (x)}{p (x)} \frac {Q (x | H)}{q (x)}\right) \leq 0 (51) \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\frac {f (x \mid H)}{p (x)} = 1 - \frac {q (x)}{p (x)} \frac {Q (x \mid H)}{q (x)} \leq 0 \tag {52}
+$$
+
+By assumption,  $\frac{q(x)}{p(x)} \leq \frac{q(y)}{p(y)}$ . By the  $q$ -convexity of  $Q$ , we have  $\frac{Q(x|H)}{q(x)} \leq \frac{Q(y|H \cup \{x\})}{q(y)}$ . Therefore,
+
+$$
+\frac {q (y)}{p (y)} \frac {Q (y | H \cup \{x \})}{q (y)} \geq \frac {q (x)}{p (x)} \frac {Q (x | H)}{q (x)} \tag {53}
+$$
+
+It follows that:
+
+$$
+\begin{array}{l} \frac {f (y \mid H \cup \{x \})}{p (y)} = 1 - \frac {q (y)}{p (y)} \frac {Q (y \mid H \cup \{x \})}{q (y)} (54) \\ \leq \frac {f (x | H)}{p (x)} \leq 0 (55) \\ \end{array}
+$$
+
+![](images/3e8f241fd25d5ff389bcb5829830fdec995189a9b0e7bc9b85530f46659486ee.jpg)
+
+Proof of Theorem 7. We first prove that the acceptance rate of the greedy method is:
+
+$$
+\begin{array}{l} \alpha^ {\text {G r e e d y}} (p, p _ {\text {d r a f t}}) (56) \\ = \sum_ {i \in \Sigma} \sum_ {\bar {i} \in A _ {i}} \pi_ {p, p _ {\text {d r a f t}}} ^ {\text {G r e e d y}} (i, \bar {i}) (57) \\ = \sum_ {i \in \Sigma} \sum_ {\bar {i} \in A _ {i}} \pi_ {p, p _ {\mathrm {d r a f t}}} ^ {\text {G r e e d y}} (i | \bar {i}) p _ {\mathrm {d r a f t}} (\bar {i}) (58) \\ = \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} \sum_ {\bar {i} \in A _ {i}} \pi_ {p, p _ {\mathrm {d r a f t}}} ^ {\text {G r e e d y}} (i | \bar {i}) p _ {\mathrm {d r a f t}} (\bar {i}) (59) \\ + \sum_ {i \in \Sigma \backslash \operatorname {T o p} _ {n - 1} (q)} \sum_ {\bar {i} \in A _ {i}} \pi_ {p, p _ {\text {d r a f t}}} ^ {\text {G r e e d y}} (i | \bar {i}) p _ {\text {d r a f t}} (\bar {i}) (60) \\ = \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} \sum_ {\bar {i} _ {n} \in \Sigma} \pi_ {p, q ^ {- \top \operatorname {T o p} _ {n - 1} (q)}} ^ {*} (i | \bar {i} _ {n}) q ^ {- \top \operatorname {T o p} _ {n - 1} (q)} (\bar {i} _ {n}) (61) \\ + \sum_ {i \in \Sigma \backslash \operatorname {T o p} _ {n - 1} (q)} \pi_ {p, p _ {\mathrm {d r a f t}}} ^ {\text {G r e e d y}} (i | (\operatorname {T o p} _ {n - 1} (q), i)) q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i) (62) \\ = \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} p (i) (63) \\ + \sum_ {i \in \Sigma \backslash \operatorname {T o p} _ {n - 1} (q)} \pi_ {p, q ^ {- \top} \operatorname {T o p} _ {n - 1} (q)} ^ {*} (i | i) q ^ {- \top} \operatorname {T o p} _ {n - 1} (q) (i) (64) \\ = \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} p (i) + \sum_ {i \in \Sigma \backslash \operatorname {T o p} _ {n - 1} (q)} \min  (p (i), q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i)) (65) \\ \end{array}
+$$
+
+Note that  $\sum_{i\in \mathrm{Top}_{n - 1}(q)}\min (p(i),q^{-\mathrm{Top}_{n - 1}(q)}(i)) = \sum_{i\in \mathrm{Top}_{n - 1}(q)}\min (p(i),0) = 0.$
+
+Next, we compute the optimal acceptance rate. Note that when  $\operatorname{Top}_{n-1}(q) \not\subseteq H$ , we must have  $Q(H) = 0$ . When  $\operatorname{Top}_{n-1}(q) \subseteq H$ , we have  $Q(H) = \sum_{i \in H} q^{\neg \operatorname{Top}_{n-1}(q)}(i)$ . Therefore,
+
+$$
+\begin{array}{l} \alpha^ {*} (p, p _ {\mathrm {d r a f t}}) (66) \\ = 1 + \min  _ {H \subset \Sigma} P (H) - Q (H) (67) \\ = 1 + \min  _ {H \subset \Sigma , \text {s . t .} \operatorname {T o p} _ {n - 1} (q) \subseteq H} \sum_ {i \in H} p (i) - q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i) (68) \\ \end{array}
+$$
+
+The optimal set is  $H^{*} = \{i\in \Sigma |q^{-\mathrm{Top}_{n - 1}(q)}(i)\geq p(i)\} \cup \mathrm{Top}_{n - 1}(q)$ . In this case,
+
+$$
+\alpha^ {*} (p, p _ {\mathrm {d r a f t}}) \tag {69}
+$$
+
+$$
+\begin{array}{l} = 1 - \sum_ {i \in \Sigma} \left(q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i) - p (i)\right) _ {+} + \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} p (i) (70) \\ = \sum_ {i \in \Sigma} \min  (p (i), q ^ {- \operatorname {T o p} _ {n - 1} (q)} (i)) _ {+} + \sum_ {i \in \operatorname {T o p} _ {n - 1} (q)} p (i) (71) \\ \end{array}
+$$
+
+![](images/9fdb7cbd9214de1ff51e9f4cd716099985f3afc33c2eaf02a66b71c2e793f942.jpg)
+
+# C.1 DERIVATION OF THE DUAL PROBLEM
+
+We start from the primal problem (9):
+
+$$
+\max  _ {S \in \mathbb {R} ^ {\Sigma \times \Sigma^ {n}}} \sum_ {i \in \Sigma} \sum_ {\bar {i} \in \Sigma^ {n}} S _ {i, \bar {i}}
+$$
+
+$$
+\sum_ {\bar {i} \in \Sigma^ {n}} S _ {i, \bar {i}} \leq p (i) \quad \forall i \in \Sigma \tag {72}
+$$
+
+$$
+\sum_ {i \in \Sigma} S _ {i, \bar {i}} \leq p _ {\text {d r a f t}} (\bar {i}) \quad \forall \bar {i} \in \Sigma^ {n}
+$$
+
+$$
+S _ {i, \bar {i}} \geq 0 \quad \forall i \in \Sigma , \bar {i} \in \Sigma^ {n}
+$$
+
+$$
+S _ {i, \bar {i}} = 0 \quad \forall i \in \Sigma , \bar {i} \notin A _ {i}
+$$
+
+We introduce dual variables  $y_{i}$  for each constraint  $\sum_{\bar{i} \in \Sigma^n} S_{i,\bar{i}} \leq p(i)$  and  $z_{\bar{i}}$  for each constraint  $\sum_{i \in \Sigma} S_{i,\bar{i}} \leq p_{\mathrm{draft}}(\bar{i})$ . The Lagrangian function is:
+
+$$
+\begin{array}{l} L (S, y, z) = \sum_ {i \in \Sigma} \sum_ {\bar {i} \in \Sigma^ {n}} S _ {i, \bar {i}} (73) \\ + \sum_ {i \in \Sigma} y _ {i} (p (i) - \sum_ {\tilde {i} \in \Sigma^ {n}} S _ {i, \tilde {i}}) (74) \\ + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} \left(p _ {\text {d r a f t}} (\bar {i}) - \sum_ {i \in \Sigma} S _ {i, \bar {i}}\right) (75) \\ \end{array}
+$$
+
+The dual function is:
+
+$$
+g (y, z) = \max  _ {S \in \mathbb {R} ^ {\Sigma \times \Sigma^ {n}}} L (S, y, z)
+$$
+
+$$
+s. t. S _ {i, \bar {i}} \geq 0 \quad \forall i \in \Sigma , \bar {i} \in \Sigma^ {n} \tag {76}
+$$
+
+$$
+S _ {i, \bar {i}} = 0 \quad \forall i \in \Sigma , \bar {i} \notin A _ {i}
+$$
+
+Rearranging the Lagrangian function:
+
+$$
+\begin{array}{l} L (S, y, z) = \sum_ {i \in \Sigma} \sum_ {\bar {i} \in \Sigma^ {n}} \left(1 - y _ {i} - z _ {\bar {i}}\right) S _ {i, \bar {i}} (77) \\ + \sum_ {i \in \Sigma} y _ {i} p (i) + \sum_ {\bar {i} \in \Sigma^ {n}} z _ {\bar {i}} p _ {\text {d r a f t}} (\bar {i}) (78) \\ \end{array}
+$$
+
+For the dual function to be bounded, we must have:
+
+$$
+1 - y _ {i} - z _ {\bar {i}} \leq 0, \forall i \in \Sigma , \bar {i} \in A _ {i} \tag {79}
+$$
+
+Therefore, the dual problem is:
+
+$$
+\min  _ {y \in \mathbb {R} ^ {\Sigma}, z \in \mathbb {R} ^ {\Sigma^ {n}}} \sum_ {i \in \Sigma} y _ {i} p (i) + \sum_ {i \in \Sigma^ {n}} z _ {i} p _ {\text {d r a f t}} (\bar {i})
+$$
+
+$$
+\mathrm {s . t .} y _ {i} + z _ {\bar {i}} \geq 1 \quad \forall i \in \Sigma , \bar {i} \in A _ {i} \tag {80}
+$$
+
+$$
+y _ {i} \geq 0 \quad \forall i \in \Sigma
+$$
+
+$$
+z _ {i} \geq 0 \quad \forall \bar {i} \in \Sigma^ {n}
+$$
+
+This completes the derivation of the dual problem.
+
+# D ADDITIONAL ILLUSTRATION
+
+Illustration of single-step draft tokens generation and verification:
+
+![](images/4b2ff4bc095112630b593e35fc776fea7c42d7667e58de97f627f1a99907b662.jpg)
+
+Pseudo code for apply multi-draft speculative sampling for multiple steps, with arbitrary tree topology.
+
+```txt
+def multi_draft_speculative_decoding(prompt, tree topology, draft_model, target_model):  
+    Multi-Draft Speculative Decoding algorithm for accelerating language model inference.  
+Example tree topology:  
+tree topology = [  
+[0], [1], [2], [3], # First level: 4 branches  
+[0,0], [0,1], [0,2], [1,0], [1,1],  
+[2,0], [2,1], [3,0], # Second level  
+[0,0,0], [0,0,1], [0,0,2], [0,1,0],  
+[0,1,1], [0,2,0], [0,2,1], [1,0,0], # Third level  
+[0,0,0,0], [0,0,0,1], [0,0,0,2], # Fourth level  
+[0,0,0,0,0], [0,0,0,0,1] # Fifth level  
+]  
+This is the default EAGLE tree structure where each number represents which draft token to use at each level.
+```
+
+```coffeescript
+# Initialize dictionaries to store drafts and distributions
+drafts = {}
+draft_distributions = {}
+target_distributions = {}
+# Generate drafts for each prefix in the tree topology
+for prefix in [[]] + tree topo:logy:
+    children = get_children-tree topo:logy, prefix)
+    if not children:
+        continue # Skip if no expandable children paths
+    # Generate drafts and corresponding distributions
+    # e.g. sampling with/without replacement or Section 5.1
+    (  draftstuple(prefix)],  draft_distributions[tuple(prefix])
+) = generate_draft_tokens(draft_model, prefix)
+# Compute probability distributions from target model for all drafts
+target_distributions = compute_target_distributions(target_model, drafts)
+# Start verification and generation process
+prefix = []
+while True:
+    if tuple(prefix) not in drafts:
+        break # End if no available drafts
+    # e.g. RRS or K-Seq or Section 5.2
+        output_token = verification(
+            draftstuple(prefix]), draft_distributions[tuple(prefix)], target_distributions[tuple(prefix])
+) 
+    prefix.append(output_token)
+# Return the generated sequence
+return prefix
+def get_children-tree topo:logy, prefix):
+    Get all child paths in tree topo:logy that extend the given
+prefix by one token.
+Examples:
+tree topo:logy = [[0], [1], [0,0], [0,1], [1,0]]
+get_children([[], tree topo:logy) -> [[0], [1]])
+get_children([0], tree topo:logy) -> [[0,0], [0,1]]
+get_children([1], tree topo:logy) -> [[1,0]])
+get_children([0,0], tree topo:logy) -> []
+return [ path for path in tree topo:logy
+if len(path) == len(prefix) + 1 and path[:len(prefix)] == prefix ]
+```
+
+Unfolded definition of verify algorithm for greedy draft construction.
+
+$$
+\begin{array}{l} \pi_ {p, p _ {\text {d r a f t}}} ^ {\text {G r e e d y}} (i | \bar {i}) \\ = \pi_ {p, q} ^ {*} _ {\neg \operatorname {T o p} _ {n - 1} (q)} \left(i | \bar {i} _ {n}\right) \\ = \left\{ \begin{array}{l l} \min  \left(\frac {p (i) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right)}{q (i)}, 1\right) & i = \bar {i} _ {n} \\ \left(1 - \frac {p (\bar {i} _ {n}) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right)}{q (i _ {n})}\right) + \frac {p (i) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right)}{\sum_ {z \in \Sigma} (p (z) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right) - \mathbb {I} (z \notin \operatorname {T o p} _ {n - 1} (q)) q (z)) _ {+}} & i \in \operatorname {T o p} _ {n - 1} (q) \\ \left(1 - \frac {p (\bar {i} _ {n}) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right)}{q (\bar {i} _ {n})}\right) + \frac {(p (i) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right) - q (i)) _ {+}}{\sum_ {z \in \Sigma} (p (z) \left(1 - \sum_ {j \in \operatorname {T o p} _ {n - 1} (q)} q (j)\right) - \mathbb {I} (z \notin \operatorname {T o p} _ {n - 1} (q)) q (z)) _ {+}} & i \neq \bar {i} _ {n}, i \notin \operatorname {T o p} _ {n - 1} (q) \end{array} \right. \tag {81} \\ \end{array}
+$$
+
+# E SUMMARY OF NOTATIONS
+
+-  $\Sigma$ : The vocabulary set  
+-  $\Delta_{\Sigma}$ : The probability simplex over vocabulary  $\Sigma$  
+-  $[n]$ : The set  $1, \ldots, n$  
+-  $P_{\mathrm{target}}(\cdot | x_1, x_2, \dots, x_m)$ : The target model, a probabilistic model that predicts the probability of the next word given the context  
+-  $P_{\mathrm{draft}}(\cdot | x_1, x_2, \dots, x_m)$ : The draft model used to generate candidate tokens  
+-  $P_{\mathrm{verify}}(\cdot |\widehat{x}^{(1)}, \ldots, \widehat{x}^{(n)})$ : The verification algorithm that selects the final output token from the draft tokens  
+-  $p(\cdot) = P_{\text{target}}(\cdot | x_1, x_2, \ldots, x_m)$ : Shorthand for the target distribution  
+-  $p_{\mathrm{draft}}$ : The distribution of draft tokens  
+-  $\pi \in \Pi(p, p_{\mathrm{draft}})$ : A joint distribution with marginal distributions  $p$  and  $p_{\mathrm{draft}}$ .  
+-  $\pi_{p,p_{\mathrm{draft}}}^* \in \Pi(p, p_{\mathrm{draft}})$ : The optimal transport joint distribution  
+-  $\alpha^{*}(p, p_{\mathrm{draft}})$ : The optimal acceptance rate  
+-  $A_{i} \coloneqq \{\vec{i} \in \Sigma^{n} | \exists j \in [n], \vec{i}_{j} = i\}$ : The incidence set for token  $i$  
+-  $q(\cdot)$ : The shorthand notation of the output distribution of the draft model  
+-  $q^{-\hat{\bar{t}}_1, \dots, \hat{\bar{t}}_j - 1}(x)$ : The probability of token  $x$  when sampling without replacement, excluding previously selected tokens  
+-  $\operatorname{Res}^{p - q} \in \Delta_{\Sigma}$ : The residual distribution  
+-  $\pi_{p,p_{\mathrm{draft}}}^{\mathrm{RRS,w}}$ ,  $\pi_{p,p_{\mathrm{draft}}}^{\mathrm{RRS,wo}}$ : The RRS verification algorithms for with/without replacement sampling  
+-  $\alpha^{\mathrm{RRS,w}}(p, p_{\mathrm{draft}})$ ,  $\alpha^{\mathrm{RRS,wo}}(p, p_{\mathrm{draft}})$ : Acceptance rates for RRS with/without replacement  
+-  $\beta_{p,q}(\rho)$ : A function used in the K-SEQ algorithm  
+-  $\pi_{p,p_{\mathrm{draft}}}^{\mathrm{K - SEQ}}$  : The K-SEQ verification algorithm  
+-  $\alpha^{\mathrm{K - SEQ}}(p,p_{\mathrm{draft}})$  : Acceptance rate for the K-SEQ algorithm  
+-  $P(H) = \sum_{i \in H} p(i)$ : Sum of target probabilities over set  $H$  
+-  $Q(H) = \sum_{\bar{i} \in H^n} p_{\mathrm{draft}}(\bar{i})$ : Sum of draft probabilities over set  $H^n$  
+-  $f(H) = P(H) - Q(H)$ : Difference between target and draft probabilities over set  $H$  
+-  $\pi_{p,p\mathrm{draft}}$ : The greedy verification algorithm  
+-  $\alpha^{\mathrm{Greedy}}(p, p_{\mathrm{draft}})$ : Acceptance rate for the greedy draft sampling method  
+-  $C_{i,j}$ : Matrix representation of joint distribution  
+-  $(p(i) - p_{\mathrm{draft}}(i))_+$ : The positive part of the difference  
+-  $G_H(t)$ : Generating function defined as  $\prod_{i \in H} (1 + q(i)t)$  
+-  $W_{n,H}$ : Coefficient of  $t^n$  in  $G_H(t)$
+
+-  $f(x|H)$ : The marginal value of element  $x$  with respect to set  $H$  
+-  $\operatorname{Top}_{n - 1}(q)$ : The top n-1 tokens according to probability in  $q$  
+-  $S \in \mathbb{R}^{\Sigma \times \Sigma^n}$ : Variables in the equivalent LP formulation  
+-  $y \in \mathbb{R}^{\Sigma}, z \in \mathbb{R}^{\Sigma^n}$ : Dual variables  
+-  $\pi (\cdot |\cdot j)$ : The conditional distribution given  $j$  
+-  $\pi_{i,j}$ : Individual elements of the joint distribution matrix  
+-  $\bar{i}_{2}$ : The sequence  $\bar{i}$  with the first element removed  
+-  $H_{i}$ : Sets constructed by adding elements one by one  
+-  $\sigma$ : An ordering of  $\Sigma$  used in the efficient computation algorithm  
+- Coeff  $t^n$ : Coefficient of  $t^n$  in a generating function
+
+# F ADDITIONAL EXPERIMENTS
+
+Table 3: Average generation length  $\tau$  of different MDSD methods and their ratio  $\Delta$  on MT-Bench based on Eagle framework.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2"># Drafts = 2, # Steps = 4</td><td colspan="2"># Drafts = 4, # Steps = 3</td><td colspan="2">EAGLE default sparse tree</td></tr><tr><td>τ</td><td>Δ</td><td>τ</td><td>Δ</td><td>τ</td><td>Δ</td></tr><tr><td colspan="7">T = 0.1</td></tr><tr><td>RRS w/ replacement</td><td>3.04 ± 0.02</td><td>-</td><td>2.83 ± 0.02</td><td>-</td><td>3.19 ± 0.03</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>3.27 ± 0.02</td><td>1.07 (± 0.02) ×</td><td>2.96 ± 0.02</td><td>1.05 (± 0.01) ×</td><td>3.42 ± 0.03</td><td>1.07 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>3.63 ± 0.02</td><td>1.19 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>3.62 ± 0.02</td><td>1.19 (± 0.02) ×</td><td>3.39 ± 0.01</td><td>1.20 (± 0.02) ×</td><td>3.70 ± 0.03</td><td>1.16 (± 0.02) ×</td></tr><tr><td colspan="7">T = 0.6</td></tr><tr><td>RRS w/ replacement</td><td>3.22 ± 0.02</td><td>-</td><td>3.11 ± 0.01</td><td>-</td><td>3.41 ± 0.02</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>3.52 ± 0.02</td><td>1.09 (± 0.02) ×</td><td>3.39 ± 0.01</td><td>1.09 (± 0.01) ×</td><td>3.71 ± 0.02</td><td>1.09 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>3.52 ± 0.02</td><td>1.09 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>3.52 ± 0.02</td><td>1.09 (± 0.02) ×</td><td>3.45 ± 0.01</td><td>1.11 (± 0.01) ×</td><td>3.66 ± 0.02</td><td>1.07 (± 0.02) ×</td></tr><tr><td colspan="7">T = 1.0</td></tr><tr><td>RRS w/ replacement</td><td>3.14 ± 0.02</td><td>-</td><td>3.09 ± 0.01</td><td>-</td><td>3.22 ± 0.02</td><td>-</td></tr><tr><td>RRS w/o replacement</td><td>3.22 ± 0.02</td><td>1.02 (± 0.02) ×</td><td>3.25 ± 0.01</td><td>1.05 (± 0.01) ×</td><td>3.43 ± 0.02</td><td>1.06 (± 0.02) ×</td></tr><tr><td>SpecHub</td><td>3.35 ± 0.02</td><td>1.07 (± 0.02) ×</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Greedy</td><td>3.33 ± 0.02</td><td>1.06 (± 0.02) ×</td><td>3.34 ± 0.01</td><td>1.08 (± 0.01) ×</td><td>3.33 ± 0.02</td><td>1.03 (± 0.02) ×</td></tr></table>
+
+Remark 8. For # Drafts = 2, # Steps = 4, and  $T = 0.6$ , three methods - RRS without replacement, SpecHub, and Greedy - show similar average generation lengths. After truncating to two decimal places, they appear to be the same. However, they are actually different numbers: 3.51781, 3.51944, 3.51975.

2025/Towards Optimal Multi-draft Speculative Decoding/images.zip

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2025/Towards Out-of-Modal Generalization without Instance-level Modal Correspondence/67f64d8a-a30a-48c0-ae44-a555a23f08e5_origin.pdf

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2025/Towards Out-of-Modal Generalization without Instance-level Modal Correspondence/full.md

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+# TOWARDS OUT-OF-MODAL GENERALIZATION WITHOUTINSTANCE-LEVEL MODAL CORRESPONDENCE
+
+Zhuo Huang $^{1}$ , Gang Niu $^{2}$ , Bo Han $^{3,2}$ , Masashi Sugiyama $^{2,4}$ , Tongliang Liu $^{1,2,*}$
+
+$^{1}$ Sydney AI Centre, The University of Sydney  $^{2}$ RIKEN Center for Advanced Intelligence Project
+
+$^{3}$ Hong Kong Baptist University  $^{4}$ The University of Tokyo
+
+# ABSTRACT
+
+The world is understood from various modalities, such as appearance, sound, and language. Since each modality only partially represents objects in a certain meaning, leveraging additional ones is beneficial in both theory and practice. However, exploiting novel modalities normally requires cross-modal pairs corresponding to the same instance, which is extremely resource-consuming and sometimes even impossible, making knowledge exploration of novel modalities largely restricted. To seek practical multi-modal learning, here we study Out-of-Modal (OOM) Generalization as an initial attempt to generalize to an unknown modality without given instance-level modal correspondence. Specifically, we consider Semi-Supervised and Unsupervised scenarios of OOM Generalization, where the first has scarce correspondences and the second has none, and propose Connect&Explore (COX) to solve these problems. COX first connects OOM data and known In-Modal (IM) data through a variational information bottleneck framework to extract shared information. Then, COX leverages the shared knowledge to create emergent correspondences, which is theoretically justified from an information-theoretic perspective. As a result, the label information on OOM data emerges along with the correspondences, which helps explore the OOM data with unknown knowledge, thus benefiting generalization results. We carefully evaluate the proposed COX method under various OOM generalization scenarios, verifying its effectiveness and extensibility. The code is available at https://github.com/tmllab/2025_ICLR_COX.
+
+# 1 INTRODUCTION
+
+To understand the world, we use various data modalities, such as image (He et al., 2016; 2017; Ren et al., 2015) and text (Devlin et al., 2018; Vaswani et al., 2017). Each modality describes objects through a certain physical perspective, thus contributing to understanding objects. Therefore, multi-modal learning (MML) (Alayrac et al., 2022; Ngiam et al., 2011; Radford et al., 2021; Socher et al., 2013) which learns from multiple modality data has been a core research topic in AI. Thanks to the utilization of various modalities, the learning performance has shown to be beneficial on various tasks compared to uni-modal learning (Huang et al., 2021; Lu, 2024; Radford et al., 2021; Sun et al., 2020), such as cross-modal retrieval and generation (Yasunaga et al., 2023; Zhang et al., 2021; Zhen et al., 2019), human-computer interaction (Pantic & Rothkrantz, 2003; Rahman et al., 2022), and robotics (Jiang et al., 2023; Yu et al., 2023).
+
+![](images/402ccc5a186755ac2b451071b4cd21b64cbaa10514d1ef150cc85b9b44e25080.jpg)  
+Figure 1: AI is enhanced as more modalities are incorporated, so how can we teach AI to learn from novel modalities based on the ones it already know?
+
+However, existing states of the art are not satisfactory, and emerging modalities need to be leveraged effectively just like the relatively new data modalities of geomagnetic fields (Hashimoto, 1926), sound waves (Harley et al., 2003), and electromagnetic waves (Weinstein, 1988). Therefore, emerging technologies have constantly leveraged new sensors to enhance their performance. For example, Embodied AIs (Savva et al., 2019) already possess abilities like 3D vision and language, but they are still exploring novel skills, such as tactile and bio-sensing. Since it is hard to leverage such uncommon and inexperienced skills in practice, adapting the knowledge from common modalities to
+
+understand the novel ones could be beneficial, as shown in Figure 1. In practice, most existing MML investigations (Radford et al., 2021; Girdhar et al., 2023; Wang et al., 2024d; Zhu et al., 2023) require instance-level modal correspondence, i.e., multi-modal data are paired with the same instance, which is often hard to satisfy in real-world scenarios when facing novel modalities (Liang et al., 2023; Sun et al., 2020; Xia et al., 2024). For a robotic example, some modalities are common and easy to acquire, e.g., vision and language. However, others like tactility need special sensors to resample from the same objects seen or spoken. Unfortunately, the resample could no longer be accessible in practice. As a result, the new modalities usually have incomplete or even no correspondence, which could seriously block the knowledge interaction across modalities and hinder the benefits brought by MML. Hence, a question naturally occurs: Do we really need instance-level modal correspondence to explore novel modalities?
+
+This paper studies a practical yet unexplored problem named Out-of-Modal (OOM) Generalization. Particularly, given several modalities, i.e., In-Modal (IM) data, the goal is to generalize to an unknown modality without or sometimes only with scarce correspondence. Such a setting implies the real-world utilization of novel modalities: Even though human perception is limited to certain modalities, e.g., touch, sight, sound, and smell, we can still understand unperceivable ones such as magnetism by utilizing inherently-possessed senses, e.g., feel the force when pulling two magnets together; or see the magnetic field by observing the alignment of iron filings around a magnet.
+
+Based on this insight, we utilize IM perceptrons that contain prior knowledge to encode known IM data, which can be implemented using existing MML models (Radford et al., 2021; Girdhar et al., 2023; Zhu et al., 2023; Wang et al., 2024d), and an OOM learner which learns novel modalities without any prior knowledge. By analyzing the interactions between latent features, we show theoretically and empirically that the OOM learner can be trained to gradually discover the OOM knowledge, as shown in Figure 2. First, we consider semi-supervised OOM generalization where few correspondences are given. Based on the correspondence, we can capture the prior probability distribution and learn mappings that connect OOM data and IM data. Through an information-theoretic perspective, we propose Connect&Explore (COX), which encourages the agreement on mappings across modalities, further sharing the cross-modal knowledge and exploring OOM knowledge. Then, we extend COX to an unsupervised OOM generalization scenario where there is no instance-level correspondence at all. To tackle such a challenge, we enhance the OOM-IM connections by maximizing cross-modal interaction. First, we select data pairs from cross-modal mappings according to feature similarity. By assuming that the data pairs closing to OOM mappings can be considered as correspondence, we can create emerging correspondence and solve the unsupervised case via the semi-supervised solution. To validate the proposed COX, we carefully design experiments using various multi-modal datasets to validate its effectiveness. Moreover, we provide extensive analyses in various scenarios to understand our method and inspire future research. To sum up, our contributions are three-fold:
+
+- We discover a novel and practical problem named Out-of-Modal Generalization, which aims to explore a novel modality using the knowledge from known modalities.  
+- We consider two typical situations: Semi-Supervised OOM generalization and Unsupervised OOM generalization, and propose a Connect&Explore framework to tackle both problems from an information-theoretic perspective.  
+- We conduct extensive experiments to tackle the OOM generalization on various datasets and provide intuitive insights to help inspire future research.
+
+# 2 RELATED WORK
+
+Modality Generalization (Liu et al., 2024) generally focuses on leveraging the knowledge from some modalities and generalizing to another one. Existing studies are conducted in different settings and with various tasks. Cross-modal fine-tuning mimics transfer learning by adapting the distribution of IM data to OOM data using the same model. Shen et al. (2023) proposed to conduct distribution alignment to achieve this goal which requires both pre-trained knowledge and labeled target modality data. Based on a similar problem setting, Cai et al. (2024c) designed a gradual modality generation scheme that selects the top- $k$  active feature patches from target modalities, and replaces them with source modalities patches. Such a progressive strategy can align target modal data to ensure generalization. Cross-Modal Generalization uses separate encoders and focus on generalizing to a different modality data from the same instance. Liang et al. (2021) used meta-learning
+
+Table 1: A comparison of different MML problems and their corresponding settings.  
+
+<table><tr><td>Problem</td><td>References</td><td>IM Knowledge</td><td>OOM Knowledge</td><td>Correspondence</td></tr><tr><td>Cross-Modal Fine-Tuning</td><td>Shen et al. (2023); Cai et al. (2024c)</td><td>pre-trained &amp; labeled</td><td>labeled</td><td>X</td></tr><tr><td rowspan="2">Cross-Modal Generalization</td><td>Liang et al. (2021)</td><td>pre-trained &amp; labeled</td><td>pre-trained</td><td>✓</td></tr><tr><td>Xia et al. (2024)</td><td>pre-trained &amp; labeled</td><td>pre-trained &amp; labeled</td><td>✓</td></tr><tr><td>MML w/o labeled Multi-Modal Data</td><td>Liang et al. (2023)</td><td>partially labeled</td><td>partially labels</td><td>✓</td></tr><tr><td rowspan="2">OOM Generalization</td><td>Semi-Supervised case (Section 3.3)</td><td>pre-trained &amp; labeled</td><td>scarcely labeled</td><td>A few</td></tr><tr><td>Unsupervised case (Section 3.4)</td><td>pre-trained &amp; labeled</td><td>X</td><td>X</td></tr></table>
+
+to align OOM data to IM space and generalize to OOM tasks dynamically. Xia et al. (2024) studied a different setting where IM and OOM data are both known during training. Then, a unified representation space is learned to help with the downstream generalization of OOM data. Some other studies consider generalization when all modalities are available, Ma et al. (2019) studied cross-modal generalization without paired data, Wang et al. (2023) applied the information bottleneck to CLIP training, Fang et al. (2024) conducted multi-modal fusion under limited clinical data, and Dong et al. (2023) considered domain generalization with fully-paired multi-modal data. A recent study MML without Labeled Multi-Modal Data (Liang et al., 2023) proposed a different setting where both IM and OOM data have labels, but they are not paired. Instead, additional unlabeled paired multi-modal data is given for learning the interaction between modalities. Moreover, Xue et al. (2022) understood the interactions and applied it to knowledge distillation. Except for cross-modal fine-tuning which follows transfer learning, existing MML works mostly require instance-level correspondence. This work proposes OOM Generalization, where there is no correspondence and the OOM knowledge is barely provided. The comparison of related works is shown in Table 1.
+
+Modality Binding aims to learn a joint embedding space across different modalities. Contrastive Language-Image Pre-training CLIP (Radford et al., 2021) is the first work that aligns image with language data. Then, ImageBind (Girdhar et al., 2023) proposed to use vision modalities to bind various modalities into the same representation space. Further, LanguageBind (Zhu et al., 2023) proposed using language as an alternative solution, which binds various modalities similarly. Recently, FreeBind (Wang et al., 2024d) extended the existing unified space into an additional expert space. Specifically, two types of binding were considered, namely space displacement bond and space combination bind. Since modality binding often requires a large amount of data with correspondence, the selected modalities are often quite common. Therefore, the OOM generalization problem can take advantage of the development of modality binding by leveraging the encoders as our IM perceptrs to learn novel modalities.
+
+# 3 OOM GENERALIZATION
+
+In this section, we first formalize the OOM generalization setting. Then, we demonstrate the proposed method. Further, we consider a Semi-Supervised case where a few correspondences are available and an Unsupervised scenario where there is no correspondence, showing that the proposed method can successfully tackle both settings and effectively leverage unpaired OOM data.
+
+# 3.1 PROBLEM SETTING
+
+In OOM generalization, we are given a set of known modalities  $\{\mathcal{M}_1^{\mathrm{I}},\ldots ,\mathcal{M}_K^{\mathrm{I}}\}$  where  $\mathcal{M}_{k\in \{1,\dots,K\}}^{\mathrm{I}} = \{(x_{k,i}^{\mathrm{I}},y_{k,i}^{\mathrm{I}})_{i = 1}^{N}\in \mathcal{X}\times \mathcal{Y}\}$  is composed of  $N$  number of labeled IM examples with its subscript  $i$  denoting the correspondence across different modalities. Moreover, we have an unknown modality  $\mathcal{M}^{\mathrm{O}} = \{(x_j^{\mathrm{O}})_{j = 1}^M\}$  containing  $M$  unlabeled OOM examples. In some cases, it is possible to obtain few correspondences with IM data, then our OOM data could be  $\mathcal{M}^{\mathrm{O}} = \{(x_i^{\mathrm{O}},y_i^{\mathrm{O}})\}_{i = 1}^L\cup \{(x_j^{\mathrm{O}})\}_{j = L + 1}^M$ , where  $L\ll M$  and the subscript  $i$  traces the corresponding IM data instance and label.
+
+To tackle OOM generalization, we propose a learning framework as shown in Figure 2. Particularly, we use a set of IM perceptrs  $\{g_1^{\mathrm{I}},\dots ,g_K^{\mathrm{I}}\}$  to perceive IM data, which can be realized by many existing modality-binding models, such as ImageBind (Girdhar et al., 2023) and LanguageBind (Zhu et al., 2023).
+
+![](images/2ab556ddfac505ab6a466ba3ed1ab5dbf145a617074040fa35f28c61e95d52f7.jpg)  
+Figure 2: Learning framework of our OOM generalization.
+
+Then, the features of IM data are obtained via  $z_{k}^{\mathrm{I}} = g_{k}^{\mathrm{I}}(x_{k}^{\mathrm{I}})$ . Moreover, we use an OOM learner  $g^{\mathrm{O}}$  to learn features  $z^{\mathrm{O}}$  from OOM data through  $z^{\mathrm{O}} = g^{\mathrm{O}}(x^{\mathrm{O}})$ . Our goal is to effectively generalize to OOM data by exploring the relationships between the OOM feature  $z^{\mathrm{O}}$  and IM features  $\{z_k^{\mathrm{I}}\}_{k=1}^K$ . Note that we only focus on the generalization performance of OOM data, the improvement of learning IM data is not the goal of this paper. Therefore, we freeze the parameters of all IM perceptrs and only train the OOM learner during experiments. On top of the above models, we further define classifiers  $h^{\mathrm{O}}(x^{\mathrm{O}}) \coloneqq h^{\mathrm{O}}(x^{\mathrm{O}}; g^{\mathrm{O}})$  and  $h_{k}(x_{k}^{\mathrm{I}}) \coloneqq h_{k}(x_{k}^{\mathrm{I}}; g_{k}^{\mathrm{I}})$  that make predictions.
+
+# 3.2 METHODOLOGY: CONNECT&EXPLORE (COX)
+
+Here we elucidate the proposed method based on the interactive relationship between modalities (Liang et al., 2023; Williams & Beer, 2010). Specifically, the total information of two modalities under a certain task is decomposed into 1) commonality<sup>1</sup> which indicates common attributes across modalities, 2) uniqueness that is only presented in each modality, and 3) synergy denoting the emerging information when modalities are presented together. Note that we do not consider 3) in this paper as our goal is generalizing to OOM data.
+
+To generalize to an unknown modality based on common ones, we aim to extract the commonality that can help partially comprehend OOM data based on IM data. Then, we model the posterior distribution of OOM data by selecting anchor points with minimum uniqueness. To this end, the OOM generalization can be successfully established. The proposed COX method comprises two steps: 1) learning connections by mapping IM data to OOM data to extract commonality, and 2) exploring high uniqueness OOM data by matching their posterior to high-commonality OOM data.
+
+Connection through Commonality aims to capture common knowledge across modalities using generative models (Lu, 2024). Here we follow the variational information bottleneck (VIB) framework (Alemi et al., 2016) to achieve this goal. We assume that given IM data  $X^{\mathrm{I}}$  and OOM data  $X^{\mathrm{O}}$ , the latent variable  $V$  extracted from  $X^{12}$ , and label  $Y$ , the joint distribution is factorized as
+
+$$
+p \left(X ^ {\mathrm {I}}, X ^ {\mathrm {O}}, V, Y\right) = p \left(V, Y \mid X ^ {\mathrm {O}}, X ^ {\mathrm {I}}\right) p \left(X ^ {\mathrm {O}} \mid X ^ {\mathrm {I}}\right) P \left(X ^ {\mathrm {I}}\right), \tag {1}
+$$
+
+where we assume  $p(V, Y|X^{\mathrm{O}}, X^{\mathrm{I}}) = p(V|X^{\mathrm{I}})p(Y|X^{\mathrm{I}})$ , corresponding to the Markov chains  $V \leftrightarrow X^{\mathrm{I}} \leftrightarrow X^{\mathrm{O}}$  and  $X^{\mathrm{I}} \leftrightarrow Y \nleftrightarrow X^{\mathrm{O}}$ . Such an assumption means that  $V$  is not related to  $X^{\mathrm{O}}$  (Alemi et al., 2016) and the given label  $Y$  is not directly connected to  $X^{\mathrm{O}}$  under our OOM setting. Intuitively, given an IM datum, i.e., dog image, it is sufficient to infer the label "dog", and the same for inferring from an unknown OOM datum, i.e., dog bark. Thus, in common multi-modal settings, the label prediction using IM information dog image is not further conditioned on OOM knowledge dog bark, because here the OOM knowledge is redundant when IM data is given.
+
+Our goal is to extract valuable knowledge from IM data to leverage OOM data by maximizing the information commonality (Liang et al., 2023; Williams & Beer, 2010):
+
+$$
+\max  I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}; Y\right) = I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) - I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}} \mid Y\right), \tag {2}
+$$
+
+where  $I(X^{\mathrm{O}};X^{\mathrm{I}};Y)$  denotes the mutual information between  $X^{\mathrm{O}}$  and  $X^{\mathrm{I}}$  regarding the task  $Y$ , i.e., the label; and  $I(X^{\mathrm{O}};X^{\mathrm{I}}|Y)$  indicates the conditional mutual information irrelevant to  $Y$ . We start with the first term:
+
+$$
+I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) = \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}\right) \log \frac {p \left(x ^ {\mathrm {O}} , x ^ {\mathrm {I}}\right)}{p \left(x ^ {\mathrm {O}}\right) p \left(x ^ {\mathrm {I}}\right)} = \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}\right) \log \frac {p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right)}{p \left(x ^ {\mathrm {O}}\right)}, \tag {3}
+$$
+
+where  $p(x^{\mathrm{O}}|x^{\mathrm{I}}) = \int dvp(x^{\mathrm{O}},v|x^{\mathrm{I}}) = \int dvp(x^{\mathrm{O}}|v)p(v|x^{\mathrm{I}})$  can be approximated via a decoder  $q(x^{\mathrm{O}}|v)$ . Since the Kullback Leibler (KL) divergence is always non-negative, we have  $\mathrm{KL}[p(X^{\mathrm{O}}|V)\parallel q(X^{\mathrm{O}}|V)]\geq 0\Rightarrow \int dx^{\mathrm{O}}p(x^{\mathrm{O}}|v)\log p(x^{\mathrm{O}}|v)\geq \int dx^{\mathrm{O}}p(x^{\mathrm{O}}|v)\log q(x^{\mathrm{O}}|v)$ , and leveraging Jensen's inequity, we can have
+
+$$
+\begin{array}{l} I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}\right) \log \frac {\int d v q \left(x ^ {\mathrm {O}} | v\right) p \left(v | x ^ {\mathrm {I}}\right)}{p \left(x ^ {\mathrm {O}}\right)} (4) \\ \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d v p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}\right) \log q \left(x ^ {\mathrm {O}} | v\right) p \left(v \mid x ^ {\mathrm {I}}\right) + H \left(X ^ {\mathrm {O}}\right), (5) \\ \end{array}
+$$
+
+where the last term is independent of our optimization process. Further, we rewrite  $p(x^{\mathrm{O}}, x^{\mathrm{I}}) = \int dvp(x^{\mathrm{O}}, x^{\mathrm{I}}, v) = \int dvp(x^{\mathrm{I}})p(x^{\mathrm{O}}|x^{\mathrm{I}})p(v|x^{\mathrm{I}})$ . Then, we have the following lower bound:
+
+$$
+I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d v p \left(x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(v \mid x ^ {\mathrm {I}}\right) \log q \left(x ^ {\mathrm {O}} \mid v\right) p \left(v \mid x ^ {\mathrm {I}}\right), \tag {6}
+$$
+
+which is realized by sampling from the joint data distribution, the latent variable from our encoder  $p(v|x^{\mathrm{I}})$ , and the tractable variational approximation  $q(x^{\mathrm{O}}|v)$ .
+
+Similarly, we can upper-bound the second term  $I(X^{\mathrm{O}};X^{\mathrm{I}}|Y)$  (details shown in Appendix A):
+
+$$
+I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}} | Y\right) \leq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}, y\right) \log p (y | x ^ {\mathrm {I}}) p \left(x ^ {\mathrm {O}} | x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {I}}\right) - \log h ^ {\mathrm {O}} \left(y | x ^ {\mathrm {O}}\right), \tag {7}
+$$
+
+where  $h^{\mathrm{O}}(y|x^{\mathrm{O}})$  is our classifier model for predicting OOM data. To this end, we can lower-bound our objective by combining Eqs. 6 and 7:
+
+$$
+\begin{array}{l} I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}; Y\right) \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d v p \left(x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(v \mid x ^ {\mathrm {I}}\right) \log q \left(x ^ {\mathrm {O}} \mid v\right) p \left(v \mid x ^ {\mathrm {I}}\right) \tag {8} \\ - \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p (x ^ {\mathrm {O}}, x ^ {\mathrm {I}}, y) \log p (y | x ^ {\mathrm {I}}) p (x ^ {\mathrm {O}} | x ^ {\mathrm {I}}) p (x ^ {\mathrm {I}}) + \log h ^ {\mathrm {O}} (y | x ^ {\mathrm {O}}) = \mathcal {L} _ {\mathrm {c o n}}. \\ \end{array}
+$$
+
+The above lower bound contains two parts: 1) OOM data reconstruction where we reconstruct  $X^{\mathrm{O}}$  using the latent  $V$  and 2) OOM data label prediction where we model the label distribution  $Y$ . In practice, we can approximate  $p(x^{\mathrm{O}}, x^{\mathrm{I}}, y)$  using empirical samples from IM and OOM data. Moreover, we use encoder  $p(v|x^{\mathrm{I}})$  without any prior assumptions because we can leverage the feature distribution from the pre-trained IM perceptrs. Additionally, a classifier  $h(y|x^{\mathrm{O}})$  is optimized to categorize OOM data based on given labels. Empirically, we can minimize
+
+$$
+\mathcal {L} _ {\text {c o n}} := \frac {1}{M} \sum_ {i = 1} ^ {M} \| x _ {i} ^ {\mathrm {O}} - q \left(x _ {i} ^ {\mathrm {O}} \mid v _ {i}\right) p \left(v _ {i} \mid x _ {i} ^ {\mathrm {I}}\right) \| _ {2} ^ {2} - \log h ^ {\mathrm {O}} \left(y _ {i} \mid x _ {i} ^ {\mathrm {O}}\right), \tag {9}
+$$
+
+where we use the reconstruction error  $\| \cdot \|_2^{23}$  to realize the log-likelihood  $q(x^{\mathrm{O}}|v)p(v|x^{\mathrm{I}})$ , as similarly done by Kingma & Welling (2013). After building the connections, we can ensure the taskrelevant information shared across modalities is learned, which helps partially understand OOM data regarding its commonality. However, note that the second term in Eq. 8 is not fully leveraged which contains  $p(y|x^{\mathrm{I}})$  modeled by the IM perceptrs. Take a step further, we can obtain  $-\int dx^{\mathrm{O}}dx^{\mathrm{I}}dyp(x^{\mathrm{O}},x^{\mathrm{I}},y)\log \frac{p(y|x^{\mathrm{I}})p(x^{\mathrm{O}}|x^{\mathrm{I}})p(x^{\mathrm{I}})}{h^{\mathrm{O}}(y|x^{\mathrm{O}})}$ . Since  $p(x^{\mathrm{O}}|x^{\mathrm{I}})p(x^{\mathrm{I}})$  is fixed in label prediction, we can derive  $-\mathrm{KL}(p(y|x^{\mathrm{I}})\parallel h^{\mathrm{O}}(y|x^{\mathrm{O}}))$  which implies that the label information related IM data can be harnessed to explore commonality. Next, we demonstrate how the commonality helps OOM generalization, and provide a solution to explore uniqueness.
+
+Exploration of Uniqueness can be achieved via selecting and exploring the OOM data with high uniqueness. To identify these data, we can leverage the agreement and disagreement achieved by the optimal classifiers from various IM data. Our final goal is to optimize via
+
+$$
+\min  _ {h ^ {\mathrm {O}}} \operatorname {K L} \left(h ^ {\mathrm {O}} \left(y \mid x _ {d} ^ {\mathrm {O}}\right) \| h ^ {\mathrm {O}} \left(y \mid x _ {a} ^ {\mathrm {O}}\right)\right), \text {w h e r e} x _ {d} ^ {\mathrm {O}} \in \mathcal {D}, x _ {a} ^ {\mathrm {O}} \in \mathcal {A}, \tag {10}
+$$
+
+in which  $h_1^*$  and  $h_2^*$  denote the optimal classifiers found in two IM data  $x_1^{\mathrm{I}}$  and  $x_2^{\mathrm{I}}$ , respectively, and  $x_d^{\mathrm{O}}$  and  $x_a^{\mathrm{O}}$  are selected from OOM data with modality disagreement  $\mathcal{D} := \{x^{\mathrm{O}} : h_1^*(x^{\mathrm{O}}) \neq h_2^*(x^{\mathrm{O}})\}$  and agreement  $\mathcal{A} := \{x^{\mathrm{O}} : h_1^*(x^{\mathrm{O}}) = h_2^*(x^{\mathrm{O}})\}$ , respectively. Here we use two inmodalities for simplicity, but the conclusion can be extended to multiple modalities. Moreover, the data with agreement is considered anchor points that guide the exploration of those with disagreement. This objective aims to match the posterior of OOM data with uniqueness  $h^{\mathrm{O}}(y|x_d^{\mathrm{O}})$  to the one of anchor points  $h^{\mathrm{O}}(y|x_a^{\mathrm{O}})$ . To justify this, we first define modality disagreement:
+
+Definition 1 (Modality disagreement). Given  $X_{1}, X_{2}$  and target  $Y$ , as well as their corresponding optimal classifiers  $h_{1}^{*}$  and  $h_{2}^{*}$ , their modality disagreement is defined as  $\alpha(h_{1}^{*}, h_{2}^{*}) = \mathbb{E}_{p(x_{1}, x_{2})}[d(h_{1}^{*}, h_{2}^{*})]$  where  $d: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R}^{+}$  is a distance function in the label space scoring the disagreement between  $h_{1}^{*}$  and  $h_{2}^{*}$ .
+
+Theorem 1. Given two Bayes' optimal classifiers  $h_1^*$  and  $h_2^*$  from two in-modalities, under relaxed triangle inequality, inverse Lipschitz condition, and classifier optimality assumptions (Sridharan & Kakade, 2008), the modalities disagreement is upper-bounded by (see details in Appendix B)
+
+$$
+\alpha \left(h _ {1} ^ {*}, h _ {2} ^ {*}\right) \leq I \left(X ^ {\mathrm {O}}, X _ {2} ^ {\mathrm {I}}, Y \mid X _ {1} ^ {\mathrm {I}}\right) + I \left(X ^ {\mathrm {O}}, X _ {1} ^ {\mathrm {I}}, Y \mid X _ {2} ^ {\mathrm {I}}\right) + 2 I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right). \tag {11}
+$$
+
+Finally, based on the decomposition of the task-related mutual information of  $X^{\mathrm{O}}$ :  $I(X^{\mathrm{O}}, Y) = I(X^{\mathrm{O}}, X_2^{\mathrm{I}}, Y|X_1^{\mathrm{I}}) + I(X^{\mathrm{O}}, X_1^{\mathrm{I}}, Y|X_2^{\mathrm{I}}) + I(X^{\mathrm{O}}, Y|X_1^{\mathrm{I}}, X_2^{\mathrm{I}}) + I(X^{\mathrm{O}}, X_1^{\mathrm{I}}, X_2^{\mathrm{I}}, Y)$ , as shown in Figure 3, we can achieve
+
+$$
+\alpha \left(h _ {1} ^ {*}, h _ {2} ^ {*}\right) \leq I \left(X ^ {\mathrm {O}}, Y\right) - I \left(X ^ {\mathrm {O}}, X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, Y\right) + I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right), \tag {12}
+$$
+
+where the first term denotes the overall information, the second term indicates the commonality shared between all modalities, and the third term stands for the uniqueness only preserved in OOM data. Intuitively, when we try to increase the modality disagreement, the commonality is decreased and OOM uniqueness is increased, which successfully justifies our learning objective: In order to explore the uniqueness of OOM data, we can explore the ones with high modality disagreement; conversely, the OOM data with high commonality and low uniqueness is found where agreement is achieved among  $h_1^*$  and  $h_2^*$ . Therefore, we select such
+
+![](images/1308088f8ffd279464c13b0795592203c9e81c3ca9e79b9030d8cc7a3de127e1.jpg)  
+Figure 3: Decomposition of  $I(X^{\mathrm{O}}, Y)$ .
+
+data as anchor points that provide informative guidance to help explore uniqueness.
+
+Next, we consider two realistic scenarios of OOM generalization and demonstrate how the proposed COX method can tackle them.
+
+# 3.3 SEMI-SUPERVISED OOM GENERALIZATION
+
+We start with a semi-supervised case where a few correspondences are available in OOM data, as shown in Figure 4 (a). Based on the VIB framework proposed in Section 3.2, we first leverage the OOM data  $\{(x_i^{\mathrm{O}},y_i^{\mathrm{O}})\}_{i = 1}^{L}$  corresponding to IM data  $\{(x_{k,i}^{\mathrm{I}},y_{k,i}^{\mathrm{O}})\}_{i = 1}^{L},\forall k\in$ $\{1,\dots ,K\}$  to build  $K$  connections using additional generative models that can be trained via a point-topoint mapping. As a result, the map
+
+![](images/bc69f06b5c159e93ae942f44ae5b517be2b8f42986322fc6aea23a58f2f8a18b.jpg)  
+(a)
+
+![](images/41c1fcd9908452a5e11040761fcbbd95bf37892ec94d58f537463c2266162d87.jpg)  
+(b)  
+Figure 4: Two scenarios: (a) Semi-Supervised OOM Generalization and (b) Unsupervised OOM Generlaizaiton.
+
+pings on the OOM feature space can successfully match the OOM feature distribution, which allows us to directly apply IM data posteriors to select and explore the uniqueness of OOM data. Hence, we formulate our objective as
+
+$$
+\min  _ {h ^ {\mathrm {O}}} \mathcal {L} _ {\mathrm {s s l}} := \frac {1}{L} \sum_ {i = 1} ^ {L} \operatorname {C E} \left(h ^ {\mathrm {O}} \left(x _ {i} ^ {\mathrm {O}}\right), y _ {i} ^ {\mathrm {O}}\right) + \frac {1}{L + | \mathcal {D} |} \sum_ {x _ {d, j} \in \mathcal {D}} \sum_ {x _ {i = 1} ^ {L}} \operatorname {K L} \left(h ^ {\mathrm {O}} \left(x _ {d, j} ^ {\mathrm {O}}\right) \| h ^ {\mathrm {O}} \left(x _ {i} ^ {\mathrm {O}}\right); h _ {1} ^ {*}, h _ {2} ^ {*}\right), \tag {13}
+$$
+
+where the first term exploits labeled OOM data with correspondence and the second term explores OOM data  $\mathcal{D}$  with modality disagreement by minimizing its KL divergence from the label posterior. Through the above objective, we can maximally exploit the uniqueness of OOM data to achieve effective OOM generalization.
+
+# 3.4 UNSUPERVISED OOM GENERALIZATION
+
+As for the unsupervised case, we propose two-phase training: 1) we first conduct a warm-up training to initialize the OOM feature space and the connection, and 2) then, we enhance the connection by creating emergent correspondence and further exploring OOM data.
+
+Specifically, we select anchor points from OOM data by directly applying modality agreement among all Bayes' optimal classifiers from IM data via
+
+$$
+\mathcal {A} _ {\text {s o r t e d}} = \operatorname {S O R T} _ {T} (\mathcal {A}, \frac {1}{K} \sum_ {k = 1} ^ {K} \max  h _ {k} ^ {*} (x ^ {\mathrm {O}})), \text {w h e r e} \mathcal {A} = \left\{\forall x ^ {\mathrm {O}} \in \mathcal {M} ^ {\mathrm {O}}: h _ {1} ^ {*} (x ^ {\mathrm {O}}) = \dots = h _ {K} ^ {*} (x ^ {\mathrm {O}}) \right\}, \tag {14}
+$$
+
+where the  $\mathrm{SORT}_T(\cdot, \cdot)$  is a sort function, which ranks each element  $x^{\mathrm{O}}$  in  $\mathcal{A}$  based on the value of  $\frac{1}{K}\sum_{k=1}^{K}\max h_k^*(x^{\mathrm{O}})$  from large to small. Here, we select anchor points with the top- $T$  largest likelihood averaged over all  $K$  IM classifiers. Then, we warm up the OOM learner via minimizing cross-entropy loss  $\min \frac{1}{T}\sum_{x^{\mathrm{O}}\in \mathcal{A}_{\mathrm{sorted}}} \mathrm{CE}(h^{\mathrm{O}}(x^{\mathrm{O}}), \arg \max h_k^*(x^{\mathrm{O}}))$ . Additionally, we also warm up the connection by leveraging class-wise information. Specifically, we compute the cluster centroids for each modality via  $\frac{1}{|\mathcal{C}_y|}\sum_{x_i^{\mathrm{O}}\in \mathcal{C}_y: = \{x^{\mathrm{O}}: h^{\mathrm{O}}(x^{\mathrm{O}}) = y, y \in \mathcal{Y}\}}z_i^{\mathrm{O}}$  and pair them to each IM centroid correspondingly. To this end, we can build up initial connections by following the VIB framework.
+
+After the warm-up, we aim to further enhance both our connection and OOM exploration by creating emergent correspondence, as shown in Figure 4 (b). To tackle this, we map all IM data into the OOM feature space. If an OOM feature is close to all mappings  $v_{k,i}, \forall k = \{1,\dots ,K\}$ , then they can form a strong correspondence. Further, we select such OOM data as anchor points, which is further labeled the same as the corresponding IM data. Formally, we optimize OOM learners via
+
+$$
+\min  _ {h ^ {\mathrm {O}}} \mathcal {L} _ {\text {u n s}} := \frac {1}{| \mathcal {A} |} \sum_ {\left(x _ {a} ^ {\mathrm {O}}, y\right) \in \mathcal {A}} \operatorname {C E} \left(h ^ {\mathrm {O}} \left(x _ {a} ^ {\mathrm {O}}\right), y\right) + \frac {1}{| \mathcal {A} | + | \mathcal {D} |} \sum_ {x _ {d} ^ {\mathrm {O}} \in \mathcal {D}} \sum_ {x _ {a} ^ {\mathrm {O}} \in \mathcal {A}} \operatorname {K L} \left(h ^ {\mathrm {O}} \left(x _ {d} ^ {\mathrm {O}}\right) \| h ^ {\mathrm {O}} \left(x _ {a} ^ {\mathrm {O}}\right); h _ {1} ^ {*}, h _ {2} ^ {*}\right), \tag {15}
+$$
+
+where  $\mathcal{A}$  denotes the updated anchor points which are realized by sorting the Euclidean distance:  $\mathcal{A} := \mathrm{SORT}_S(\{(x_j^{\mathrm{O}},y_i^{\mathrm{I}})\}_{j = 1}^{M}, - \min_{i\in \{1,\dots,N\}}\frac{1}{K}\sum_{k = 1}^{K}\| z_j^{\mathrm{O}} - v_{k,i}\|),$  where the first term computes the cross-entropy loss from the anchor points, and the second term calculates the KL divergence between the OOM data with modality disagreement and the OOM anchor points.
+
+After these two steps, we can effectively tackle the unsupervised OOM generalization. In practice, we connect modalities and select anchor points in the feature space, and hence our application to both two scenarios can be efficient. In the next section, we carefully conduct extensive experiments to justify the effectiveness and extendibility of the proposed COX method under various settings.
+
+# 4 EXPERIMENTS
+
+In our experiments, we first elucidate the experimental details. Then, we provide performance comparisons to various baseline methods on different datasets. Finally, we conduct empirical analyses to provide an intuitive understanding of the proposed method.
+
+# 4.1 IMPLEMENTATION DETAILS
+
+Datasets. We consider datasets with at least three modalities: 1) TVL dataset (Fu et al., 2024) contains tactile sensing, RGB image, and class name which can be transformed into language; 2) LLVIP (Jia et al., 2021) dataset has infrared thermal data, RGB image, and annotations for pedestrian detection. We follow Zhu et al. (2023) to crop the pedestrian and background which stand for two classes. Further, we use the OpenAI template (Radford et al., 2021) to create language description; 3) NYU-D dataset (Silberman et al., 2012) contains RGB image, depth data, and class name that can be transformed into language description as well; 4) VGGS dataset (Chen et al., 2020a) includes video data, corresponding sound, and the language description; 5) MSR-VTT (Xu et al., 2016) includes videos and text description, we break down the videos into video frames and the audio data; 6) MOSEI dataset (Zadeh et al., 2018) contains videos from 7 classes of emotions, we extract audio data from the videos and use the emotion type to create language descriptions.
+
+Models. We employ two types of IM perceptrons, namely ImageBind (Girdhar et al., 2023) and LanguageBind (Zhu et al., 2023) which correspondingly contain 6 and 5 encoders to process different modalities. We select one modality for each experiment as OOM and then choose the rest as IM. For IM data, we use the existing encoders to extract their features. As for OOM data, we conduct preprocessing to ensure its compatibility. Then, we initialize an OOM learner from scratch using ViT-T/16 to learn from the OOM data using the guidance from IM perceptrors. Note that for the TVL dataset, there are no existing encoders to process tactile modality. Therefore, when the tactile modality is chosen as IM data, we fine-tune the encoder using contrastive learning on the training
+
+Table 2: Classification performance comparison of different methods across multiple datasets with different OOM modalities.  
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">IM Perceptor</td><td rowspan="2">Method</td><td colspan="3">TVL</td><td colspan="3">LLVIP</td><td colspan="3">NYU-D</td><td colspan="3">VGGS</td></tr><tr><td>RGB</td><td>Lan</td><td>Tac</td><td>RGB</td><td>Lan</td><td>The</td><td>RGB</td><td>Dep</td><td>Lan</td><td>Aud</td><td>Vid</td><td>Lan</td></tr><tr><td rowspan="10">Semi-Supervised</td><td rowspan="5">ImageBind</td><td>Random</td><td>0.4</td><td>0.3</td><td>0.2</td><td>48.2</td><td>47.3</td><td>51.0</td><td>10.2</td><td>11.3</td><td>10.2</td><td>0.3</td><td>0.3</td><td>0.3</td></tr><tr><td>ERM</td><td>23.1</td><td>19.5</td><td>22.7</td><td>54.6</td><td>53.1</td><td>54.1</td><td>45.2</td><td>44.5</td><td>38.1</td><td>9.3</td><td>10.2</td><td>8.4</td></tr><tr><td>EntMin</td><td>24.0</td><td>21.8</td><td>23.6</td><td>56.7</td><td>57.0</td><td>55.4</td><td>48.0</td><td>46.3</td><td>39.3</td><td>10.5</td><td>13.3</td><td>8.9</td></tr><tr><td>COX</td><td>31.2</td><td>25.3</td><td>26.5</td><td>59.2</td><td>58.3</td><td>58.3</td><td>52.3</td><td>50.7</td><td>44.2</td><td>16.8</td><td>18.4</td><td>11.7</td></tr><tr><td>aligned</td><td>79.5</td><td>29.8</td><td>35.8</td><td>65.4</td><td>61.8</td><td>63.4</td><td>61.8</td><td>54.0</td><td>52.7</td><td>27.8</td><td>29.3</td><td>19.1</td></tr><tr><td rowspan="5">LanguageBind</td><td>Random</td><td>0.4</td><td>0.3</td><td>0.2</td><td>48.2</td><td>47.3</td><td>51.0</td><td>10.2</td><td>11.3</td><td>10.2</td><td>0.3</td><td>0.3</td><td>0.3</td></tr><tr><td>ERM</td><td>23.6</td><td>20.1</td><td>22.6</td><td>56.5</td><td>54.9</td><td>58.3</td><td>44.8</td><td>44.5</td><td>39.9</td><td>9.8</td><td>13.7</td><td>9.9</td></tr><tr><td>EntMin</td><td>25.7</td><td>23.1</td><td>25.1</td><td>59.8</td><td>60.0</td><td>62.2</td><td>49.4</td><td>47.3</td><td>42.7</td><td>11.9</td><td>14.5</td><td>12.8</td></tr><tr><td>COX</td><td>33.5</td><td>26.3</td><td>27.3</td><td>61.2</td><td>62.3</td><td>66.4</td><td>58.8</td><td>53.5</td><td>48.4</td><td>18.3</td><td>22.1</td><td>13.4</td></tr><tr><td>aligned</td><td>81.6</td><td>31.2</td><td>38.3</td><td>74.1</td><td>73.2</td><td>87.2</td><td>68.6</td><td>65.1</td><td>57.7</td><td>38.6</td><td>32.5</td><td>20.9</td></tr><tr><td rowspan="8">Unsupervised</td><td rowspan="4">ImageBind</td><td>Random</td><td>0.4</td><td>0.3</td><td>0.2</td><td>48.2</td><td>47.3</td><td>51.0</td><td>10.2</td><td>11.3</td><td>10.2</td><td>0.3</td><td>0.3</td><td>0.3</td></tr><tr><td>SSL</td><td>6.3</td><td>4.3</td><td>5.1</td><td>52.3</td><td>56.1</td><td>52.4</td><td>14.6</td><td>13.6</td><td>18.9</td><td>2.5</td><td>6.9</td><td>3.8</td></tr><tr><td>COX</td><td>18.9</td><td>15.4</td><td>17.1</td><td>54.8</td><td>57.2</td><td>53.8</td><td>21.7</td><td>22.0</td><td>19.5</td><td>9.3</td><td>10.2</td><td>10.5</td></tr><tr><td>aligned</td><td>79.5</td><td>29.8</td><td>35.8</td><td>65.4</td><td>61.8</td><td>63.4</td><td>61.8</td><td>54.0</td><td>52.7</td><td>27.8</td><td>29.3</td><td>19.1</td></tr><tr><td rowspan="4">LanguageBind</td><td>Random</td><td>0.4</td><td>0.3</td><td>0.2</td><td>48.2</td><td>47.3</td><td>51:0</td><td>10.2</td><td>11.3</td><td>10.2</td><td>0.3</td><td>0.3</td><td>0.3</td></tr><tr><td>SSL</td><td>6.8</td><td>6.5</td><td>5.1</td><td>54.6</td><td>57.8</td><td>53.8</td><td>16.9</td><td>18.1</td><td>16.3</td><td>7.2</td><td>5.6</td><td>4.8</td></tr><tr><td>COX</td><td>19.3</td><td>19.2</td><td>18.6</td><td>55.0</td><td>56.4</td><td>55.7</td><td>24.5</td><td>23.1</td><td>20.4</td><td>10.0</td><td>11.6</td><td>10.4</td></tr><tr><td>aligned</td><td>81.6</td><td>31.2</td><td>38.3</td><td>74.1</td><td>73.2</td><td>87.2</td><td>68.6</td><td>65.1</td><td>57.7</td><td>38.6</td><td>32.5</td><td>20.9</td></tr></table>
+
+set. For ImageBind, the tactile encoder is aligned with the image encoder, and for LanguageBind, it is aligned with the language encoder, which is the same as the original training process. For training the connection between modalities, we employ multi-layer perceptrons to realize the variational information bottleneck framework. Moreover, to obtain the optimal classifier from each in-modality, we utilize the extracted features and train a linear layer as classification heads.
+
+Setup. We consider two scenarios of OOM generalization: For the semi-supervised case, we sample  $10\%$  of the training data as labeled data with each class having a balanced number of labels. For the unsupervised case, we have no labels at all. For selecting the number of anchor points, we choose the same number of examples for the warm-up and training phases, which is  $10\%$  of the total training set. To train the OOM learner, we use the Adam optimizer with an initial learning rate of  $1e - 3$  with weight decay  $1e - 5$ , and train the model for 50 epochs.
+
+Baseline methods. Since there is no existing baseline method to compare with under our setting, we implement four methods for comparison, namely: Random where the model is randomly initialized, ERM where only labeled data is used to minimize the empirical risk, EntMin (Grandvalet & Bengio, 2004) which minimize the entropy of unlabeled data meanwhile conduct ERM, SSL which conducts self-supervised learning using Gaussian noise perturbation on the input, and MoCo He et al. (2020) which updates model parameters with ensembling and meanwhile conducts contrastive learning. Note that we use MoCo for comparison for the retrieval task in Table 3 because it is not for classification, and it is combined with EntMin in the semi-supervised case. Moreover, we use a pretrained encoder as an upper-limit baseline "aligned". Next, we carefully compare the performance of our COX to these baseline methods.
+
+# 4.2 PERFORMANCE COMPARISON
+
+For performance comparisons, we conduct classification and cross-modal retrieval to validate the proposed COX. There are seven modalities are considered, namely RGB image, language, tactile, thermal, depth, audio, and video which are simplified as RGB, Lan, Tac, The, Dep, Aud, and Vid, respectively. For each column, we choose one modality as OOM data, the rest modalities are selected IM data. For the retrieval task, we report the recall rate in both top 1 (R@1) and top 5 (R@5). The results are shown in Tables 2 and 3. We can see that the proposed COX clearly shows the best performance in both scenarios. Specifically, COX can achieve more than  $5\%$  performance improvement for most of the OOM setting, which justifies that leveraging the knowledge from IM perceptrors can indeed help OOM generalization compared to using OOM data alone. Moreover, even though the performance is relatively limited compared to the fully pre-trained baseline under the unsupervised case, considering it is an extremely challenging setting, we can still largely improve the performance for over  $10\%$  compared to the Random baseline, which demonstrates that the unsupervised OOM generalization is indeed learnable further leads to a novel research direction for improving the gen
+
+Table 3: Cross-modal retrieval performance comparison of different methods across multiple datasets with different OOM modalities.  
+
+<table><tr><td rowspan="3">Setting</td><td rowspan="3">IM Perceptor</td><td rowspan="3">Method</td><td colspan="6">MSR-VTT</td><td colspan="6">MOSEI</td></tr><tr><td colspan="2">Aud</td><td colspan="2">Lan</td><td colspan="2">Vid</td><td colspan="2">Aud</td><td colspan="2">Lan</td><td colspan="2">Vid</td></tr><tr><td>R@1</td><td>R@5</td><td>R@1</td><td>R@5</td><td>R@1</td><td>R@5</td><td>R@1</td><td>R@5</td><td>R@1</td><td>R@5</td><td>R@1</td><td>R@5</td></tr><tr><td rowspan="12">Semi-Supervised</td><td rowspan="6">ImageBind</td><td>Random</td><td>5.4</td><td>25.1</td><td>5.0</td><td>25.4</td><td>5.4</td><td>24.2</td><td>14.3</td><td>42.5</td><td>14.4</td><td>42.8</td><td>14.1</td><td>42.1</td></tr><tr><td>ERM</td><td>15.6</td><td>30.3</td><td>16.1</td><td>35.2</td><td>18.5</td><td>38.2</td><td>28.0</td><td>45.3</td><td>29.3</td><td>47.1</td><td>33.4</td><td>48.2</td></tr><tr><td>EntMin</td><td>18.5</td><td>32.4</td><td>19.2</td><td>38.5</td><td>21.0</td><td>39.4</td><td>29.6</td><td>46.7</td><td>32.0</td><td>48.7</td><td>35.4</td><td>50.5</td></tr><tr><td>MoCo</td><td>20.5</td><td>33.9</td><td>21.1</td><td>38.9</td><td>23.4</td><td>43.5</td><td>30.1</td><td>47.3</td><td>32.7</td><td>50.1</td><td>36.2</td><td>51.0</td></tr><tr><td>COX</td><td>23.3</td><td>35.8</td><td>23.4</td><td>39.1</td><td>26.5</td><td>48.8</td><td>32.4</td><td>48.0</td><td>33.8</td><td>50.4</td><td>38.8</td><td>53.7</td></tr><tr><td>Aligned</td><td>35.5</td><td>51.5</td><td>32.3</td><td>52.4</td><td>36.8</td><td>61.8</td><td>42.9</td><td>66.4</td><td>48.2</td><td>69.4</td><td>50.5</td><td>71.6</td></tr><tr><td rowspan="6">LanguageBind</td><td>Random</td><td>5.2</td><td>24.3</td><td>5.4</td><td>25.1</td><td>5.0</td><td>25.6</td><td>13.5</td><td>43.1</td><td>14.2</td><td>42.7</td><td>14.6</td><td>41.9</td></tr><tr><td>ERM</td><td>16.3</td><td>31.1</td><td>16.5</td><td>36.2</td><td>18.7</td><td>37.9</td><td>27.3</td><td>45.5</td><td>28.4</td><td>47.6</td><td>33.4</td><td>49.3</td></tr><tr><td>EntMin</td><td>19.6</td><td>33.4</td><td>19.8</td><td>38.6</td><td>22.4</td><td>37.9</td><td>30.2</td><td>45.5</td><td>33.5</td><td>49.0</td><td>36.0</td><td>49.7</td></tr><tr><td>MoCo</td><td>21.1</td><td>34.8</td><td>20.9</td><td>39.2</td><td>24.5</td><td>38.6</td><td>31.1</td><td>46.7</td><td>34.5</td><td>50.5</td><td>37.0</td><td>51.7</td></tr><tr><td>COX</td><td>25.2</td><td>36.0</td><td>24.1</td><td>40.0</td><td>28.7</td><td>49.5</td><td>34.6</td><td>49.8</td><td>34.6</td><td>50.2</td><td>39.2</td><td>55.4</td></tr><tr><td>Aligned</td><td>42.0</td><td>53.6</td><td>38.8</td><td>58.6</td><td>44.8</td><td>70.0</td><td>44.6</td><td>68.9</td><td>49.5</td><td>67.4</td><td>51.1</td><td>68.3</td></tr><tr><td rowspan="10">Unsupervised</td><td rowspan="5">ImageBind</td><td>Random</td><td>5.4</td><td>25.1</td><td>5.0</td><td>25.4</td><td>5.4</td><td>24.2</td><td>14.3</td><td>42.5</td><td>14.4</td><td>42.8</td><td>14.1</td><td>42.1</td></tr><tr><td>SSL</td><td>8.9</td><td>28.4</td><td>9.3</td><td>28.1</td><td>10.1</td><td>29.5</td><td>17.4</td><td>48.8</td><td>16.2</td><td>45.2</td><td>16.0</td><td>45.0</td></tr><tr><td>MoCo</td><td>9.2</td><td>28.9</td><td>9.5</td><td>28.4</td><td>10.6</td><td>30.0</td><td>17.8</td><td>50.3</td><td>16.6</td><td>45.8</td><td>17.1</td><td>44.4</td></tr><tr><td>COX</td><td>13.5</td><td>30.4</td><td>16.5</td><td>32.4</td><td>15.2</td><td>34.8</td><td>20.8</td><td>53.7</td><td>18.7</td><td>46.7</td><td>18.2</td><td>48.9</td></tr><tr><td>Aligned</td><td>35.5</td><td>51.5</td><td>32.3</td><td>52.4</td><td>36.8</td><td>61.8</td><td>42.9</td><td>66.4</td><td>48.2</td><td>69.4</td><td>50.5</td><td>71.6</td></tr><tr><td rowspan="5">LanguageBind</td><td>Random</td><td>5.2</td><td>24.3</td><td>5.4</td><td>25.1</td><td>5.0</td><td>25,6</td><td>13.5</td><td>43.1</td><td>14.2</td><td>42.7</td><td>14.6</td><td>41.9</td></tr><tr><td>SSL</td><td>9.2</td><td>28.9</td><td>11.0</td><td>28.8</td><td>10.3</td><td>28.7</td><td>18.0</td><td>48.9</td><td>18.4</td><td>45.0</td><td>17.8</td><td>45.6</td></tr><tr><td>MoCo</td><td>9.6</td><td>29.4</td><td>11.1</td><td>28.5</td><td>11.0</td><td>29.3</td><td>18.8</td><td>50.7</td><td>18.5</td><td>45.2</td><td>18.0</td><td>45.5</td></tr><tr><td>COX</td><td>14.8</td><td>31.1</td><td>18.4</td><td>34.4</td><td>15.4</td><td>35.0</td><td>23.1</td><td>52.8</td><td>19.4</td><td>47.2</td><td>20.4</td><td>49.9</td></tr><tr><td>Aligned</td><td>42.0</td><td>53.6</td><td>38.8</td><td>58.6</td><td>44.8</td><td>70.0</td><td>44.6</td><td>68.9</td><td>49.5</td><td>67.4</td><td>51.1</td><td>68.3</td></tr></table>
+
+eralization performance. Additionally, note that the performance of COX is affected by the quality of IM perceptrs, as using LanguageBind shows relatively higher performance compared to using ImageBind. Thus, it would be potentially helpful to leverage sophisticated IM perceptrs to benefit the generalization performance.
+
+# 4.3 EMPIRICAL ANALYSIS
+
+To provide an intuitive justification for the proposed method, here we conduct empirical analyses using the MSR-VTT dataset on various OOM scenarios and modalities.
+
+![](images/0d376ad8f4a0c3af47f66f0edec11f6ba9545ee5652b992708a8be106c8ddfc4.jpg)  
+Figure 5: Connection effect on maximum mean discrepancy and accuracy across modalities.
+
+![](images/342379d7ab0e38cf6f1d0f23df4ec359a9ee4a89d97e489dd3ef6fb0844a8a17.jpg)
+
+![](images/751cccbbfc97a31f52f2dc2d76892ae5f3d40970c0185f1f5582479a5a2ad7b2.jpg)
+
+Connection mitigates modality gap. To understand the performance of our VIB-based connection learning, here we show its effect on generalization out-of-modal. Specifically, during connection training, we compute the maximum mean discrepancy (MMD) between the mapping of each IM data and the OOM data. Meanwhile, we evenly select 6 points during the training and extract the IM mappings which are used to learn a classification head as the optimal classifier. Based on our theoretical result, we apply the classifiers to OOM data and compute their accuracies, as shown in Figure 5. We can see that as training goes on, the MMD between each IM mapping and OOM data is decreasing and the corresponding accuracy is increasing, which shows that: 1) our connection can indeed close the modality gap between their features and 2) as the mappings of IM data getting close to OOM data, the optimal classifier shows better classification results on OOM data, which benefits the knowledge transfer from known modalities to unknown ones.
+
+Modality disagreement identifies uncertainty. To understand the effect of modality disagreement, we conduct an analysis under the semi-supervised scenario by training the OOM learner to use only labeled data for 10 epochs. Then, we leverage the modality disagreement criteria to separate OOM data into those with disagreement and agreement and show their prediction accuracies in
+
+Figure 6 (a). We can see that the accuracy for OOM data with disagreement is significantly lower than those with agreement, meaning that the prediction uncertainty, i.e., data with low accuracy, is effectively identified by the proposed modality disagreement.
+
+Modality agreement alleviates uncertainty. Further, we conduct training by following the procedure proposed in Section 3.3 and again show the accuracies of OOM data with disagreement and agreement in Figure 6 (b). We can see that the performance gap between the two types of data is largely mitigated, which justifies the methodology of exploring OOM data using the guid
+
+![](images/f480e839d5dfc99d48a8a5056c6d27fce3c8d945bd45102c35a39c3dd7a6a389.jpg)  
+Figure 6: Prediction accuracy of OOM data with modality disagreement and modalities agreement, respectively. (a) Before exploration. (b) After exploration.
+
+![](images/810b0cedb94f46c4d81004beba58c62c1767f56b1c9c9ebbf2bf6d1267e2ea37.jpg)
+
+ance of modality agreement. As a result, we can achieve almost comparable performance on both types of data, benefiting the overall generalization performance.
+
+Ablation study. Additionally, we conduct an ablation study to justify the effect of our methodology. Specifically, we consider three ablations: 1) "w/o connection" where we remove the connection and directly apply the modality disagreement criteria on the original features of IM data and OOM data, 2) "w/o exploration" where we only leverage the OOM data with agreement for training, 3) For unsupervised scenario, we consider "w/o warm-up" where we do not conduct the warm-up phase and directly
+
+training the model. The results in Table 4 show that all modules are essential for achieving effective OOM generalization. Specifically, the connection is vital for the knowledge transduction of IM data to OOM data, without which the generalization performance is largely degraded. The conclusion is consistent with the connection analysis where directly applying optimal classifiers across modalities leads to poor accuracy. Moreover, removing exploration also hinders the performance because the uniqueness of OOM data is largely ignored. Additionally, we find that the warm-up phase is essential for the unsupervised case. As initialized models have no classification capability, we need pre-training to form basic feature clusters that are consistent with IM data, further enabling effective OOM generalization.
+
+Discussion on computational efficiency. Note that we conduct the feature connection mostly on the feature space, the computational cost of training VIB framework work is quite acceptable. The main cost is training the OOM learner which is the basic training with cross-entropy loss optimization and can be implemented on a single NVIDIA 3090/4090 GPU.
+
+Table 4: Ablation study on various settings.  
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Ablation</td><td colspan="3">MSR-VTT R@1</td></tr><tr><td>Aud</td><td>Lan</td><td>Vid</td></tr><tr><td rowspan="3">Semi</td><td>w/o connection</td><td>8.7</td><td>7.9</td><td>10.3</td></tr><tr><td>w/o exploration</td><td>16.4</td><td>16.5</td><td>18.8</td></tr><tr><td>COX</td><td>25.2</td><td>24.1</td><td>40.0</td></tr><tr><td rowspan="2">Unsup.</td><td>w/o warm-up</td><td>7.4</td><td>11.5</td><td>10.5</td></tr><tr><td>COX</td><td>14.8</td><td>18.4</td><td>15.4</td></tr></table>
+
+# 5 CONCLUSION AND LIMITATION
+
+In this paper, we study a novel and promising research direction dubbed Out-of-Modal (OOM) Generalization which aims to leverage knowledge from existing modalities to generalize to an unknown modality without instance-level correspondence. We consider two scenarios where there are a few correspondences and there is no correspondence, i.e., semi-supervised and unsupervised cases, respectively. To tackle these problems, we propose a Connect&Explore (COX) method which first learns connections across modalities to extract common knowledge and then explores the unique knowledge of OOM data based on modality disagreement. Extensive experiments are conducted to justify the proposed method and intuitive insights are provided to inspire future studies. However, our research is limited to several aspects which we hope to address in the future. First, although challenging as it is, the performance is relatively limited compared to fully-aligned models, which requires more investigations to enhance generalization. Second, our OOM generalization is mostly conducted within the modalities from the same dataset. In the future, we hope to discover scenarios where the OOM data is from a different dataset with a large modality gap.
+
+# 6 ACKNOWLEDGEMENTS
+
+Bo Han was supported by RGC Young Collaborative Research Grant No. C2005-24Y and RIKEN Collaborative Research Fund. Masashi Sugiyama was supported by JST ASPIRE Grant Number JPMJAP2405. Tongliang Liu was partially supported by the following Australian Research Council projects: FT220100318, DP220102121, LP220100527, LP220200949, and IC190100031, he was also supported by JST ASPIRE Grant Number JPMJAP2405.
+
+# REFERENCES
+
+Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. In NeurIPS, volume 35, pp. 23716-23736, 2022.  
+Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv preprint arXiv:1612.00410, 2016.  
+David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In NeurIPS, volume 32, 2019.  
+Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021.  
+Chengyi Cai, Zesheng Ye, Lei Feng, Jianzhong Qi, and Feng Liu. Bayesian-guided label mapping for visual reprogramming. In NeurIPS, 2024a.  
+Chengyi Cai, Zesheng Ye, Lei Feng, Jianzhong Qi, and Feng Liu. Sample-specific masks for visual reprogramming-based prompting. In ICML, 2024b.  
+Lincan Cai, Shuang Li, Wenxuan Ma, Jingxuan Kang, Binhui Xie, Zixun Sun, and Chengwei Zhu. Enhancing cross-modal fine-tuning with gradually intermediate modality generation. arXiv preprint arXiv:2406.09003, 2024c.  
+Honglie Chen, Weidi Xie, Andrea Vedaldi, and Andrew Zisserman. Vggsound: A large-scale audiovisual dataset. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 721-725. IEEE, 2020a.  
+Jiaao Chen, Zichao Yang, and Diyi Yang. Mixtext: Linguistically-informed interpolation of hidden space for semi-supervised text classification. arXiv preprint arXiv:2004.12239, 2020b.  
+Runnan Chen, Youquan Liu, Lingdong Kong, Nenglun Chen, Xinge Zhu, Yuexin Ma, Tongliang Liu, and Wenping Wang. Towards label-free scene understanding by vision foundation models. In NeurIPS, volume 36, pp. 75896-75910, 2023a.  
+Runnan Chen, Youquan Liu, Lingdong Kong, Xinge Zhu, Yuexin Ma, Yikang Li, Yuenan Hou, Yu Qiao, and Wenping Wang. Clip2scene: Towards label-efficient 3d scene understanding by clip. In CVPR, pp. 7020-7030, 2023b.  
+Shiming Chen, Ziming Hong, Wenjin Hou, Guo-Sen Xie, Yibing Song, Jian Zhao, Xinge You, Shuicheng Yan, and Ling Shao. Transzero++: Cross attribute-guided transformer for zero-shot learning. IEEE transactions on pattern analysis and machine intelligence, 45(11):12844-12861, 2022a.  
+Shiming Chen, Ziming Hong, Yang Liu, Guo-Sen Xie, Baigui Sun, Hao Li, Qinmu Peng, Ke Lu, and Xinge You. Transzero: Attribute-guided transformer for zero-shot learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 330-338, 2022b.  
+Yongqiang Chen, Wei Huang, Kaiwen Zhou, Yatao Bian, Bo Han, and James Cheng. Understanding and improving feature learning for out-of-distribution generalization. In NeurlPS, volume 36, pp. 68221-68275, 2023c.
+
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.  
+Hao Dong, Ismail Nejjar, Han Sun, Eleni Chatzi, and Olga Fink. Simmmdg: A simple and effective framework for multi-modal domain generalization. NeurIPS, 36:78674-78695, 2023.  
+Yingying Fang, Shuang Wu, Sheng Zhang, Chaoyan Huang, Tieyong Zeng, Xiaodan Xing, Simon Walsh, and Guang Yang. Dynamic multimodal information bottleneck for multimodality classification. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 7696-7706, 2024.  
+Letian Fu, Gaurav Datta, Huang Huang, William Chung-Ho Panitch, Jaimyn Drake, Joseph Ortiz, Mustafa Mukadam, Mike Lambeta, Roberto Calandra, and Ken Goldberg. A touch, vision, and language dataset for multimodal alignment. In ICML, 2024. URL https://openreview.net/forum?id=tFEOOH9eH0.  
+Rohit Girdhar, Alaaeldin El-Nouby, Zhuang Liu, Mannat Singh, Kalyan Vasudev Alwala, Armand Joulin, and Ishan Misra. Imagebind: One embedding space to bind them all. In CVPR, 2023.  
+Madelyn Glymour, Judea Pearl, and Nicholas P Jewell. Causal inference in statistics: A primer. John Wiley & Sons, 2016.  
+Yves Grandvalet and Yoshua Bengio. Semi-supervised learning by entropy minimization. NeurIPS, 17, 2004.  
+Heidi E Harley, Erika A Putman, and Herbert L Roitblat. Bottlenose dolphins perceive object features through echolocation. Nature, 424(6949):667-669, 2003.  
+Masukichi Hashimoto. Origin of the compass. Memoirs of the Research Department of the Toyo Bunko (The Oriental Library), 1:69-92, 1926.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.  
+Kaiming He, Georgia Gkioxari, Piotr Dólar, and Ross Girshick. Mask r-cnn. In Proceedings of the IEEE international conference on computer vision, pp. 2961-2969, 2017.  
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, pp. 9729-9738, 2020.  
+Rundong He, Yicong Dong, Lan-Zhe Guo, Yilong Yin, and Tailin Wu. Re-evaluating the impact of unseen-class unlabeled data on semi-supervised learning model. In ICLR, 2025.  
+Dan Hendrycks, Steven Basart, Norman Mu, Saurav Kadavath, Frank Wang, Evan Dorundo, Rahul Desai, Tyler Zhu, Samyak Parajuli, Mike Guo, et al. The many faces of robustness: A critical analysis of out-of-distribution generalization. In ICCV, pp. 8340-8349, 2021.  
+Ziming Hong, Shiming Chen, Guo-Sen Xie, Wenhan Yang, Jian Zhao, Yuanjie Shao, Qinmu Peng, and Xinge You. Semantic compression embedding for generative zero-shot learning. In IJCAI, pp. 956-963, 2022.  
+Ziming Hong, Zhenyi Wang, Li Shen, Yu Yao, Zhuo Huang, Shiming Chen, Chuanwu Yang, Mingming Gong, and Tongliang Liu. Improving non-transferrable representation learning by harnessing content and style. In ICLR, 2024.  
+Yu Huang, Chenzhuang Du, Zihui Xue, Xuanyao Chen, Hang Zhao, and Longbo Huang. What makes multi-modal learning better than single (provably). In NeurIPS, volume 34, pp. 10944-10956, 2021.  
+Zhuo Huang, Li Shen, Jun Yu, Bo Han, and Tongliang Liu. Flatmatch: Bridging labeled data and unlabeled data with cross-sharpness for semi-supervised learning. In NeurIPS, volume 36, pp. 18474-18494, 2023a.
+
+Zhuo Huang, Xiaobo Xia, Li Shen, Bo Han, Mingming Gong, Chen Gong, and Tongliang Liu. Harnessing out-of-distribution examples via augmenting content and style. In ICLR, 2023b.  
+Zhuo Huang, Muyang Li, Li Shen, Jun Yu, Chen Gong, Bo Han, and Tongliang Liu. Winning prize comes from losing tickets: Improve invariant learning by exploring variant parameters for out-of-distribution generalization. International Journal of Computer Vision, pp. 1-19, 2024.  
+Xinyu Jia, Chuang Zhu, Minzhen Li, Wenqi Tang, and Wenli Zhou. Llvip: A visible-infrared paired dataset for low-light vision. In ICCV, pp. 3496-3504, 2021.  
+Yunfan Jiang, Agrim Gupta, Zichen Zhang, Guanzhi Wang, Yongqiang Dou, Yanjun Chen, Li Fei-Fei, Anima Anandkumar, Yuke Zhu, and Linxi Fan. Vima: Robot manipulation with multimodal prompts. In ICML, pp. 14975–15022. PMLR, 2023.  
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.  
+Mingyu Li, Tao Zhou, Zhuo Huang, Jian Yang, Jie Yang, and Chen Gong. Dynamic weighted adversarial learning for semi-supervised classification under intersectional class mismatch. ACM Transactions on Multimedia Computing, Communications and Applications, 20(4):1-24, 2024a.  
+Muyang Li, Runze Wu, Haoyu Liu, Jun Yu, Xun Yang, Bo Han, and Tongliang Liu. Instant: Semi-supervised learning with instance-dependent thresholds. In NeurIPS, volume 36, pp. 2922-2938, 2023.  
+Xiu-Chuan Li and Tongliang Liu. Efficient and trustworthy causal discovery with latent variables and complex relations. In ICLR, 2025.  
+Xiu-Chuan Li, Kun Zhang, and Tongliang Liu. Causal structure recovery with latent variables under milder distributional and graphical assumptions. In ICLR, 2024b.  
+Xiu-Chuan Li, Jun Wang, and Tongliang Liu. Recovery of causal graph involving latent variables via homologous surrogates. In ICLR, 2025.  
+Paul Pu Liang, Peter Wu, Liu Ziyin, Louis-Philippe Morency, and Ruslan Salakhutdinov. Cross-modal generalization: Learning in low resource modalities via meta-alignment. In Proceedings of the 29th ACM International Conference on Multimedia, pp. 2680–2689, 2021.  
+Paul Pu Liang, Chun Kai Ling, Yun Cheng, Alex Obolenskiy, Yudong Liu, Rohan Pandey, Alex Wilf, Louis-Philippe Morency, and Ruslan Salakhutdinov. Multimodal learning without labeled multimodal data: Guarantees and applications. arXiv preprint arXiv:2306.04539, 2023.  
+Runqi Lin, Chaojian Yu, Bo Han, and Tongliang Liu. On the over-memorization during natural, robust and catastrophic overfitting. arXiv preprint arXiv:2310.08847, 2023a.  
+Runqi Lin, Chaojian Yu, and Tongliang Liu. Eliminating catastrophic overfitting via abnormal adversarial examples regularization. NeurIPS, 36:67866-67885, 2023b.  
+Runqi Lin, Chaojian Yu, Bo Han, Hang Su, and Tongliang Liu. Layer-aware analysis of catastrophic overfitting: revealing the pseudo-robust shortcut dependency. arXiv preprint arXiv:2405.16262, 2024.  
+Yexiong Lin, Yu Yao, Xiaolong Shi, Mingming Gong, Xu Shen, Dong Xu, and Tongliang Liu. Csisolate: Extracting hard confident examples by content and style isolation. In NeurIPS, volume 36, pp. 58556-58576, 2023c.  
+Yexiong Lin, Yu Yao, and Tongliang Liu. Learning the latent causal structure for modeling label noise. In NeurIPS, volume 37, pp. 120549-120577, 2025.  
+Xiaohao Liu, Xiaobo Xia, Zhuo Huang, and Tat-Seng Chua. Towards modality generalization: A benchmark and prospective analysis. arXiv preprint arXiv:2412.18277, 2024.  
+Youquan Liu, Lingdong Kong, Jun Cen, Runnan Chen, Wenwei Zhang, Liang Pan, Kai Chen, and Ziwei Liu. Segment any point cloud sequences by distilling vision foundation models. In NeurIPS, volume 36, pp. 37193-37229, 2023.
+
+Zhou Lu. A theory of multimodal learning. NeurIPS, 36, 2024.  
+Shuang Ma, Daniel McDuff, and Yale Song. Unpaired image-to-speech synthesis with multimodal information bottleneck. In ICCV, pp. 7598-7607, 2019.  
+Jiquan Ngiam, Aditya Khosla, Mingyu Kim, Juhan Nam, Honglak Lee, and Andrew Y Ng. Multimodal deep learning. In ICML, pp. 689-696, 2011.  
+Maja Pantic and Leon JM Rothkrantz. Toward an affect-sensitive multimodal human-computer interaction. Proceedings of the IEEE, 91(9):1370-1390, 2003.  
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In ICML, pp. 8748-8763. PMLR, 2021.  
+Muhammad Arifur Rahman, David J Brown, Nicholas Shopland, Andrew Burton, and Mufti Mahmud. Explainable multimodal machine learning for engagement analysis by continuous performance test. In International Conference on Human-Computer Interaction, pp. 386-399. Springer, 2022.  
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. NeurIPS, 28, 2015.  
+Leslie Rice, Eric Wong, and Zico Kolter. Overfitting in adversarially robust deep learning. In ICML, pp. 8093-8104. PMLR, 2020.  
+Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A platform for embodied air research. In ICCV, pp. 9339-9347, 2019.  
+Junhong Shen, Liam Li, Lucio M Dery, Corey Staten, Mikhail Khodak, Graham Neubig, and Ameet Talwalkar. Cross-modal fine-tuning: Align then refine. In ICML, pp. 31030-31056. PMLR, 2023.  
+Nathan Silberman, Derek Hoiem, Pushmeet Kohli, and Rob Fergus. Indoor segmentation and support inference from rgbd images. In Computer Vision-ECCV 2012: 12th European Conference on Computer Vision, Florence, Italy, October 7-13, 2012, Proceedings, Part V 12, pp. 746-760. Springer, 2012.  
+Richard Socher, Milind Ganjoo, Christopher D Manning, and Andrew Ng. Zero-shot learning through cross-modal transfer. NeurIPS, 26, 2013.  
+Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raffel, Ekin Dogus Cubuk, Alexey Kurakin, and Chun-Liang Li. Fixmatch: Simplifying semi-supervised learning with consistency and confidence. In NeurIPS, volume 33, pp. 596-608, 2020.  
+Karthik Sridharan and Sham M Kakade. An information theoretic framework for multi-view learning. In  $COLT$ , number 114, pp. 403-414, 2008.  
+Xinwei Sun, Yilun Xu, Peng Cao, Yuqing Kong, Lingjing Hu, Shanghai Zhang, and Yizhou Wang. Tcgm: An information-theoretic framework for semi-supervised multi-modality learning. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part III 16, pp. 171-188. Springer, 2020.  
+Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971, 2023.  
+Weijie Tu, Weijian Deng, and Tom Gedeon. A closer look at the robustness of contrastive language-image pre-training (CLIP). In Thirty-seventh Conference on Neural Information Processing Systems, 2023. URL https://openreview.net/forum?id=wMNpMe0vp3.  
+Weijie Tu, Weijian Deng, Dylan Campbell, Stephen Gould, and Tom Gedeon. An empirical study into what matters for calibrating vision-language models. In ICML, 2024a. URL https://openreview.net/forum?id=qoxuPshrZb.
+
+Weijie Tu, Weijian Deng, and Tom Gedeon. Toward a holistic evaluation of robustness in clip models. arXiv preprint arXiv:2410.01534, 2024b.  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. NeurIPS, 30, 2017.  
+Haoyu Wang, Zhuo Huang, Zhiwei Lin, and Tongliang Liu. Noisegpt: Label noise detection and rectification through probability curvature. In NeurIPS, 2024a.  
+Qikai Wang, Rundong He, Yongshun Gong, Chunxiao Ren, Haoliang Sun, Xiaoshui Huang, and Yi-long Yin. Diverse teacher-students for deep safe semi-supervised learning under class mismatch. arXiv preprint arXiv:2405.16093, 2024b.  
+Qizhou Wang, Yong Lin, Yongqiang Chen, Ludwig Schmidt, Bo Han, and Tong Zhang. A sober look at the robustness of clips to spurious features. In NeurIPS, 2024c.  
+Qizhou Wang, Bo Han, Yang Liu, Chen Gong, Tongliang Liu, and Jiming Liu. W-doc: Wasserstein distribution-agnostic outlier exposure. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025.  
+Yidong Wang, Hao Chen, Qiang Heng, Wenxin Hou, Yue Fan, Zhen Wu, Jindong Wang, Marios Savvides, Takahiro Shinozaki, Bhiksha Raj, et al. Freematch: Self-adaptive thresholding for semi-supervised learning. arXiv preprint arXiv:2205.07246, 2022.  
+Ying Wang, Tim GJ Rudner, and Andrew G Wilson. Visual explanations of image-text representations via multi-modal information bottleneck attribution. NeurIPS, 36:16009-16027, 2023.  
+Zehan Wang, Ziang Zhang, Xize Cheng, Rongjie Huang, Luping Liu, Zhenhui Ye, Haifeng Huang, Yang Zhao, Tao Jin, Peng Gao, et al. Freebind: Free lunch in unified multimodal space via knowledge fusion. In ICML, 2024d.  
+LA Weinstein. Electromagnetic waves. Radio i svyaz', Moscow, 1988.  
+Paul L Williams and Randall D Beer. Nonnegative decomposition of multivariate information. arXiv preprint arXiv:1004.2515, 2010.  
+Yuhao Wu, Jiangchao Yao, Bo Han, Lina Yao, and Tongliang Liu. Unraveling the impact of heterophilic structures on graph positive-unlabeled learning. In ICML, 2024a.  
+Yuhao Wu, Jiangchao Yao, Xiaobo Xia, Jun Yu, Ruxin Wang, Bo Han, and Tongliang Liu. Mitigating label noise on graphs via topological sample selection. In ICML, 2024b.  
+Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In NeurIPS, volume 32, 2019.  
+Xiaobo Xia, Tongliang Liu, Bo Han, Nannan Wang, Mingming Gong, Haifeng Liu, Gang Niu, Dacheng Tao, and Masashi Sugiyama. Part-dependent label noise: Towards instance-dependent label noise. In NeurIPS, volume 33, pp. 7597-7610, 2020.  
+Xiaobo Xia, Bo Han, Nannan Wang, Jiankang Deng, Jiatong Li, Yinian Mao, and Tongliang Liu. Extended t: Learning with mixed closed-set and open-set noisy labels. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.  
+Yan Xia, Hai Huang, Jieming Zhu, and Zhou Zhao. Achieving cross modal generalization with multimodal unified representation. NeurIPS, 36, 2024.  
+Jun Xu, Tao Mei, Ting Yao, and Yong Rui. Msr-vtt: A large video description dataset for bridging video and language. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5288-5296, 2016.  
+Zihui Xue, Zhengqi Gao, Sucheng Ren, and Hang Zhao. The modality focusing hypothesis: Towards understanding crossmodal knowledge distillation. arXiv preprint arXiv:2206.06487, 2022.
+
+Yu Yao, Tongliang Liu, Bo Han, Mingming Gong, Jiankang Deng, Gang Niu, and Masashi Sugiyama. Dual t: Reducing estimation error for transition matrix in label-noise learning. NeurIPS, 33:7260-7271, 2020.  
+Yu Yao, Tongliang Liu, Mingming Gong, Bo Han, Gang Niu, and Kun Zhang. Instance-dependent label-noise learning under a structural causal model. In NeurIPS, volume 34, pp. 4409-4420, 2021.  
+Michihiro Yasunaga, Armen Aghajanyan, Weijia Shi, Richard James, Jure Leskovec, Percy Liang, Mike Lewis, Luke Zettlemoyer, and Wen-tau Yih. Retrieval-augmented multimodal language modeling. In ICML, 2023.  
+Youngjae Yu, Jiwan Chung, Heeseung Yun, Jack Hessel, Jae Sung Park, Ximing Lu, Rowan Zellers, Prithviraj Ammanabrolu, Ronan Le Bras, Gunhee Kim, et al. Fusing pre-trained language models with multimodal prompts through reinforcement learning. In CVPR, pp. 10845-10856, 2023.  
+Suqin Yuan, Lei Feng, and Tongliang Liu. Late stopping: Avoiding confidently learning from mislabeled examples. In ICCV, pp. 16079-16088, 2023.  
+Suqin Yuan, Lei Feng, and Tongliang Liu. Early stopping against label noise without validation data. In ICLR, 2024.  
+AmirAli Bagher Zadeh, Paul Pu Liang, Soujanya Poria, Erik Cambria, and Louis-Philippe Morency. Multimodal language analysis in the wild: Cmu-mosei dataset and interpretable dynamic fusion graph. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 2236-2246, 2018.  
+Han Zhang, Jing Yu Koh, Jason Baldridge, Honglak Lee, and Yinfei Yang. Cross-modal contrastive learning for text-to-image generation. In CVPR, pp. 833-842, 2021.  
+Liangli Zhen, Peng Hu, Xu Wang, and Dezhong Peng. Deep supervised cross-modal retrieval. In CVPR, pp. 10394-10403, 2019.  
+Jiyang Zheng, Weihao Li, Jie Hong, Lars Petersson, and Nick Barnes. Towards open-set object detection and discovery. In CVPR, pp. 3961-3970, 2022.  
+Jiyang Zheng, Yu Yao, Bo Han, Dadong Wang, and Tongliang Liu. Enhancing contrastive learning for ordinal regression via ordinal content preserved data augmentation. In ICLR, 2024.  
+Bin Zhu, Bin Lin, Munan Ning, Yang Yan, Jiaxi Cui, HongFa Wang, Yatian Pang, Wenhao Jiang, Junwu Zhang, Zongwei Li, et al. Languagebind: Extending video-language pretraining to n-modality by language-based semantic alignment. arXiv preprint arXiv:2310.01852, 2023.
+
+In the appendix, we first provide a detailed derivation of our VIB framework. Then, we carefully prove the modality disagreement in Theorem 1. Further, we provide additional experiments to evaluate our method. Finally, we give a discussion on the relationship between existing studies and provide a prospective outlook for future research.
+
+# A LOWER BOUND OF OUR VIB FRAMEWORK
+
+Recall that we have the following factorization:
+
+$$
+p \left(X ^ {\mathrm {I}}, X ^ {\mathrm {O}}, V, Y\right) = p \left(V, Y \mid X ^ {\mathrm {O}}, X ^ {\mathrm {I}}\right) p \left(X ^ {\mathrm {O}} \mid X ^ {\mathrm {I}}\right) P \left(X ^ {\mathrm {I}}\right), \tag {16}
+$$
+
+with Markov chains  $V \leftrightarrow X^{\mathrm{I}} \leftrightarrow X^{\mathrm{O}}$  and  $X^{\mathrm{I}} \leftrightarrow Y \nleftrightarrow X^{\mathrm{O}}$ . Our goal is to maximize the information redundancy (Liang et al., 2023; Williams & Beer, 2010):
+
+$$
+\max  I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}; Y\right) = I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) - I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}} \mid Y\right), \tag {17}
+$$
+
+where the first term is lower-bounded by:
+
+$$
+I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}\right) \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d v p \left(x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(v \mid x ^ {\mathrm {I}}\right) \log q \left(x ^ {\mathrm {O}} \mid v\right) p \left(v \mid x ^ {\mathrm {I}}\right), \tag {18}
+$$
+
+Then, we consider the second term  $I(X^{\mathrm{O}};X^{\mathrm{I}}|Y)$ :
+
+$$
+\begin{array}{l} I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}} \mid Y\right) = \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}, y\right) \log \frac {p \left(x ^ {\mathrm {O}} , x ^ {\mathrm {I}} \mid y\right)}{p \left(x ^ {\mathrm {O}} \mid y\right) p \left(x ^ {\mathrm {I}} \mid y\right)} (19) \\ = \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p \left(x ^ {\mathrm {O}}, x ^ {\mathrm {I}}, y\right) \log \frac {p \left(x ^ {\mathrm {O}} , x ^ {\mathrm {I}} , y\right)}{p \left(y \mid x ^ {\mathrm {O}}\right)} - H (Y) + H (Y \mid X ^ {\mathrm {I}}) + H \left(X ^ {\mathrm {O}}\right) + H \left(X ^ {\mathrm {I}}\right). (20) \\ \end{array}
+$$
+
+Note that we use the factorization  $p(x^{\mathrm{O}}, x^{\mathrm{I}}, y) = p(y|x^{\mathrm{I}})p(x^{\mathrm{O}}|x^{\mathrm{I}})p(x^{\mathrm{I}})$ , and further ignore the entropy terms<sup>4</sup>, then we have:
+
+$$
+I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}} | Y\right) \leq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p (y | x ^ {\mathrm {I}}) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {I}}\right) \log p (y | x ^ {\mathrm {I}}) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {I}}\right) - \log h \left(y \mid x ^ {\mathrm {O}}\right), \tag {21}
+$$
+
+which is based on the positivity of KL divergence between our classifier  $h(y|x^{\mathrm{O}})$  and  $p(y|x^{\mathrm{O}})$ .
+
+To this end, we can lower-bound our objective by combining Eqs. 18 and 21:
+
+$$
+\begin{array}{l} I \left(X ^ {\mathrm {O}}; X ^ {\mathrm {I}}; Y\right) \geq \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d v p \left(x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(v \mid x ^ {\mathrm {I}}\right) \log q \left(x ^ {\mathrm {O}} \mid v\right) p \left(v \mid x ^ {\mathrm {I}}\right) (22) \\ - \int d x ^ {\mathrm {O}} d x ^ {\mathrm {I}} d y p (y | x ^ {\mathrm {I}}) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {I}}\right) \log p (y | x ^ {\mathrm {I}}) p \left(x ^ {\mathrm {O}} \mid x ^ {\mathrm {I}}\right) p \left(x ^ {\mathrm {I}}\right) + \log h (y | x ^ {\mathrm {O}}) = \mathcal {L} _ {c o n}. (23) \\ \end{array}
+$$
+
+# B PROOF OF THEOREM 1
+
+Proof.
+
+Assumption 1 (Relaxed triangle inequality). For the distance function  $d:\mathcal{V}\times \mathcal{V}\to \mathbb{R}^{+}$ , there exists  $c_{d}\geq 1$  such that  $\forall \hat{y}_1,\hat{y}_2,\hat{y}_3\in \hat{\mathcal{V}} d(\hat{y}_1,\hat{y}_2)\leq c_d(d(\hat{y}_1,\hat{y}_3) + d(\hat{y}_2,\hat{y}_3))$
+
+Assumption 2 (Inverse Lipschitz condition). For the function  $d$ , it holds that  $\forall h, \mathbb{E}[d(h(x_1,x_2),h^* (x_1,x_2))] \leq |\mathcal{L}(h) - \mathcal{L}(h^*)|$ , where  $h^*$  is the Bayes optimal classifier on both  $x_{1}$  and  $x_{2}$ ; and  $\mathbb{E}[d(h(x),h^* (x))] \leq |\mathcal{L}(h) - \mathcal{L}(h^*)|$ , where  $h^*$  is the Bayes optimal classifier on  $x$ .
+
+Assumption 3 (Classifier optimality). For any classifiers  $h$  in comparison to the Bayes' optimal classifier  $h^*$ , there exists constants  $\epsilon > 0$  such that  $|\mathcal{L}(h) - \mathcal{L}(h^*)|^2 \leq \epsilon$ .
+
+To bridge  $h_1^*$  and  $h_2^*$ , we use  $h_{1,2}^*$  and  $h^*$  to denote the Bayes' optimal classifier on both IM data and all data, respectively. Then, we capture the relationship between the uniqueness of OOM data given both IM data and the difference in their Bayes' optimal prediction errors:
+
+$$
+\begin{array}{l} \left| \mathcal {L} \left(h _ {1, 2} ^ {*}\right) - \mathcal {L} \left(h ^ {*}\right) \right| ^ {2} = \left| \mathbb {E} _ {X} \mathbb {E} _ {Y | X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, X ^ {\mathrm {O}}} \ell \left(h ^ {*} \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right), y\right) - \mathbb {E} _ {X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}} \mathbb {E} _ {Y | X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}} \ell \left(h _ {1} ^ {*} \left(x _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, y\right) \right. \right| ^ {2} (24) \\ \leq \left| \mathbb {E} _ {Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, X ^ {\mathrm {O}}} \ell \left(h ^ {*} \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right), y\right) - \mathbb {E} _ {Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}} \ell \left(h _ {1} ^ {*} \left(x _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right), y\right) \right| ^ {2} (25) \\ \leq \operatorname {K L} \left(p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right) \| p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}\right)\right) (26) \\ \leq \mathbb {E} _ {X} \mathrm {K L} \left(p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right) \| p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}\right)\right) (27) \\ = I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right). (28) \\ \end{array}
+$$
+
+Then, we first capture the redundancy between one IM data and OOM data given another IM data:
+
+$$
+\begin{array}{l} \left| \mathcal {L} \left(h _ {1} ^ {*}\right) - \mathcal {L} \left(h ^ {*}\right) \right| ^ {2} = \left| \mathbb {E} _ {X} \mathbb {E} _ {Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, X ^ {\mathrm {O}}} \ell \left(h ^ {*} \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right), y\right) - \mathbb {E} _ {X _ {1} ^ {\mathrm {I}}} \mathbb {E} _ {Y \mid X _ {1} ^ {\mathrm {I}}} \ell \left(h _ {1} ^ {*} \left(x _ {1} ^ {\mathrm {I}}\right), y\right) \right| ^ {2} (29) \\ \leq \left| \mathbb {E} _ {Y | X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, X ^ {\mathrm {O}}} \ell \left(h ^ {*} \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right), y\right) - \mathbb {E} _ {Y | X _ {1} ^ {\mathrm {I}}} \ell \left(h _ {1} ^ {*} \left(x _ {1} ^ {\mathrm {I}}\right), y\right) \right| ^ {2} (30) \\ \leq \mathrm {K L} \left(p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right) \| p \left(y \mid x _ {1} ^ {\mathrm {I}}\right)\right) (31) \\ \leq \mathbb {E} _ {X} \mathrm {K L} \left(p \left(y \mid x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right) \| p \left(y \mid x _ {1} ^ {\mathrm {I}}\right)\right) (32) \\ = I \left(X ^ {\mathrm {O}}, X _ {2} ^ {\mathrm {I}}, Y \mid X _ {1} ^ {\mathrm {I}}\right). (33) \\ \end{array}
+$$
+
+Further leveraging triangle inequality through the Bayes' optimal classifier  $h^*$  and the inverse Lipschitz condition, we have:
+
+$$
+\begin{array}{l} \mathbb {E} _ {p \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right)} \left[ d \left(h _ {1} ^ {*}, h _ {1, 2} ^ {*}\right) \right] \leq \mathbb {E} _ {p \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right)} \left[ d \left(h _ {1} ^ {*}, h ^ {*}\right) \right] + \mathbb {E} _ {p \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}, x ^ {\mathrm {O}}\right)} \left[ d \left(h ^ {*}, h _ {1, 2} ^ {*}\right) \right] (34) \\ \leq \left| \mathcal {L} \left(h _ {1} ^ {*}\right) - \mathcal {L} \left(h ^ {*}\right) \right| ^ {2} + \left| \mathcal {L} \left(h ^ {*}\right) - \mathcal {L} \left(h _ {1, 2} ^ {*}\right) \right| ^ {2} (35) \\ \leq I \left(X ^ {\mathrm {O}}, X _ {2} ^ {\mathrm {I}}, Y \mid X _ {1} ^ {\mathrm {I}}\right) + I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right). (36) \\ \end{array}
+$$
+
+Symmetrically, we can have  $|\mathcal{L}(h_2^*) - \mathcal{L}(h^*)|^2 \leq I(X^{\mathrm{O}},X_1^{\mathrm{I}},Y|X_2^{\mathrm{I}})$  and further obtain  $\mathbb{E}_{p(x_2,x_2^{\mathrm{I}},x^{\mathrm{O}})}[d(h_2^*,h_{1,2}^*)]\leq I(X^{\mathrm{O}},X_1^{\mathrm{I}},Y|X_2^{\mathrm{I}}) + I(X^{\mathrm{O}},Y|X_1^{\mathrm{I}},X_2^{\mathrm{I}})$ . Then combining with Eq. 36:
+
+$$
+\mathbb {E} _ {p \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}\right)} \left[ d \left(h _ {1} ^ {*}, h _ {2} ^ {*}\right) \right] \leq I \left(X ^ {\mathrm {O}}, X _ {2} ^ {\mathrm {I}}, Y \mid X _ {1} ^ {\mathrm {I}}\right) + I \left(X ^ {\mathrm {O}}, X _ {1} ^ {\mathrm {I}}, Y \mid X _ {2} ^ {\mathrm {I}}\right) + 2 I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right) \tag {37}
+$$
+
+Finally, based on the decomposition of the task-related mutual information of  $X^{\mathrm{O}}$ :  $I(X^{\mathrm{O}}, Y) = I(X^{\mathrm{O}}, X_2^{\mathrm{I}}, Y|X_1^{\mathrm{I}}) + I(X^{\mathrm{O}}, X_1^{\mathrm{I}}, Y|X_2^{\mathrm{I}}) + I(X^{\mathrm{O}}, Y|X_1^{\mathrm{I}}, X_2^{\mathrm{I}}) + I(X^{\mathrm{O}}, X_1^{\mathrm{I}}, X_2^{\mathrm{I}}, Y)$ , as shown in Figure 3, we can achieve:
+
+$$
+\alpha \left(h _ {1} ^ {*}, h _ {2} ^ {*}\right) := \mathbb {E} _ {p \left(x _ {1} ^ {\mathrm {I}}, x _ {2} ^ {\mathrm {I}}\right)} \left[ d \left(h _ {1} ^ {*}, h _ {2} ^ {*}\right) \right] \leq I \left(X ^ {\mathrm {O}}, Y\right) - I \left(X ^ {\mathrm {O}}, X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}, Y\right) + I \left(X ^ {\mathrm {O}}, Y \mid X _ {1} ^ {\mathrm {I}}, X _ {2} ^ {\mathrm {I}}\right), \tag {38}
+$$
+
+![](images/b124e317d88561e87a8b0c6157daa6d88cd9ac7bfe6cd6e0d8f361bec3a61c18.jpg)
+
+# C ADDITIONAL EXPERIMENTS
+
+We conduct additional experiments to further justify the proposed COX. First, we study the performance benefits brought by COX under various correspondence rates in OOM data. Specifically, we choose MSR-VTT and NYU-D datasets and use Vid and Dep as OOM modalities, respectively, and show the result in Figure 7. First of all, we observe that COX brings more benefits when correspondence is more scarce. This is because sufficient correspondence can maximally uncover the knowledge of OOM data. As correspondence gets less, the knowledge that can be explored from correspondence decreases. However, COX leverages the knowledge from IM data which brings more benefits even with less correspondence. Thus, the increased benefits of COX under low-correspondence scenarios demonstrate its effectiveness in tackling OOM generalization without correspondence.
+
+Moreover, we testify how varied performance levels of IM perceptrs could affect the OOM performance. To achieve this, we change the number of IM data in each dataset as  $10\%$ ,  $40\%$ ,  $70\%$ , and
+
+![](images/304dc83210804ca39c6ba08360146b334708c715ed492b4c2407adfa16246c28.jpg)  
+Figure 7: Performance benefits brought by COX under various correspondence rate in OOM data.
+
+![](images/682eabdc72fa8a2aa5bf122c36c5b8d3bbd35d65d663df4b9eb0a5b576efec1d.jpg)
+
+![](images/64b78eefe8090d5c336a2de2918fd55081e2155039f91784e265ec08311be799.jpg)  
+Figure 8: Effect of Varying IM Perceptor Performance Level.
+
+![](images/5a5827508466a160ef06ce5cdb5012aa565db3b65c251025b87637ae189ef392.jpg)
+
+<table><tr><td>MSR-VTT</td><td>Vision</td><td>NYU-D</td><td>Language</td></tr><tr><td>FreeMatch</td><td>45.2</td><td>MixText</td><td>21.2</td></tr><tr><td>COX</td><td>52.3</td><td>COX</td><td>23.4</td></tr></table>
+
+Table 5: Comparison with competitive unimodal methods from Vision and Language.  
+
+<table><tr><td>Setting</td><td>Method</td><td>MSR-VTT</td><td>NYU-D</td></tr><tr><td rowspan="2">Unsup</td><td>MoCo</td><td>30.0</td><td>15.7</td></tr><tr><td>MoCo+COX</td><td>35.4</td><td>23.8</td></tr></table>
+
+Table 6: Combining COX with MoCo for knowledge extraction from OOM to IM data.
+
+$100\%$ , and test the OOM performance of COX, as shown in Figure 8. We can see that the OOM performance is significantly affected by the accuracy level of IM perceptrs. When the performance of IM perceptrs improves, the OOM performance of COX is also enhanced. Therefore, improving the performance of IM perceptrs is vital for OOM generalization using COX.
+
+Further, to understand the contribution of COX on uni-modal study, we conduct comparison and combination with uni-modal methods. First, we consider two uni-modalities vision and language from MSR-VTT and NYU-D datasets, respectively. By comparing to FreeMatch (Wang et al., 2022) and MixText (Chen et al., 2020b), two competitive semi-supervised learning methods that correspondingly deal with vision and language data, we show the performance of COX in Table 5. Even though the two baselines were effective under their original setting, their performance is still limited when applied to challenging multi-modal datasets with scarce knowledge. As we can see, COX still shows very effective performance compared to them, again justifying the benefits from COX by leveraging IM data.
+
+Then, we consider combining COX with the well-known unsupervised method MoCo (He et al., 2020) and show the performance benefits brought by COX for enhancing unknown modality. We show the result in Table 6. Delightfully, we observe significant performance improvement on both
+
+Vid from MSR-VTT and RGB from the NYU-D dataset. This is because COX unleashes the potential label information from IM data to enhance label prediction of OOM data, i.e., Vid and RGB here. Such a finding implies that COX can extract knowledge from other modalities to enhance new ones, which is the main goal of this study.
+
+# D RETROSPECTIVE AND PROSPECTIVE DISCUSSION
+
+OOM generalization without instance-level correspondence presents challenges that could be related to existing studies but differentiates from them. Here we identify the relationship between OOM generalization and existing fields and discuss how we can be inspired to solve OOM generalization in future studies.
+
+- Out-of-Distribution (OOD) Generalization (Hendrycks et al., 2021; Chen et al., 2023c; Huang et al., 2023b; 2024; Wang et al., 2025) where the goal is generalizing to unknown data distributions given several existing ones. The difference between OOM and OOD generalization lies in the data gap: the former faces modality gap while the latter deals with distribution gap. Intuitively, the modality gap demonstrates the change of learning space, thus it is more challenging than the distribution gap where data is still sampled within the same learning space, i.e., same dimension and formats. To solve OOM generalization, a comprehensive understanding of how modalities are generated is required. For example, the surface of an object affects the reflection of light, thus deciding its visual information. Meanwhile, the texture of the surface also influences its tactile sensing. Therefore, we hypothesize that some modalities are commonly influenced by a hidden factor (Glymour et al., 2016; Li et al., 2024b; 2025; Li & Liu, 2025; Lin et al., 2023c; 2025; Zheng et al., 2024). By identifying such a factor, the shared information across modalities could be discovered, further benefiting OOM generalization.
+
+- Trustworthy Machine Learning aims to develop reliable, robust, explainable models under realistic scenarios, which considers various types of problems, such as Semi-Supervised Learning (SSL) (Berthelot et al., 2019; Sohn et al., 2020; Huang et al., 2023a; He et al., 2025; Li et al., 2023; 2024a; Wang et al., 2024b), Learning with Noisy Labels (LNL) (Xia et al., 2019; 2020; 2022; Yao et al., 2021; 2020; Yuan et al., 2023; 2024; Wang et al., 2024a; Wu et al., 2024b;a), and Robustness (Rice et al., 2020; Lin et al., 2023a;b; 2024; Hong et al., 2024). Compared to the existing studies, OOM generalization without instance-level correspondence requires linking between data across modalities. During this process, unpaired data and noisy correspondence could occur, further hindering the generalization performance. To solve this problem, progressively generating pseudo correspondence and denoising the noisy ones as done by SSL and LNL might be helpful to complement the correspondence. Therefore, future studies for pairing multi-modal data through trustworthy machine learning techniques could be a promising direction.
+
+- Foundation Models (Chen et al., 2023b;a; Radford et al., 2021; Touvron et al., 2023; Bommasani et al., 2021; Liu et al., 2023; Cai et al., 2024b;a) are pre-trained on large-scale data to possess powerful zero-shot generalization abilities (Wang et al., 2024c; Hong et al., 2022; Chen et al., 2022b;a; Tu et al., 2023; 2024a;b; Zheng et al., 2022) have been widely used in practice nowadays. In OOM generalization, we rely on existing foundation models to encode modalities into a shared latent space. Therefore, the performance of our study is highly related to the capabilities of the foundation models. Common strategies either fuse all modalities into a uni-modal model or separately encode and align. Because of the cost of multi-modal correspondence, the former multi-modal fusion strategy might be impractical for large-scale applications. Therefore, alignment between pre-trained encoders could be a promising direction. To effectively organize multiple foundation models to achieve OOM generalization, more comprehensive studies on addressing the modality gap, modality imbalance, and alignment strategy are needed in future research.

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+# TOWARDS REALISTIC DATA GENERATION FOR REALWORLD SUPER-RESOLUTION
+
+Long Peng $^{1,2}$ , Wenbo Li $^{2\dagger}$ , Renjing Pei $^{2}$ , Jingjing Ren $^{3}$ , Jiaqi Xu $^{4}$ , Yang Wang $^{1,5\dagger}$ , Yang Cao $^{1}$ , Zheng-Jun Zha $^{1}$
+
+<sup>1</sup> USTC, <sup>2</sup> Huawei Noah's Ark Lab, <sup>3</sup> HKUST(GZ), <sup>4</sup> CUHK, <sup>5</sup> Chang'an University {longp2001@mail., ywang120@}ustc.edu.cn, liwenbo50@huawei.com
+
+# ABSTRACT
+
+Existing image super-resolution (SR) techniques often fail to generalize effectively in complex real-world settings due to the significant divergence between training data and practical scenarios. To address this challenge, previous efforts have either manually simulated intricate physical-based degradations or utilized learning-based techniques, yet these approaches remain inadequate for producing large-scale, realistic, and diverse data simultaneously. In this paper, we introduce a novel Realistic Decoupled Data Generator (RealDGen), an unsupervised learning data generation framework designed for real-world super-resolution. We meticulously develop content and degradation extraction strategies, which are integrated into a novel content-degradation decoupled diffusion model to create realistic low-resolution images from unpaired real LR and HR images. Extensive experiments demonstrate that RealDGen excels in generating large-scale, high-quality paired data that mirrors real-world degradations, significantly advancing the performance of popular SR models on various real-world benchmarks.
+
+# 1 INTRODUCTION
+
+Real-world image Super-Resolution (Real SR) is a fundamental problem in image processing, aiming to enhance the resolution and quality of images in real-world scenarios (Chen et al., 2022; Yu et al., 2024; Liu et al., 2023; Zhang et al., 2023c; Sun et al., 2023; Zhang et al., 2024). It has a wide range of applications across various fields, including photography (Chen et al., 2019) and medical imaging (Li et al., 2021), which enhance human visual perception and the robustness of vision systems (Haris et al., 2021; Noor et al., 2019; Gunturk et al., 2003; Chen et al., 2020a). However, traditional bicubic-interpolation-based Real SR methods have proven less effective in complex real-world scenarios due to the significant discrepancy between the bicubic pattern and real degradation (Chen et al., 2022; Liu et al., 2023; Cai et al., 2019; Wang et al., 2020; Chen et al., 2024). Consequently, substantial efforts have been directed towards developing methods for generating more realistic data to improve the generalization ability of Real SR models (Cai et al., 2019; Wei et al., 2020; Zhang et al., 2021; Wang et al., 2021b; Zhang et al., 2023b; Park et al., 2023; Li et al., 2022b; Xiao et al., 2020; Wolf et al., 2021; Luo et al., 2024; Sun & Chen, 2024; Hendrycks & Dietterich, 2019; Deng et al., 2023).
+
+To explore what kind of data contributes most to the SR model's capability, we synthesize different sets of training data using (Elad & Feuer, 1997) and evaluate the performance of FSRCNN (Dong et al., 2016). As shown in Figure 1(a), the red triangles represent test samples of the target domain, while rectangular boxes indicate training sets with different blur kernels and noise levels. The results clearly demonstrate that the closer the training data distribution is to the test data, the better the model's performance. This underscores the importance of designing a method that can adaptively generate accurate data for different target domains. Therefore, an ideal data generation system for Real SR should meet the following criteria: I) Large-scale, to satisfy the extensive data requirements for training deep learning models (Raicu et al., 2008; Maeda, 2020; Sehwag et al., 2022; Wu et al., 2023); II) Realistic, to enable Real SR models to accurately learn the characteristics of real-world degradation (Cai et al., 2019; Wei et al., 2020; Zhang et al., 2021; Ji et al., 2020; Liu et al., 2022);
+
+![](images/1557247922c000518afe1fcefa3f97d4eb936ea9ada378e88682c6773f2d667a.jpg)  
+(a) SR performance on different train data
+
+![](images/c3f0ce5b3bebccab1d26795d2b73094527c13de7a1c5380fcea5b8a651c654f9.jpg)  
+(c) Our proposed RealDGen
+
+![](images/6917a2d147d637f7bb5b7da56573ed6f7cee38b6d0fb30b53ab64d1fcfa31ffa.jpg)  
+(b) Degradation distribution  
+Figure 1: (a) and (b) are SR performance on different train data and degradation distribution of different methods. (c) is the pipeline of our unsupervised data generation framework RealDGen.
+
+and III) Adaptive, to flexibly generate data with arbitrary given degradation patterns, improving generalization in target domains (Liu et al., 2023; Zhang et al., 2023b; Chen et al., 2019; Lugmayr et al., 2019b; Mou et al., 2022).
+
+Existing data generation methods for Real SR (Cai et al., 2019; Wei et al., 2020; Zhang et al., 2021; Wang et al., 2021b; Maeda, 2020; Bulat et al., 2018; Yang et al., 2023; Lugmayr et al., 2020a; Yuan et al., 2018; Ignatov et al., 2017), as shown in Figure 1(b), can be broadly categorized into the following: a) Manual Collection via Focal Length Adjustment: This approach involves using digital single-lens reflex cameras (DSLRs) with varying focal lengths to capture images, followed by alignment (Cai et al., 2019; Wei et al., 2020). While it can produce realistic paired data, it is labor-intensive and often results in scene monotony and image misalignment, failing to meet the large-scale data requirement. b) Hand-Crafted Physical-Based Degradation Modeling: This method employs various degradation models (e.g., noise, blur, bicubic, JPEG) applied in single-order or higher-order combinations (Zhang et al., 2021; Wang et al., 2021b). Although efficient in generating large data quantities, the synthetic data often fails to accurately reflect the complex degradation patterns of real-world images, and its lack of adaptability to specific domains limits effectiveness. c) Learning-Based Methods: Techniques involving Generative Adversarial Networks (GANs) (Bulat et al., 2018) and diffusion models (Yang et al., 2023) are proposed to simulate realistic real-world degradation for low-resolution (LR) images. While these methods produce more realistic data compared to hand-crafted approaches, they often struggle to generalize to new and diverse domains, limiting their applicability in real-world scenarios. In summary, existing data generation methods face challenges in achieving both realism and adaptability: tailoring models to specific target domains may hinder their adaptability to new domains, and vice versa. Overcoming these challenges is crucial for advancing the field of Real SR.
+
+In this paper, we introduce a novel unsupervised learning framework Realistic Decoupled Data Generator (RealDGen) to meet the large-scale, realistic, and adaptive data generation criteria, as shown in Figure 1(c). RealDGen enhances degradation realism and content fidelity by separately modeling content and degradation through unsupervised learning and integrating them into a diffusion model to generate paired data. The training involves two steps: first, pre-training degradation and content extractors using contrastive and reconstruction learning to improve representation robustness; second, using these pre-trained extractors to condition the diffusion model with real LR degradation and content representations. To improve generalization to unknown LR distributions, the partial parameters of the extractors are fine-tuned. During data generation, unpaired HR and real LR images are used to extract content and degradation representations, which are then combined in the diffusion model. This process produces data that marries HR content with arbitrary LR degradation, improving adaptability to new domains. Extensive experiments show that RealDGen outperforms previous methods in generating realistic paired data and enhancing the performance of SR models in real-world scenarios.
+
+![](images/95b33d00c554d8622f59f3ba28e58e844bbe083080984ae7a383a0c84dac2ce7.jpg)  
+Figure 2: An overview of the training pipeline of our proposed RealDGen. We first train on the content and degradation extractors, then train Decoupled DDPM while fine-tuning the partial parameters of the extractors. RealDGen adaptively generates realistic LR images with arbitrarily given real LR images and unpaired HR images.
+
+![](images/3a76ed369e2d7f2f1e6ae4e1886c2257d6152727958c0c38602e45f3e9baa72e.jpg)
+
+The contributions of this paper can be summarized as follows:
+
+- We propose a novel unsupervised Realistic Decoupled Data Generator (RealDGen) to adaptively generate large-scale, realistic, and diverse data for real-world super-resolution.  
+- We introduce well-designed content and degradation extraction strategies and a novel content-degradation decoupled diffusion model to generate realistic LR with arbitrary unpaired LR and HR conditions.  
+- Compared with previous methods, our method significantly advances the generalization ability of popular SR models, achieving the best performance on real-world benchmarks.
+
+# 2 METHOD
+
+In this paper, we introduce a novel Realistic Decoupled Data Generator (RealDGen) for real-world super-resolution to adaptively generate large-scale, realistic, and diverse real paired data. In particular, well-designed content and degradation extractor learning strategies are proposed to capture robust content and degradation representations in the real world. A novel content-degradation decoupled diffusion model is proposed to generate realistic LR with arbitrary unpaired LR and HR conditions. The training process is divided into two distinct phases: (a) Content and Degradation Extractor Pre-training and (b) Decoupled DDPM Training and Extractor Fine-tuning, as shown in Figure 2.
+
+# 2.1 CONTENT AND DEGRADATION EXTRACTOR PRE-TRAINING
+
+To capture content and degradation representations, we propose dedicated degradation and content extractors, denoted as  $E_{\text{deg}}$  and  $E_{\text{cont}}$ , respectively. We employ reconstruction learning for training  $E_{\text{cont}}$ , as shown in the left of Figure 2. Specifically, for a given high-resolution (HR) image  $\mathcal{X} \in \mathbb{R}^{C \times H \times W}$ , we degrade it to a low-resolution (LR) counterpart  $\mathcal{X}_{lr} \in \mathbb{R}^{C \times h \times w}$  by Real-ESRGAN (Wang et al., 2021b) degradation model  $\mathcal{D}$  with diverse synthetic degradations. Subsequently,  $E_{\text{cont}}$  is engaged to extract the content representation  $F_{\text{cont}}$  from  $\mathcal{X}_{lr}$ . Thereafter, a HR reconstruction network  $\text{Rec}_{hr}$  is harnessed to reconstruct the HR image  $\hat{\mathcal{X}} \in \mathbb{R}^{C \times H \times W}$  from  $F_{\text{cont}}$ , as follows:
+
+$$
+F _ {\text {c o n t}} = E _ {\text {c o n t}} \left(\mathcal {X} _ {l r}\right), \hat {\mathcal {X}} = \operatorname {R e c} _ {h r} \left(F _ {\text {c o n t}}\right). \tag {1}
+$$
+
+The objective is to minimize the reconstruction loss  $\mathcal{L}_{rh}$  between the reconstructed image  $\hat{\mathcal{X}}$  and the original high-resolution image  $\mathcal{X}$ , as follows:
+
+$$
+\mathcal {L} _ {r h} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(\hat {\chi} _ {i} - \chi_ {i}\right) ^ {2} \tag {2}
+$$
+
+<table><tr><td colspan="2">Algorithm 1 Decoupled DDPM Training</td></tr><tr><td colspan="2">1: repeat</td></tr><tr><td colspan="2">2: xlr ~ q(xlr)</td></tr><tr><td colspan="2">3: t ~ Uniform({1,...,T})</td></tr><tr><td colspan="2">4: ε ~ N(0,I)</td></tr><tr><td colspan="2">5: Fcont = Econt(xlr)</td></tr><tr><td colspan="2">6: Fdeg = Edeg(xlr)</td></tr><tr><td colspan="2">7: c = M(Fcont, Fdeg)</td></tr><tr><td colspan="2">8: Take gradient descent step on ∇θ ||ε - εθ(√αt x0 + √1 - αt ε, c, t)||</td></tr><tr><td colspan="2">9: until converged</td></tr></table>
+
+<table><tr><td colspan="2">Algorithm 2 Data Generation</td></tr><tr><td colspan="2">1: xlr ~ q(xlr), xhr ~ p(xhr)</td></tr><tr><td colspan="2">2: c = M(Edeg(xlr), Econt(xhr))</td></tr><tr><td colspan="2">3: τ ~ Uniform({1,...,T})</td></tr><tr><td colspan="2">4: xt = (√αtD(xlr) + √1-αtε), ε ~ N(0,I)</td></tr><tr><td colspan="2">5: for t = τ,...,1 do</td></tr><tr><td colspan="2">6: z ~ N(0,I) if t &gt; 1, else z = 0</td></tr><tr><td colspan="2">7: xt-1 = 1/√αt(xt - 1-αt/√1-αtεθ(xt, c, t)) + σtz</td></tr><tr><td colspan="2">8: end for</td></tr><tr><td colspan="2">9: return x0</td></tr></table>
+
+where  $N$  denotes the batch size, which we empirically set to 64. After using LR-HR paired training,  $E_{cont}$  is able to learn a robust content representation under diverse degradation and real scenarios.
+
+Considering the variability of degradation in diverse scenarios and imaging devices in the real world, we advocate for contrastive learning (Chen et al., 2020b; Hermans et al., 2017) to curate positive and negative samples for training  $E_{deg}$ , which guarantees the uniqueness of the degradation representations. Specifically, for a HR image  $\mathcal{X} \in \mathbb{R}^{C \times H \times W}$ , we generate an LR image  $\mathcal{X}_{lr} \in \mathbb{R}^{C \times h \times w}$  by  $\mathcal{D}$  with parameter  $\theta$  as the anchor  $\mathcal{X}_{anc}$ . We further obtain a set of negative samples  $\mathcal{X}_{neg_i} \in \mathbb{R}^{C \times h \times w}$  by applying  $\mathcal{D}$  with different parameters  $\theta_i$  to  $\mathcal{X}$ , and a set of positive samples  $\mathcal{X}_{pos_i} \in \mathbb{R}^{C \times h \times w}$  by applying the degradation  $\mathcal{D}$  with same parameter  $\theta$  to different HR images  $\mathcal{X}_i' \in \mathbb{R}^{C \times H \times W}$ , as follows:
+
+$$
+\mathcal {X} _ {n e g} = \left\{\mathcal {D} \left(\mathcal {X}, \theta_ {1}\right), \mathcal {D} \left(\mathcal {X}, \theta_ {2}\right), \dots , \mathcal {D} \left(\mathcal {X}, \theta_ {n}\right) \right\}, \tag {3}
+$$
+
+$$
+\mathcal {X} _ {p o s} = \left\{\mathcal {D} \left(\mathcal {X} _ {1} ^ {\prime}, \theta\right), \mathcal {D} \left(\mathcal {X} _ {2} ^ {\prime}, \theta\right), \dots , \mathcal {D} \left(\mathcal {X} _ {n} ^ {\prime}, \theta\right) \right\}.
+$$
+
+The objective is to minimize the contrastive loss  $\mathcal{L}_{cl}$  to drive  $E_{deg}$  learn the uniqueness of the degradation representations in LR images, suppressing the interruption of content, as follows:
+
+$$
+\mathcal {L} _ {c l} = \sum_ {i = 1} ^ {n} \max  \left(0, d \left(E _ {d e g} \left(\mathcal {X} _ {a n c}\right), E _ {d e g} \left(\mathcal {X} _ {p o s _ {i}}\right)\right) - d \left(E _ {d e g} \left(\mathcal {X} _ {a n c}\right), E _ {d e g} \left(\mathcal {X} _ {n e g _ {i}}\right)\right) + m a r g i n\right). \tag {4}
+$$
+
+where  $d$  symbolizes the L2 distance,  $n$  is the number of samples, and we empirically set  $n$  and margin to 3 and 0.01, respectively. Furthermore, to drive  $E_{deg}$  to learn the complete degradation representations, we utilize the reconstruction learning strategy as aforementioned for supervising  $E_{deg}$ . Specifically, we utilize the pre-trained  $E_{cont}$  to learn the content representation of HR images  $\mathcal{X}_{hr}$  and  $E_{deg}$  to learn the degradation representations of  $\mathcal{X}_{lr}$ . We employ a low-resolution reconstruction network,  $Rec_{lr}$ , to combine these representations and reconstruct the LR image,  $\hat{\mathcal{X}}_{lr}$ , as follows:
+
+$$
+\hat {\mathcal {X}} _ {l r} = \operatorname {R e c} _ {l r} \left(E _ {\text {d e g}} \left(\mathcal {X} _ {l r}\right), E _ {\text {c o n t}} \left(\mathcal {X} _ {h r}\right)\right). \tag {5}
+$$
+
+The objective is to minimize the reconstruction loss  $\mathcal{L}_{rl}$  to drive  $E_{deg}$  learn the completeness of the degradation representations as follows:
+
+$$
+\mathcal {L} _ {r l} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(\hat {\chi} _ {l r _ {i}} - \chi_ {l r _ {i}}\right) ^ {2}. \tag {6}
+$$
+
+After training with well-designed learning strategies, our extractors effectively capture robust content and degradation representations. More details and analysis are provided in Appendix A.1 and A.2.
+
+# 2.2 DECOUPLED DDPM TRAINING AND EXTRACTOR FINE-TUNING
+
+We introduce a content-degradation Decoupled Diffusion Probabilistic Model (Decoupled DDPM) to generate real LR images. In detail, given real LR images  $\mathcal{X}_{lr}$  from real-world  $q$  encompassing various degradations, we extract their robust content representation  $F_{cont}$  and degradation representation  $F_{deg}$  by pre-trained  $E_{cont}$  and  $E_{deg}$ , respectively. To enhance the generalization of  $E_{deg}$  and  $E_{cont}$  on unseen real distributions, partial parameters are fine-tuned, as shown in Figure 2. A modulation block  $\mathcal{M}$  is introduced to adequately incorporate degradation representation into the content, formulated as:
+
+$$
+\mathbf {c} = \mathcal {M} \left(E _ {\text {d e g}} \left(\mathbf {x} _ {l r}\right), E _ {\text {c o n t}} \left(\mathbf {x} _ {l r}\right)\right). \tag {7}
+$$
+
+Then, this fused image representation is utilized as a condition c for controlling our Decouple DDPM to generate LR images. To make it clear, we illustrate the detailed training procedure of the Decouple DDPM, as shown in Algorithm 1. More details of fine-tuning and analysis of  $E_{cont}$  and  $E_{deg}$  are provided in Sections A.3, A.4 and A.5 of the appendix.
+
+# 2.3 DATA GENERATION
+
+We propose a novel strategy to generate realistic LR images using unpaired LR and HR images by decoupling content and degradation. First, we extract the degradation representation from a real-world LR image and the content representation from an HR image. These representations are combined in the modulation module  $\mathcal{M}$  to serve as the condition c for the diffusion model to generate LR images. The generated LR images retain the content of the HR image and the degradation of the real LR image, as shown on the right of Figure 1. To enhance fidelity, following (Meng et al., 2021), we denoise from an initial LR image  $\mathbf{x}_t$  with  $t$  steps of noise rather than from pure Gaussian noise. This initial LR image is degraded by  $\mathcal{D}$ . In Section 3.4, we analyze the step number  $T$ . Details of our data generation pipeline are in Algorithm 2.
+
+Although using content and degradation conditions improves the controllability and fidelity of Decoupled DDPM, the inherent stochasticity of the diffusion model (Ho et al., 2020; Rombach et al., 2022) can still introduce tiny artifacts and content distortion. To mitigate this, we propose a filtering mechanism. For each generated LR image, we re-extract content and degradation representations, then calculate the degradation error with the real LR image and the content error with the HR image. By selecting samples with the smallest errors, we reduce diffusion stochasticity and produce higher-fidelity LR images. More details and analysis are presented in Appendix A.3, A.5 and A.6.
+
+# 3 EXPERIMENTS AND ANALYSIS
+
+# 3.1 EXPERIMENTS SETTINGS
+
+Training details. We collect about 152,000 real low-resolution images from both public datasets (Wei et al., 2020; Cai et al., 2019; Ignatov et al., 2017) and those captured using smartphones to train RealDGen. Extractors and Decouple DDPM are trained with learning rate  $1 \times 10^{-4}$  and batch size 16 on 16 NVIDIA V100 GPUs. More details of the training setting are presented in Section A.1 and A.3 of the appendix. To ensure a fair comparison, we compare our approach with existing methods by using the widely-used DIV2K dataset (Agustsson & Timofte, 2017) as HR images and real LR images as degradation references to create the paired training data for training various popular SR models. To maintain a rigorous and fair comparison, we maintain consistency in all experimental settings and environments, with the exception of data generation. We utilize the public BasicSR for training Real-SR methods with 16 NVIDIA V100 GPUs.
+
+Compared data generation methods. We compare our methods with some state-of-the-art real-world data generation methods, including Hand-Crafted Physical-Based Degradation Models (BSRGAN (Zhang et al., 2021) and Real-ESRGAN (Wang et al., 2021b)) and learning-based degradation diffusion models proposed by (Yang et al., 2023), denoted as SynDiff for convenience.
+
+Real-SR models. To comprehensively validate the effectiveness of the generated data, we select five classic and representative backbone architectures for evaluation, including CNN-based model RRDB (Wang et al., 2018), transformer-based model SwinIR (Liang et al., 2021) and HAT (Chen et al., 2023b), diffusion-based model ResShift (Yue et al., 2024) and lightweight model SwinIR-L (Liang et al., 2021). To be fair, we conduct comparative evaluations under consistent experimental conditions and settings. We utilize L1Loss (Chen et al., 2023b; Liang et al., 2021) and perception GAN loss (Wang et al., 2021b; Johnson et al., 2016; Wang et al., 2018) for training PNSR-oriented and Perception-oriented Real SR models, respectively.
+
+Metrics. For the PNSR-oriented Real SR model, we adopt PSNR (Huynh-Thu & Ghanbari, 2008) and SSIM (Wang et al., 2004) to quantitatively evaluate the performance. For the perception-oriented Real-SR model, we adopt LPIPS (Zhang et al., 2018) and FID (Heusel et al., 2017) to quantitatively evaluate the performance. DISTS (Ding et al., 2020) and CLIP-Score (Radford et al., 2021) are further introduced to evaluate the accuracy of generated data. Note that the higher the PSNR, SSIM, and CLIP-Score, the better, and the lower the LPIPS, FID, and DISTS, the better.
+
+Table 1: Quantitative comparisons of PSNR-oriented and Perceptual-oriented training SR models on three real-world image super-resolution benchmarks. The best results are highlighted in bold.  
+
+<table><tr><td rowspan="2">PSNR-oriented Training</td><td colspan="2">RealSR</td><td colspan="2">DRealSR</td><td colspan="2">SmartPhone</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>SwinIR (Real-ESRGAN)</td><td>24.395</td><td>0.7760</td><td>26.944</td><td>0.8308</td><td>27.395</td><td>0.8338</td></tr><tr><td>SwinIR (BSRGAN)</td><td>25.852</td><td>0.7808</td><td>27.985</td><td>0.8308</td><td>28.049</td><td>0.8407</td></tr><tr><td>SwinIR (SynDiff)</td><td>25.589</td><td>0.7687</td><td>28.301</td><td>0.8309</td><td>28.566</td><td>0.8453</td></tr><tr><td>SwinIR (Ours)</td><td>26.094</td><td>0.7822</td><td>28.721</td><td>0.8341</td><td>28.737</td><td>0.8489</td></tr><tr><td>RRDB (Real-ESRGAN)</td><td>24.579</td><td>0.7614</td><td>27.131</td><td>0.8193</td><td>27.841</td><td>0.8378</td></tr><tr><td>RRDB (BSRGAN)</td><td>25.406</td><td>0.7685</td><td>27.523</td><td>0.8017</td><td>28.029</td><td>0.8278</td></tr><tr><td>RRDB (SynDiff)</td><td>25.488</td><td>0.7691</td><td>28.078</td><td>0.8257</td><td>28.303</td><td>0.8426</td></tr><tr><td>RRDB (Ours)</td><td>26.238</td><td>0.7747</td><td>28.727</td><td>0.8340</td><td>28.754</td><td>0.8507</td></tr><tr><td>HAT (Real-ESRGAN)</td><td>24.893</td><td>0.7726</td><td>27.339</td><td>0.8215</td><td>27.781</td><td>0.8336</td></tr><tr><td>HAT (BSRGAN)</td><td>25.997</td><td>0.7816</td><td>28.135</td><td>0.8273</td><td>28.137</td><td>0.8369</td></tr><tr><td>HAT (SynDiff)</td><td>25.790</td><td>0.7584</td><td>28.506</td><td>0.8286</td><td>28.508</td><td>0.8471</td></tr><tr><td>HAT (Ours)</td><td>26.140</td><td>0.7832</td><td>28.802</td><td>0.8345</td><td>28.767</td><td>0.8489</td></tr><tr><td>SwinIR-L (Real-ESRGAN)</td><td>24.367</td><td>0.7723</td><td>27.018</td><td>0.8244</td><td>27.581</td><td>0.8409</td></tr><tr><td>SwinIR-L (BSRGAN)</td><td>25.651</td><td>0.7800</td><td>27.813</td><td>0.8301</td><td>28.118</td><td>0.8437</td></tr><tr><td>SwinIR-L (SynDiff)</td><td>25.281</td><td>0.7516</td><td>28.170</td><td>0.8244</td><td>28.474</td><td>0.8487</td></tr><tr><td>SwinIR-L (Ours)</td><td>26.025</td><td>0.7810</td><td>28.869</td><td>0.8328</td><td>28.868</td><td>0.8522</td></tr><tr><td rowspan="2">Perceptual-oriented Training</td><td colspan="2">RealSR</td><td colspan="2">DRealSR</td><td colspan="2">SmartPhone</td></tr><tr><td>LPIPS↓</td><td>FID↓</td><td>LPIPS↓</td><td>FID↓</td><td>LPIPS↓</td><td>FID↓</td></tr><tr><td>SwinIR (Real-ESRGAN)</td><td>0.3037</td><td>69.965</td><td>0.3219</td><td>39.175</td><td>0.4053</td><td>78.242</td></tr><tr><td>SwinIR (BSRGAN)</td><td>0.2945</td><td>79.833</td><td>0.3023</td><td>38.541</td><td>0.3043</td><td>76.871</td></tr><tr><td>SwinIR (SynDiff)</td><td>0.3835</td><td>103.179</td><td>0.3801</td><td>54.588</td><td>0.3129</td><td>83.485</td></tr><tr><td>SwinIR (Ours)</td><td>0.2536</td><td>69.736</td><td>0.2660</td><td>38.257</td><td>0.2964</td><td>74.778</td></tr><tr><td>RRDB (Real-ESRGAN)</td><td>0.3480</td><td>82.056</td><td>0.3551</td><td>39.310</td><td>0.3480</td><td>77.573</td></tr><tr><td>RRDB (BSRGAN)</td><td>0.3041</td><td>77.412</td><td>0.3127</td><td>36.528</td><td>0.3381</td><td>78.812</td></tr><tr><td>RRDB (SynDiff)</td><td>0.4004</td><td>98.798</td><td>0.4017</td><td>56.573</td><td>0.3511</td><td>88.431</td></tr><tr><td>RRDB (Ours)</td><td>0.2972</td><td>76.973</td><td>0.3077</td><td>36.259</td><td>0.3125</td><td>76.723</td></tr><tr><td>HAT (Real-ESRGAN)</td><td>0.3066</td><td>79.209</td><td>0.3219</td><td>41.862</td><td>0.4022</td><td>87.950</td></tr><tr><td>HAT (BSRGAN)</td><td>0.2852</td><td>80.192</td><td>0.2835</td><td>41.723</td><td>0.3049</td><td>81.247</td></tr><tr><td>HAT (SynDiff)</td><td>0.3332</td><td>93.763</td><td>0.3465</td><td>50.808</td><td>0.3171</td><td>85.587</td></tr><tr><td>HAT (Ours)</td><td>0.2457</td><td>67.573</td><td>6.2587</td><td>41.319</td><td>0.2816</td><td>76.873</td></tr><tr><td>SwinIR-L (Real-ESRGAN)</td><td>0.3108</td><td>79.491</td><td>0.3234</td><td>42.986</td><td>0.4021</td><td>87.531</td></tr><tr><td>SwinIR-L (BSRGAN)</td><td>0.3013</td><td>84.195</td><td>0.2978</td><td>43.246</td><td>0.3146</td><td>83.444</td></tr><tr><td>SwinIR-L (SynDiff)</td><td>0.3793</td><td>98.646</td><td>0.3748</td><td>52.464</td><td>0.3190</td><td>80.708</td></tr><tr><td>SwinIR-L (Ours)</td><td>0.2795</td><td>75.779</td><td>0.2862</td><td>42.542</td><td>0.3047</td><td>78.632</td></tr></table>
+
+Evaluation. We utilize two public benchmarks to evaluate the performance of real-world image super-resolution methods, including the real-world dataset RealSR (Cai et al., 2019) and DRealSR (Wei et al., 2020) captured by digital single-lens reflex cameras (DSLRs). To further improve the diversity and quantity of real degradation scenarios, we have collected 891 pairs of data captured by smartphones for evaluation, denoted as SmartPhone.
+
+# 3.2 QUANTITATIVE RESULTS
+
+Generalization ability of Real SR models. We compare our method with the existing data generation methods on both PSNR-oriented and Perceptual-oriented Real SR models to validate the superiority of our method in boosting the generalization capabilities for real-world image super-resolution, and the results are shown in Table 1. We can observe that our method comprehensively improves the performance of PSNR-oriented and Perceptual-oriented SR models across three benchmarks. It's worth noting that our approach achieves a significant performance improvement, including  $0.75\mathrm{dB}$  in RRDB (Wang et al., 2018) on the PSNR of the RealSR benchmark;  $0.296\mathrm{dB}$  and  $0.699\mathrm{dB}$  in
+
+Table 2: Performance comparison of the diffusion-based ResShift model trained on our generated data versus Real-ESRGAN's simulated data on the SmartPhone and DRealSR benchmarks.  
+
+<table><tr><td>Benchmark</td><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>CLIP-IQA↑</td></tr><tr><td rowspan="2">SmartPhone</td><td>ResShift (Yue et al., 2024)</td><td>27.05</td><td>0.806</td><td>0.352</td><td>0.546</td></tr><tr><td>ResShift (Ours)</td><td>27.27</td><td>0.818</td><td>0.345</td><td>0.557</td></tr><tr><td rowspan="2">DRealSR</td><td>ResShift (Yue et al., 2024)</td><td>26.19</td><td>0.755</td><td>0.413</td><td>0.574</td></tr><tr><td>ResShift (Ours)</td><td>26.32</td><td>0.772</td><td>0.378</td><td>0.622</td></tr></table>
+
+![](images/b9ff6d3b77fdb835e925d5b73b98f93d6b1c310ef324b0ad8b468c0d67b8e357.jpg)  
+HR
+
+![](images/78e2ecb23e4a25f112dfcce15213cb3c75dd3d71ebfce4534afa5b1acbbb73ad.jpg)  
+BSRGAN
+
+![](images/cbac5445e223d333d2dd3c8e16c7cc7fbb751808b332a71c24d0dc3d837ac253.jpg)  
+Real-ESRGAN
+
+![](images/8973bd63e69a497cb79e1449c8c52e6b07d9e4373d12b56022c3ee8fac59a401.jpg)  
+SynDiff
+
+![](images/eee2f10b6dc4ab3c5ed812f86d00c937b600b1a012a39fb9e5f9ad3173a46b87.jpg)  
+Ours
+
+![](images/52e3633a538fe3e8f82843354d5eb97da6ac6ee07df4685c9bdad794d5b4a6a5.jpg)  
+Real LR
+
+![](images/9f479f28720d99935434649df18c78b6a9e5b475612e22e1b19abbc32f74c7e2.jpg)  
+HR
+
+![](images/432698187d651fd2957dddea500aad28a397a3a454a37da6d34c15dca53c1472.jpg)  
+BSRGAN
+
+![](images/2577687e7eb20d2ed952af60d7c730f7a1a71ad52ef08a37239255ee1bc073ba.jpg)  
+Real-ESRGAN
+
+![](images/498932a71832ca3ed33e4921db894d4553871d5b686e147a1d66a92826120f32.jpg)  
+SynDiff
+
+![](images/5d87c692e042861b087eda940051e18deab0f9e5980ee3662871305076a258e4.jpg)  
+Ours
+
+![](images/502be7ba7d13527f45aa4659fb99de20e788c45ed5442945ee3ccfa3f63a7299.jpg)  
+Real LR
+
+![](images/d0c6f17e0c8cc731c0a08d811a4af62e2b1b65f96dbf8a1768769d775c084be9.jpg)  
+HR
+
+![](images/51b17cf18489a1c631098df67c4fb2812e9ff3fbd1f6dc8b77430be9160f1aac.jpg)  
+BSRGAN
+
+![](images/089a675aa6287d5916954cf300ce71ae79964bcc52d591f75017ddc7efcab2da.jpg)  
+Real-ESRGAN
+
+![](images/912f875b93e5985c468581b65744b36f9683071c8778950dba40c6907d477b19.jpg)  
+SynDiff
+
+![](images/dddb7d20e5149bae5a63ea046a3944967fa313b2f0c4ba00a3ce999957e709e7.jpg)  
+Ours  
+Figure 3: Visual comparison of generated LR. Our method achieves the best visual results with realistic degradation and high fidelity. Please zoom in for better visualization.
+
+![](images/091226010d07745e2e4b48b0be9deea6c1081336137ab9ac4dac63b13f3762f0.jpg)  
+Real LR
+
+SwinIR (Liang et al., 2021) and light-weight SwinIR-L on the PSNR of the RealSR benchmark. Furthermore, our methods also significantly improve LPIPS and FID in four SR models, including 0.0395 and 12.619 in HAT (Chen et al., 2023b) on the LPIPS and FID of the RealSR benchmark. Furthermore, we also conduct experiments on the diffusion-based model ResShift (Yue et al., 2024). Specifically, the official implementation of ResShift is trained on a large-scale dataset using RealESRGAN degradation with 500,000 iterations, and we perform a quick fine-tuning with 10,000 iterations using our generated data. As shown Table 2, our proposed method effectively generates accurate and realistic data in the target domain and helps ResShift quickly adapt to the new domain, yielding consistent improvements on both DRealSR and SmartPhone benchmarks. The experiments demonstrate that our method significantly improves performance across various SR approaches, including CNN-based, Transformer-based, and diffusion-based methods.
+
+Accuracy of generated real LR. To validate our method's superiority in generating accurately realistic and matched real LR, we conduct comparisons with existing methods in terms of the accuracy of the generated data on six metrics for evaluation. Specifically, we utilize HR images in the three real-world datasets as the content reference and employ the existing methods to synthesize generated LR images, while our method is able to utilize the degradation reference from real LR. As shown in Table 3, we can observe that our method achieves the best performance on three datasets
+
+Table 3: Quantitative comparisons of the accuracy of generating real LR images on three real-world benchmarks under six distinct similarity metrics.  
+
+<table><tr><td>Benchmark</td><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>FID↓</td><td>DISTS↓</td><td>CLIP-Score↑</td></tr><tr><td rowspan="4">RealSR</td><td>Real-ESRGAN</td><td>24.3974</td><td>0.6798</td><td>0.3881</td><td>157.5237</td><td>0.2618</td><td>0.7552</td></tr><tr><td>BSRGAN</td><td>23.2665</td><td>0.6043</td><td>0.5093</td><td>177.9996</td><td>0.3048</td><td>0.6783</td></tr><tr><td>SynDiff</td><td>25.2461</td><td>0.7588</td><td>0.2371</td><td>119.9950</td><td>0.1944</td><td>0.7984</td></tr><tr><td>Ours</td><td>26.1615</td><td>0.7940</td><td>0.2226</td><td>107.2375</td><td>0.1865</td><td>0.8238</td></tr><tr><td rowspan="4">DRealSR</td><td>RealESRGAN</td><td>26.3510</td><td>0.6935</td><td>0.3842</td><td>60.2665</td><td>0.2437</td><td>0.7710</td></tr><tr><td>BSRGAN</td><td>26.1970</td><td>0.6869</td><td>0.4319</td><td>62.9189</td><td>0.2646</td><td>0.7137</td></tr><tr><td>SynDiff</td><td>28.0125</td><td>0.8136</td><td>0.1739</td><td>31.0855</td><td>0.1673</td><td>0.8462</td></tr><tr><td>Ours</td><td>28.6629</td><td>0.8360</td><td>0.1497</td><td>30.0078</td><td>0.1567</td><td>0.8577</td></tr><tr><td rowspan="4">SmartPhone</td><td>RealESRGAN</td><td>26.0680</td><td>0.5981</td><td>0.5202</td><td>106.3958</td><td>0.2855</td><td>0.7231</td></tr><tr><td>BSRGAN</td><td>27.6215</td><td>0.7179</td><td>0.4399</td><td>105.3289</td><td>0.2760</td><td>0.6771</td></tr><tr><td>SynDiff</td><td>28.5020</td><td>0.8075</td><td>0.2423</td><td>70.6379</td><td>0.2044</td><td>0.8213</td></tr><tr><td>Ours</td><td>29.0054</td><td>0.8292</td><td>0.2121</td><td>69.1162</td><td>0.1964</td><td>0.8334</td></tr></table>
+
+![](images/080ae74b6aab86d34225ae03bb3edf0097e66afd30d167d69703f2471cde6cc0.jpg)  
+LR
+
+![](images/5cbae3e84668b7f99ef807aec4654692f776e7a862b54b25db3a72979fd1caf6.jpg)  
+BSRGAN
+
+![](images/016b1588650b71220e9639ca6d4c225f63b036bde46e3d83f245f27ba32e2959.jpg)  
+Real-ESRGAN
+
+![](images/f2dd157295c4a0be8390a92e53a62ca68711e7c85d3ce53445c39dc33680f303.jpg)  
+SynDiff
+
+![](images/8f08167f77e78543b661723dc7415caee415efa3099899e89baf641efec159cb.jpg)  
+Ours
+
+![](images/507312036c6102f3ab2f880cbc3b69c781b3ef4a67bfe3d791b465ac2f67f269.jpg)  
+GT
+
+![](images/b2bb1072b666f849ed7b923ace4ecd7c914b4110e4cc0a53efd0c3b35cdb852c.jpg)  
+LR
+
+![](images/d804986feaf4e8a42e58ce40cc6de6f1cc3529847099a9be3a8b2851299874e3.jpg)  
+BSRGAN
+
+![](images/92b7e72c72190d3efe9be4d0899bd6a2786f22cc6e0058cfbb444d1ff6577e39.jpg)  
+Real-ESRGAN
+
+![](images/7dd5fcb66832c764d8b11a00deb60dcac2302b460afdd041da0694971528b52b.jpg)  
+SynDiff  
+Figure 4: Visual comparison of Real SR based on different data generation methods. Real SR training using our data achieves the best visual results. Please zoom in for better visualization.
+
+![](images/5f2adbe3901ffe50777194b5e491ef0519584770ed2952eae9e1e5d1a2376524.jpg)  
+Ours
+
+![](images/d0cbd6f7eca142297c6cae7ca68b8363eeed36d565c6359aafad0691a1f3deed.jpg)  
+GT
+
+under six metrics, including distribution distance metric: FID, image structure and texture similarity metric: DISTS, and perceptual metric: LPIPS and CLIP-score, etc. It demonstrates the superiority of our Decoupled DDPM and verifies that our generation method is closer to target real LR images.
+
+Generalization Ability on Out-of-distribution Data. Although we have collected about 152,000 real LR images for training, unseen scenarios often arise in real-world applications. To further explore our generalization capabilities on out-of-distribution data, we randomly extract 200 images from the unseen real-world video SR data (Yang et al., 2021) for evaluation, denoted as RealVSR. Then, we capture the degradation representing those images to generate the training data and compare it with existing methods on the SwinIR-L model, and the results are shown in Table 6. We can observe that our method still achieves the best performance on the out-of-distribution RealVSR benchmark, surpassing the existing state-of-the-art methods by 0.2214 and 0.0055 in PSNR and SSIM, respectively. This demonstrates the ability of our method to generalize for unseen scenarios in the real world.
+
+<table><tr><td>T</td><td>0-200</td><td>0-300</td><td>0-400</td><td>0-500</td></tr><tr><td>PSNR↑</td><td>28.762</td><td>28.868</td><td>28.542</td><td>26.713</td></tr><tr><td>SSIM↑</td><td>0.8501</td><td>0.8522</td><td>0.8435</td><td>0.7931</td></tr></table>
+
+Table 4: Ablation study on  $T$  in our proposed Decoupled DDPM during inferencing.  
+
+<table><tr><td>Method</td><td>w/o Edeg</td><td>w/o Econt</td><td>Ours</td></tr><tr><td>PSNR↑</td><td>21.642</td><td>25.745</td><td>28.868</td></tr><tr><td>SSIM↑</td><td>0.7432</td><td>0.8095</td><td>0.8522</td></tr></table>
+
+Table 5: Ablation study on degradation extractor  $E_{\text{deg}}$  and content extractor  $E_{\text{cont}}$ .
+
+![](images/7cf00eb80e42e451caa557ab59b690a6d7cee50cd62d12d99b4527e2c77a2371.jpg)
+
+![](images/e766d46531def572ce7443ac78efc83fca6a3d891846bc0e6b4a520a984844d6.jpg)
+
+![](images/fe6bf010895bea2c6375fb08f0d35ac9b5c24bf1f813e4372eedf2607ce748a1.jpg)
+
+![](images/88680a4d436c73425d60f6cab0318c57368f9e688c560933e3b720f19f9865d9.jpg)
+
+![](images/e55306906b99f4794840c2b4f80c5398f4b445a9ac7d0416334b546e4abf1c48.jpg)
+
+![](images/eb32b9d3cb5d14f6ab2041c3d81c780545d2138ec6f4cd698cb76b1a02485ebd.jpg)  
+Figure 5: Visual comparison of generated low-resolution images with different reference real LR images. Please zoom in for better visualization.
+
+![](images/56fe2e29ccf88932cdbccdd015486285410783aafcc7e613fa70df575c742c28.jpg)
+
+![](images/f75f51c5fc6ecfb9b40de6ec639b7689a84b05ecdb5a24885d83682c303f5bb5.jpg)
+
+![](images/d589d35f064923c8ab934fd8a27023ed69861676fd7339907507960f7262846c.jpg)
+
+# 3.3 QUALITATIVE RESULTS
+
+We visualize the results of the generated LR images, as shown in Figure 3. It is clear that the Real-ESRGAN and BSRGAN, which are unable to perceive real degradation, exhibit a significant gap compared to real LR images. The SynDiff also struggles to adaptively capture the real degradation and content representation, resulting in unreal degradation distribution and low fidelity, as shown in the third row in Figure 3. However, benefiting from our adaptive perception ability of the degradation of real LR image and content of HR image, our RealDGen achieves the best visual effect in terms of both the realism of degradation and the fidelity of content. To demonstrate our method's ability to capture the degradation distribution of arbitrary LR images, we simulate data using degradation representations from various real-world LR images and content representation from an HR image (Figure 5). In contrast to existing methods like BSRGAN and Real-ESRGAN, which rely on handcrafted degradation models and struggle to accurately reflect degradation in specific LR images, our approach separates content representation from the HR image and extracts degradation representation from the reference LR image, enabling more accurate and realistic data generation.
+
+To further present the effectiveness of our generated data for Real SR models, we visualize the super-resolved results of the SwinIR model on the RealSR and DRealSR benchmarks, as shown in Figure 4. Models trained on SynDiff, BSRGAN, and Real-ESRGAN data exhibit noticeable artifacts and blurring, while our method consistently delivers superior visual quality without such defects. The textual scene in the second row of Figure 4 highlights the clear advantage of our approach. Additional visual comparisons are provided in Appendix A.16.
+
+# 3.4 ABLATION STUDY
+
+Degradation and content extractor. Our core idea is utilizing the degradation extractor  $E_{deg}$  and content extractor  $E_{cont}$  to extract degradation and content representation to control Decoupled DDPM in generating realistic LR. To verify this, we eliminate components  $E_{deg}$  and  $E_{cont}$ , subsequently training the decoupled DDPM and fine-tuning the remaining extractor. For quick evaluation, we employ SwinIR-L as the baseline and evaluate on the Smartphone benchmark. The results in Table 5 demonstrate that the absence of  $E_{deg}$  and  $E_{cont}$  will cause mismatch and unreal degradation and content distortion problems and result in performance degradation, thereby affirming the indispensability of  $E_{deg}$  and  $E_{cont}$  in our method.
+
+![](images/773b2694aa1cfda65600b0146bc3c9ee3bdeb6d92a4d0271f1bebfd61510d058.jpg)  
+Figure 6: User study of generated real LR.
+
+<table><tr><td>RealVSR</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>Real-ESRGAN</td><td>22.7402</td><td>0.6939</td></tr><tr><td>BSRGAN</td><td>22.6564</td><td>0.6779</td></tr><tr><td>SynDiff</td><td>22.4607</td><td>0.6704</td></tr><tr><td>Ours</td><td>22.9616</td><td>0.6994</td></tr></table>
+
+Table 6: The performance comparison with existing data generation method on out-of-distribution RealVSR benchmark.
+
+$T$  in data generation. As illustrated in Section 2.3, in inferencing, we denoise from an initial LR image  $\mathbf{x}_t$  with  $t$ -step noise added and perform t steps of denoising to improve the fidelity of Decoupled DDPM, where  $t$  is selected from the range of 0 to  $T$ . To explore it, we set  $T$  to 200, 300, 400, and 500 and conduct ablation experiments, as presented in Table 4. It is observed that the optimal  $T$  is 300. A lower  $T$  will result in insufficient generation of degradation, while a high  $T$  will lead to a decrease in image fidelity. We propose that the value of  $T$  should be dynamically adjusted to align with the degradation level of the given LR image, rather than being manually set. In future work, we plan to develop an automatic selection mechanism for  $T$  to enhance the model's control and improve the fidelity of the generated data. More ablation studies of loss function's hyperparameters,  $n$ , and margin, are presented in Appendix A.7.
+
+# 3.5 USER STUDY
+
+To demonstrate the superiority of RealDGen in generating accurate and realistic low-resolution images, we conduct a user study involving 10 real-world LR images randomly chosen from existing benchmarks. 10 volunteers are asked to rate each scene individually (0: not similar at all; 2: not very similar; 4: slightly similar; 6: moderately similar; 8: similar; 10: extremely similar). Then, we aggregate the scores from all volunteers, and the results are shown in Figure 6. We can observe that the previous methods are unable to adaptively and accurately perceive real degradation, resulting in low fidelity in the generated real LR data, which leads to a general perception among users that there is a significant gap compared to real LR images. However, our RealDGen can adaptively capture the real degradation to accurately generate realistic LR images, resulting in an average score of 8.46 from human evaluators, surpassing previous approaches and thereby highlighting the improved visual quality that our method offers.
+
+# 4 CONCLUSION
+
+In this paper, we introduce a novel RealDGen to adaptively generate large-scale, high-quality paired data with arbitrary real LR as degradation reference and unpaired HR as content reference. Well-designed extractors and strategies are proposed to facilitate the extraction of robust content and degradation representations. A content-degradation decoupled diffusion model is proposed to adaptively generate realistic LR with given unpaired LR and HR conditions. Extensive experiments demonstrate that RealDGen not only achieves the best performance in generating realistic and accurate real LR images but also comprehensively improves the generalization ability of various popular SR models on real-world benchmarks. In addition, benefiting from the unsupervised learning of our method and the convenience of real LR image collection, it is easy to collect more real LR images with various real degradations to enhance generalization capability further.
+
+Limitation analysis. Due to the high stochasticity inherent in diffusion models, RealDGen sometimes struggles to preserve fine textures. In future work, we plan to incorporate perceptual loss during training and develop a robust mechanism for automatically selecting the optimal denoising step based on the degradation level, thereby further enhancing the fidelity of the generated data.
+
+Acknowledgments. This work was supported by the Natural Science Foundation of China under Grants 62225207,62436008 and 62206262.
+
+# REFERENCES
+
+Eirikur Agustsson and Radu Timofte. Ntire 2017 challenge on single image super-resolution: Dataset and study. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp. 126-135, 2017.  
+Adrian Bulat, Jing Yang, and Georgios Tzimiropoulos. To learn image super-resolution, use a gan to learn how to do image degradation first. In Proceedings of the European conference on computer vision (ECCV), pp. 185-200, 2018.  
+Jianrui Cai, Hui Zeng, Hongwei Yong, Zisheng Cao, and Lei Zhang. Toward real-world single image super-resolution: A new benchmark and a new model. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 3086-3095, 2019.  
+Chang Chen, Zhiwei Xiong, Xinmei Tian, Zheng-Jun Zha, and Feng Wu. Camera lens superresolution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1652-1660, 2019.  
+Du Chen, Jie Liang, Xindong Zhang, Ming Liu, Hui Zeng, and Lei Zhang. Human guided ground-truth generation for realistic image super-resolution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14082-14091, 2023a.  
+Haoyu Chen, Wenbo Li, Jinjin Gu, Jingjing Ren, Haoze Sun, Xueyi Zou, Zhensong Zhang, Youliang Yan, and Lei Zhu. Low-res leads the way: Improving generalization for super-resolution by self-supervised learning. arXiv preprint arXiv:2403.02601, 2024.  
+Honggang Chen, Xiaohai He, Linbo Qing, Yuanyuan Wu, Chao Ren, Ray E Sheriff, and Ce Zhu. Real-world single image super-resolution: A brief review. Information Fusion, 79:124-145, 2022.  
+Jin Chen, Jun Chen, Zheng Wang, Chao Liang, and Chia-Wen Lin. Identity-aware face superresolution for low-resolution face recognition. IEEE Signal Processing Letters, 27:645-649, 2020a.  
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pp. 1597-1607. PMLR, 2020b.  
+Xiangyu Chen, Xintao Wang, Jiantao Zhou, Yu Qiao, and Chao Dong. Activating more pixels in image super-resolution transformer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 22367-22377, 2023b.  
+Zeshuai Deng, Zhuokun Chen, Shuaicheng Niu, Thomas Li, Bohan Zhuang, and Mingkui Tan. Efficient test-time adaptation for super-resolution with second-order degradation and reconstruction. Advances in Neural Information Processing Systems, 36:74671-74701, 2023.  
+Xin Di, Long Peng, Peizhe Xia, Wenbo Li, Renjing Pei, Yang Cao, Yang Wang, and Zheng-Jun Zha. Qmambabsr: Burst image super-resolution with query state space model. arXiv preprint arXiv:2408.08665, 2024.  
+Keyan Ding, Kede Ma, Shiqi Wang, and Eero P Simoncelli. Image quality assessment: Unifying structure and texture similarity. IEEE transactions on pattern analysis and machine intelligence, 44(5):2567-2581, 2020.  
+Chao Dong, Chen Change Loy, and Xiaou Tang. Accelerating the super-resolution convolutional neural network. In Computer Vision-ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part II 14, pp. 391-407. Springer, 2016.  
+Michael Elad and Arie Feuer. Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images. IEEE transactions on image processing, 6(12):1646-1658, 1997.  
+Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12873-12883, 2021.
+
+Xin Gu, Guang Chen, Yufei Wang, Libo Zhang, Tiejian Luo, and Longyin Wen. Text with knowledge graph augmented transformer for video captioning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 18941-18951, 2023.  
+Xin Gu, Ming Li, Libo Zhang, Fan Chen, Longyin Wen, Tiejian Luo, and Sijie Zhu. Multi-reward as condition for instruction-based image editing. CoRR, abs/2411.04713, 2024. doi: 10.48550/ ARXIV.2411.04713. URL https://doi.org/10.48550/arXiv.2411.04713.  
+Bahadir K Gunturk, Aziz Umit Batur, Yucel Altunbasak, Monson H Hayes, and Russell M Mersereau. Eigenface-domain super-resolution for face recognition. IEEE transactions on image processing, 12(5):597-606, 2003.  
+Xu Han, Linghao Jin, Xiaofeng Liu, and Paul Pu Liang. Progressive compositionality in text-to-image generative models, 2024a. URL https://arxiv.org/abs/2410.16719.  
+Xu Han, Felix Yu, Joao Sedoc, and Benjamin Van Durme. Baby bear: Seeking a just right rating scale for scalar annotations, 2024b. URL https://arxiv.org/abs/2408.09765.  
+Muhammad Haris, Greg Shakhnarovich, and Norimichi Ukita. Task-driven super resolution: Object detection in low-resolution images. In Neural Information Processing: 28th International Conference, ICONIP 2021, Sanur, Bali, Indonesia, December 8-12, 2021, Proceedings, Part V 28, pp. 387-395. Springer, 2021.  
+Chunming He, Kai Li, Yachao Zhang, Yulun Zhang, Zhenhua Guo, Xiu Li, Martin Danelljan, and Fisher Yu. Strategic preys make acute predators: Enhancing camouflaged object detectors by generating camouflaged objects. ICLR, 2024.  
+Chunming He, Chengyu Fang, Yulun Zhang, Kai Li, Longxiang Tang, Chenyu You, Fengyang Xiao, Zhenhua Guo, and Xiu Li. Reti-diff: Illumination degradation image restoration with retina-based latent diffusion model. *ICLR*, 2025a.  
+Chunming He, Yuqi Shen, Chengyu Fang, Fengyang Xiao, Longxiang Tang, Yulun Zhang, Wangmeng Zuo, Zhenhua Guo, and Xiu Li. Diffusion models in low-level vision: A survey. TPAMI, 2025b.  
+Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. arXiv preprint arXiv:1903.12261, 2019.  
+Alexander Hermans, Lucas Beyer, and Bastian Leibe. In defense of the triplet loss for person re-identification. arXiv preprint arXiv:1703.07737, 2017.  
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.  
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.  
+Yan Huang, Shang Li, Liang Wang, Tieniu Tan, et al. Unfolding the alternating optimization for blind super resolution. Advances in Neural Information Processing Systems, 33:5632-5643, 2020.  
+Quan Huynh-Thu and Mohammed Ghanbari. Scope of validity of psnr in image/video quality assessment. *Electronics letters*, 44(13):800–801, 2008.  
+Andrey Ignatov, Nikolay Kobyshev, Radu Timofte, Kenneth Vanhoey, and Luc Van Gool. Dslr-quality photos on mobile devices with deep convolutional networks. In Proceedings of the IEEE international conference on computer vision, pp. 3277-3285, 2017.  
+Xiaozhong Ji, Yun Cao, Ying Tai, Chengjie Wang, Jilin Li, and Feiyue Huang. Real-world superresolution via kernel estimation and noise injection. In proceedings of the IEEE/CVF conference on computer vision and pattern recognition workshops, pp. 466-467, 2020.  
+Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In Computer Vision-ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part II 14, pp. 694-711. Springer, 2016.
+
+Lanqing Li, Hai Zhang, Xinyu Zhang, Shatong Zhu, Yang YU, Junqiao Zhao, and Pheng-Ann Heng. Towards an information theoretic framework of context-based offline meta-reinforcement learning. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024a. URL https://openreview.net/forum?id=QFUsZvw9mx.  
+Wenbo Li, Kun Zhou, Lu Qi, Liying Lu, and Jiangbo Lu. Best-buddy gans for highly detailed image super-resolution. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 1412-1420, 2022a.  
+Xiaoming Li, Chaofeng Chen, Xianhui Lin, Wangmeng Zuo, and Lei Zhang. From face to natural image: Learning real degradation for blind image super-resolution. In European Conference on Computer Vision, pp. 376-392. Springer, 2022b.  
+Xin Li, Deshui Miao, Zhenyu He, Yaowei Wang, Huchuan Lu, and Ming-Hsuan Yang. Learning spatial-semantic features for robust video object segmentation. arXiv preprint arXiv:2407.07760, 2024b.  
+Y Li, Bruno Sixou, and F Peyrin. A review of the deep learning methods for medical images superresolution problems. Irbm, 42(2):120-133, 2021.  
+Yawei Li, Yulun Zhang, Radu Timofte, Luc Van Gool, Lei Yu, Youwei Li, Xinpeng Li, Ting Jiang, Qi Wu, Mingyan Han, et al. Ntire 2023 challenge on efficient super-resolution: Methods and results. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1922-1960, 2023.  
+Jingyun Liang, Jiezhang Cao, Guolei Sun, Kai Zhang, Luc Van Gool, and Radu Timofte. Swinir: Image restoration using swim transformer. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 1833-1844, 2021.  
+Anran Liu, Yihao Liu, Jinjin Gu, Yu Qiao, and Chao Dong. Blind image super-resolution: A survey and beyond. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.  
+Yihao Liu, Hengyuan Zhao, Jinjin Gu, Yu Qiao, and Chao Dong. Evaluating the generalization ability of super-resolution networks. IEEE Transactions on pattern analysis and machine intelligence, 2023.  
+Shilin Lu, Yanzhu Liu, and Adams Wai-Kin Kong. Tf-Icon: Diffusion-based training-free cross-domain image composition. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2294–2305, 2023.  
+Shilin Lu, Zilan Wang, Leyang Li, Yanzhu Liu, and Adams Wai-Kin Kong. Mace: Mass concept erasure in diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6430-6440, 2024a.  
+Shilin Lu, Zihan Zhou, Jiayou Lu, Yanzhi Zhu, and Adams Wai-Kin Kong. Robust watermarking using generative priors against image editing: From benchmarking to advances. arXiv preprint arXiv:2410.18775, 2024b.  
+A Lugmayr, M Danelljan, L Van Gool, and R Timofte. Learning the super-resolution space with normalizing flow. ECCV, Srflow, 2020a.  
+Andreas Lugmayr, Martin Danelljan, and Radu Timofte. Unsupervised learning for real-world super-resolution. In 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pp. 3408-3416. IEEE, 2019a.  
+Andreas Lugmayr, Martin Danelljan, Radu Timofte, Manuel Fritsche, Shuhang Gu, Kuldeep Purohit, Praveen Kandula, Maitreya Suin, AN Rajagoapalan, Nam Hyung Joon, et al. Aim 2019 challenge on real-world image super-resolution: Methods and results. In 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pp. 3575-3583. IEEE, 2019b.  
+Andreas Lugmayr, Martin Danelljan, and Radu Timofte. Ntire 2020 challenge on real-world image super-resolution: Methods and results. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 494-495, 2020b.
+
+Fangzhou Luo, Xiaolin Wu, and Yanhui Guo. And: Adversarial neural degradation for learning blind image super-resolution. Advances in Neural Information Processing Systems, 36, 2024.  
+Shunta Maeda. Unpaired image super-resolution using pseudo-supervision. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 291-300, 2020.  
+Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. Sdedit: Guided image synthesis and editing with stochastic differential equations. arXiv preprint arXiv:2108.01073, 2021.  
+Deshui Miao, Xin Li, Zhenyu He, Huchuan Lu, and Ming-Hsuan Yang. Spatial-temporal multi-level association for video object segmentation. In European Conference on Computer Vision, pp. 91-107. Springer, 2024.  
+Chong Mou, Yanze Wu, Xintao Wang, Chao Dong, Jian Zhang, and Ying Shan. Metric learning based interactive modulation for real-world super-resolution. In European Conference on Computer Vision, pp. 723-740. Springer, 2022.  
+Seungjun Nah, Sungyong Baik, Seokil Hong, Gyeongsik Moon, Sanghyun Son, Radu Timofte, and Kyoung Mu Lee. Ntire 2019 challenge on video deblurring and super-resolution: Dataset and study. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition workshops, pp. 0-0, 2019.  
+Dewan Fahim Noor, Yue Li, Zhu Li, Shuvra Bhattacharyya, and George York. Gradient image super-resolution for low-resolution image recognition. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2332-2336. IEEE, 2019.  
+Seobin Park, Dongjin Kim, Sungyong Baik, and Tae Hyun Kim. Learning controllable degradation for real-world super-resolution via constrained flows. In International Conference on Machine Learning, pp. 27188-27203. PMLR, 2023.  
+Long Peng, Aiwen Jiang, Qiaosi Yi, and Mingwen Wang. Cumulative rain density sensing network for single image derain. IEEE Signal Processing Letters, 27:406-410, 2020.  
+Long Peng, Yang Cao, Renjing Pei, Wenbo Li, Jiaming Guo, Xueyang Fu, Yang Wang, and Zheng-Jun Zha. Efficient real-world image super-resolution via adaptive directional gradient convolution. arXiv preprint arXiv:2405.07023, 2024a.  
+Long Peng, Yang Cao, Yuejin Sun, and Yang Wang. Lightweight adaptive feature de-drifting for compressed image classification. IEEE Transactions on Multimedia, 2024b.  
+Long Peng, Wenbo Li, Jiaming Guo, Xin Di, Haoze Sun, Yong Li, Renjing Pei, Yang Wang, Yang Cao, and Zheng-Jun Zha. Unveiling hidden details: A raw data-enhanced paradigm for real-world super-resolution. arXiv preprint arXiv:2411.10798, 2024c.  
+Long Peng, Xin Di, Zhanfeng Feng, Wenbo Li, Renjing Pei, Yang Wang, Xueyang Fu, Yang Cao, and Zheng-Jun Zha. Directing mamba to complex textures: An efficient texture-aware state space model for image restoration. arXiv preprint arXiv:2501.16583, 2025.  
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748-8763. PMLR, 2021.  
+Ioan Raicu, Yong Zhao, Ian T Foster, and Alex Szalay. Accelerating large-scale data exploration through data diffusion. In Proceedings of the 2008 international workshop on Data-aware distributed computing, pp. 9-18, 2008.  
+Ali Razavi, Aaron van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2, 2019.  
+Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695, 2022.
+
+Vikash Sehwag, Caner Hazirbas, Albert Gordo, First Ozgenel, and Cristian Canton. Generating high fidelity data from low-density regions using diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11492-11501, 2022.  
+Haoze Sun, Wenbo Li, Jianzhuang Liu, Haoyu Chen, Renjing Pei, Xueyi Zou, Youliang Yan, and Yujiu Yang. Coser: Bridging image and language for cognitive super-resolution. arXiv preprint arXiv:2311.16512, 2023.  
+Wanjie Sun and Zhenzhong Chen. Learning many-to-many mapping for unpaired real-world image super-resolution and downscaling. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024.  
+Cong Wang, Jinshan Pan, and Xiao-Ming Wu. Online-updated high-order collaborative networks for single image deraining. In AAAI, volume 36, pp. 2406-2413, 2022.  
+Cong Wang, Jinshan Pan, Wanyu Lin, Jiangxin Dong, Wei Wang, and Xiao-Ming Wu. Selfpromer: Self-prompt dehazing transformers with depth-consistency. In AAAI, volume 38, pp. 5327-5335, 2024a.  
+Cong Wang, Wei Wang, Chengjin Yu, and Jie Mu. Explore internal and external similarity for single image deraining with graph neural networks. In IJCAI, 2024b.  
+Longguang Wang, Yingqian Wang, Xiaoyu Dong, Qingyu Xu, Jungang Yang, Wei An, and Yulan Guo. Unsupervised degradation representation learning for blind super-resolution. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10581-10590, 2021a.  
+Xintao Wang, Ke Yu, Shixiang Wu, Jinjin Gu, Yihao Liu, Chao Dong, Yu Qiao, and Chen Change Loy. Esrgan: Enhanced super-resolution generative adversarial networks. In Proceedings of the European conference on computer vision (ECCV) workshops, pp. 0-0, 2018.  
+Xintao Wang, Liangbin Xie, Chao Dong, and Ying Shan. Real-esrgan: Training real-world blind super-resolution with pure synthetic data. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 1905-1914, 2021b.  
+Yang Wang, Long Peng, Liang Li, Yang Cao, and Zheng-Jun Zha. Decoupling-and-aggregating for image exposure correction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 18115-18124, 2023.  
+Zhihao Wang, Jian Chen, and Steven CH Hoi. Deep learning for image super-resolution: A survey. IEEE transactions on pattern analysis and machine intelligence, 43(10):3365-3387, 2020.  
+Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600-612, 2004.  
+Pengxu Wei, Ziwei Xie, Hannan Lu, Zongyuan Zhan, Qixiang Ye, Wangmeng Zuo, and Liang Lin. Component divide-and-conquer for real-world image super-resolution. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part VIII 16, pp. 101-117. Springer, 2020.  
+Valentin Wolf, Andreas Lugmayr, Martin Danelljan, Luc Van Gool, and Radu Timofte. Deflow: Learning complex image degradations from unpaired data with conditional flows. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 94-103, 2021.  
+Weijia Wu, Yuzhong Zhao, Hao Chen, Yuchao Gu, Rui Zhao, Yefei He, Hong Zhou, Mike Zheng Shou, and Chunhua Shen. Datasetdm: Synthesizing data with perception annotations using diffusion models. Advances in Neural Information Processing Systems, 36:54683-54695, 2023.  
+Jin Xiao, Hongwei Yong, and Lei Zhang. Degradation model learning for real-world single image super-resolution. In Proceedings of the Asian Conference on Computer Vision, 2020.  
+Tao Yang, Peiran Ren, Lei Zhang, et al. Synthesizing realistic image restoration training pairs: A diffusion approach. arXiv preprint arXiv:2303.06994, 2023.
+
+Xi Yang, Wangmeng Xiang, Hui Zeng, and Lei Zhang. Real-world video super-resolution: A benchmark dataset and a decomposition based learning scheme. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4781-4790, 2021.  
+Zihan Ye, Guanyu Yang, Xiaobo Jin, Youfa Liu, and Kaizhu Huang. Rebalanced zero-shot learning. IEEE Transactions on Image Processing, 32:4185-4198, 2023. doi: 10.1109/TIP.2023.3295738.  
+Zihan Ye, Shreyank N Gowda, Shiming Chen, Xiaowei Huang, Haotian Xu, Fahad Shahbaz Khan, Yaochu Jin, Kaizhu Huang, and Xiaobo Jin. Zerodiff: Solidified visual-semantic correlation in zero-shot learning. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=wy9FRV805s.  
+Fanghua Yu, Jinjin Gu, Zheyuan Li, Jinfan Hu, Xiangtao Kong, Xintao Wang, Jingwen He, Yu Qiao, and Chao Dong. Scaling up to excellence: Practicing model scaling for photo-realistic image restoration in the wild. arXiv preprint arXiv:2401.13627, 2024.  
+Yuan Yuan, Siyuan Liu, Jiawei Zhang, Yongbing Zhang, Chao Dong, and Liang Lin. Unsupervised image super-resolution using cycle-in-cycle generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp. 701-710, 2018.  
+Zongsheng Yue, Jianyi Wang, and Chen Change Loy. Resshift: Efficient diffusion model for image super-resolution by residual shifting. Advances in Neural Information Processing Systems, 36, 2024.  
+Hai Zhang, Hang Yu, Junqiao Zhao, Di Zhang, xiao zhang, Hongtu Zhou, Chang Huang, and Chen Ye. How to fine-tune the model: Unified model shift and model bias policy optimization. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 59252-59272. Curran Associates, Inc., 2023a. URL https://proceedings.neurips.cc/paper_files/paper/2023/file/b9b4f084b2e6709a2bfad0f601271aec-Paper-Conference.pdf.  
+Hai Zhang, Boyuan Zheng, Tianying Ji, JinHang Liu, Anqi Guo, Junqiao Zhao, and Lanqing Li. Scrutinize what we ignore: Reining in task representation shift of context-based offline meta reinforcement learning. In The Thirteenth International Conference on Learning Representations, 2025a. URL https://openreview.net/forum?id=CrlX1GBGVm.  
+Kai Zhang, Jingyun Liang, Luc Van Gool, and Radu Timofte. Designing a practical degradation model for deep blind image super-resolution. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4791-4800, 2021.  
+Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 586-595, 2018.  
+Rui Zhang, Ruixu Geng, Yadong Li, Ruiyuan Song, Hanqin Gong, Dongheng Zhang, Yang Hu, and Yan Chen. RFMamba: Frequency-aware state space model for RF-based human-centric perception. In The Thirteenth International Conference on Learning Representations, 2025b. URL https://openreview.net/forum?id=1G9fjBLb6d.  
+Ruofan Zhang, Jinjin Gu, Haoyu Chen, Chao Dong, Yulun Zhang, and Wenming Yang. Crafting training degradation distribution for the accuracy-generalization trade-off in real-world superresolution. In International Conference on Machine Learning, pp. 41078-41091. PMLR, 2023b.  
+Wenlong Zhang, Xiaohui Li, Xiangyu Chen, Xiaoyun Zhang, Yu Qiao, Xiao-Ming Wu, and Chao Dong. Seal: A framework for systematic evaluation of real-world super-resolution. In The Twelfth International Conference on Learning Representations, 2023c.  
+Wenlong Zhang, Xiaohui Li, Guangyuan Shi, Xiangyu Chen, Yu Qiao, Xiaoyun Zhang, Xiao-Ming Wu, and Chao Dong. Real-world image super-resolution as multi-task learning. Advances in Neural Information Processing Systems, 36, 2024.
+
+# A APPENDIX
+
+# A.1 TRAINING DETAIL AND ANALYSIS OF CONTENT AND DEGRADATION EXTRACTOR
+
+# A.1.1 CONTENT EXTRACTOR
+
+We initially contemplate leveraging the auto-regressive architecture of VQGAN Esser et al. (2021) as our Content Extractor  $E_{cont}$  to capture content representations. However, our experimental endeavors reveal that the Generative Adversarial Network (GAN) in VQGAN impedes fine-tuning and the extraction of realistic content representations. Consequently, we elect to employ the encoder component of VQVAE Razavi et al. (2019) as  $E_{cont}$ . Given that these extractors have not been trained on degradation scenarios, they are not immediately suitable for our method. To surmount this challenge, we have meticulously crafted a fine-tuning strategy predicated on reconstruction learning. Specifically, we adopt Real-ESRGAN to degrade high-resolution (HR) images to generate paired datasets for training. During training, we maintain the decoder of VQVAE in a frozen state while fine-tuning the encoder, using HR images for supervised learning, with the objective loss function being the L2 loss. Our fine-tuning regimen is conducted with a learning rate of  $1 \times 10^{-5}$ , utilizing 8 NVIDIA V100 GPUs, which spent approximately one week for training. The reconstructed samples rendered by our fine-tuned Content Extractor are delineated in Figure 7. It is evident that after fine-tuning, VQVAE adeptly captures content representations in degraded scenarios and adeptly reconstructs high-resolution images.
+
+![](images/a8e4e365870059bb3ccf3bf4f117241561cadd428d341d2905b7a6220e43a540.jpg)  
+Figure 7: Visual comparison with input LR image, VQVAE, our pretrained VQVAE, and GT. VQVAE, after training on our designed learning strategy, can better capture the robust content representations to reconstruct high-qualified HR.
+
+Why can the content feature extractor pre-trained on LR data robustly extract content features from HR data during data generation? The proposed content extractor is designed to robustly extract content information from an arbitrary image. Given the inevitable presence of degradation in real-world HR images, we collect 152,000 real images with various types and degrees of degradation for unsupervised fine-tuning. Within this dataset, images with relatively low degradation levels exhibit a distribution closer to that of HR images. Fine-tuning the content extractor on these real images improves its generalization capability in real-world scenarios. Quantitative results (Tables 1, 2, and 5) and qualitative results (Figures 3, 4, and 5) validate that our content extractor can effectively extract robust content representations from any HR image to generate realistic and high-fidelity LR images.
+
+Why use LR-HR pairs to pre-train the content extractor? The content extractor is designed to extract robust 'pure' content information from an image while ignoring potential degradation. Using LR-LR or HR-HR pairs causes the content extractor to learn identity mapping, mixing content, and degradation representations. During inference, we aim to extract content from a relatively clear image ('HR') without involving its degradation and combine it with the degradation of another image ('LR') to generate a new LR image. Therefore, we should use LR-HR pairs to train the content extractor. Table 2 and Figure 3 validate that our content extractor can effectively extract robust content representations to generate realistic and high-fidelity LR images.
+
+# A.1.2 DEGRADATION EXTRACTOR
+
+We introduce a novel Degradation Extractor denoted as  $E_{deg}$ , which is comprised of a feature extraction layer integrated with 16 residual blocks and a mapping function facilitated by adaptive polling and a 4-layer convolutional structure, as shown in Figure 8. Specifically, given the input of a low-resolution image into the Degradation Extractor, it engenders the output degradation representation  $F_{deg} \in \mathbb{R}^{1 \times 1 \times 2048}$ . To ensure the extraction of comprehensive and unique degradation representations in the LR image, we employ both reconstruction learning and contrastive learning methodologies in training our network. The training is executed on 8 NVIDIA V100 GPUs over the course of approximately seven days, with the learning rate configured at  $1e - 4$ .
+
+![](images/0f6e6837735fd26575c84f66e2efe939e1c9020d38a04dd9b0ab39086e116e15.jpg)  
+Figure 8: Illustration of our proposed degradation extractor.
+
+# A.2 DETAIL OF HR AND LR RECONSTRUCTION NETWORK
+
+To better adapt to the content extractor, that is, the encoder of the VQVAE, the decoder of VQVAE is adopted as HR reconstruction network  $Rec_{hr}$  tasked with the reconstruction of high-resolution images. Given that the LR reconstruction network is tasked with reconstructing the LR image from content and degradation representations, we have adopted the aforementioned modulation network as our LR reconstruction network  $Rec_{lr}$ .
+
+# A.3 TRAINING DETAILS OF DECOUPLE DDPM
+
+During the training of the Decoupled DDPM, we configure the maximum diffusion step to be 500, span the training over 100 epochs, and utilize a learning rate of  $1e - 4$  for the decoupled diffusion model. For the fine-tuning of the extractors, we apply a more refined learning rate of  $1e - 6$ . The batch size is defined as 8, and the entire training regimen is on 8 NVIDIA V100 GPUs, which typically consume around 14 days to complete.
+
+We have amassed a collection of approximately 152,000 real low-resolution images sourced from publicly available datasets such as Wei et al. (2020); Cai et al. (2019); Ignatov et al. (2017) and those captured using smartphones. To elaborate, we have extracted all low-resolution images from these datasets and have cropped each to a uniform size of  $256 \times 256$ , resulting in a total of 110,000 images. Subsequently, to enhance the diversity of real-world degradation distribution, we have additionally procured 42,000 low-resolution images, each of the same  $256 \times 256$  size. Collectively, this corpus of 152,000 low-resolution images serves as the training data for our Decoupled DDPM.
+
+# A.4 DETAIL OF MODULATION BLOCK
+
+As detailed in Section 2, we commence with real low-resolution images, denoted as  $\mathcal{X}_{\mathrm{lr}}$ , which are derived from the real-world degradation distribution  $q$ . Utilizing our pre-trained extractors  $E_{cont}$  and  $E_{deg}$ , we meticulously extract the respective content and degradation representations,  $F_{cont}$  and  $F_{deg}$ . To authentically emulate the intricacies of the real imaging process, we introduce a modulation block, denoted as  $\mathcal{M}$ , which seamlessly integrates the degradation representation into the content representation. The integration is mathematically articulated as follows:
+
+$$
+\mathbf {c} = \mathcal {M} \left(E _ {\text {d e g}} \left(\mathbf {x} _ {l r}\right), E _ {\text {c o n t}} \left(\mathbf {x} _ {l r}\right)\right) \tag {8}
+$$
+
+In this section, we proceed to elucidate the intricate mechanisms underpinning the modulation block  $\mathcal{M}$ .  $\mathcal{M}$  is meticulously constructed from a series of four modulation layers, each comprising a convolutional layer, an activation function, and a modulation unit, as shown in Figure 9. The inputs to  $\mathcal{M}$  encompass both the degradation and content representations. Within each modulation unit, the degradation representation is subjected to a sophisticated fusion process. Ultimately, the block culminates in its output through a final convolutional operation, synthesizing the enhanced representation.
+
+![](images/59ee51eae397997326b3ba16cfafb73da90536fef8ff28fbaf3c1afe56cb3499.jpg)  
+Figure 9: Illustration of our modulation block.
+
+# A.5 ANALYSIS AND DETAILS OF FINE-TUNING EXTRACTOR
+
+During the second phase of training, we froze the majority of parameters in the content extractor, allowing only the first block within the VQVAE to undergo fine-tuning. Simultaneously, we applied a similar approach to the degradation extractor, restricting fine-tuning to the final convolutional layer. To validate the effectiveness of fine-tuning, we conduct an ablation experiment. In the second phase, we completely freeze the extractor and train only the Decoupled DDPM. We find that the realism of the generated degradation decreases because the extractor is trained exclusively on synthetic datasets, which results in its inability to extract real degradation representations and content. Consequently, the subsequent DDPM also has difficulty fitting the real LR accurately. Specifically, we test on the RealSR dataset using SwinIR-L, resulting in a 0.25dB performance degradation in PSNR with fine-tuning.
+
+# A.6 STOCHASTICITY OF DIFFUSION
+
+As discussed in Sec. 4, to mitigate the stochasticity of diffusion, we propose a filtering mechanism to eliminate outlier data. Specifically, we extract the degradation and content representations of the three simulated LR images, compare their similarities with the input LR's degradation and the HR's content representations, and select the best one to enhance the fidelity and quality of the generated LR images.
+
+<table><tr><td>RealSR</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>FID↓</td></tr><tr><td>w/o Filtering</td><td>25.8965</td><td>0.7886</td><td>0.2316</td><td>116.8970</td></tr><tr><td>Ours</td><td>26.1615</td><td>0.7940</td><td>0.2226</td><td>107.2375</td></tr></table>
+
+Table 7: The similarity results of ablation study on proposed filtering mechanic.  
+
+<table><tr><td>SwinIR-L</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>w/o Filtering</td><td>25.8747</td><td>0.7783</td></tr><tr><td>Ours</td><td>26.0250</td><td>0.7810</td></tr></table>
+
+To validate this, we conduct experiments on the RealSR dataset, calculating the similarity and data distribution compared to the real LR images in RealSR. Additionally, we verify the effectiveness using the downstream SR network SwinIR-L, with results presented in Tables 7 and 8. It can be observed that the data generated without the filtering mechanism exhibits lower similarity and further divergence from the real LR images, as well as reduced performance in the downstream SR network. This confirms the effectiveness and practicality of our proposed filtering mechanism in real-world scenarios.
+
+# A.7 MORE ABLATION STUDIES
+
+Table 8: The performance of SwinIR-L on RealSR benchmark.  
+
+<table><tr><td>n</td><td>1</td><td>0.1</td><td>0.01</td><td>0.001</td><td>0.0001</td></tr><tr><td>PSNR↑</td><td>28.34</td><td>28.43</td><td>28.66</td><td>28.52</td><td>28.60</td></tr><tr><td>FID↓</td><td>31.37</td><td>30.78</td><td>30.00</td><td>30.49</td><td>30.15</td></tr></table>
+
+Table 9: Ablation studies on  $n$  .  
+
+<table><tr><td>margin</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>PSNR↑</td><td>28.35</td><td>28.50</td><td>28.66</td><td>28.61</td><td>28.60</td></tr><tr><td>FID↓</td><td>30.97</td><td>30.61</td><td>30.00</td><td>30.16</td><td>30.15</td></tr></table>
+
+Table 10: Ablation studies on margin.
+
+In the loss function of the manuscript, we set the number of samples  $n$  and margin to 3 and 0.01, respectively, based on our experimental results. As shown in Tables 9 and 10, evaluating PSNR and FID between synthetic LR and real LR images on the DRealSR dataset, this configuration yields the best performance. Since it is impractical to exhaustively test all possible configurations, we empirically chose these values.
+
+# A.8 MORE DOWNSAMPLE SCALE
+
+Table 11: Comparison of data generation performance of different methods at different downsample scales on RealSR datasets.  
+
+<table><tr><td>Downsample Scale</td><td>Metric</td><td>BSRGAN</td><td>Real-ESRGAN</td><td>SynDiff</td><td>Ours</td></tr><tr><td>×4</td><td>PSNR (dB)↑</td><td>23.26</td><td>24.39</td><td>25.24</td><td>26.16</td></tr><tr><td>×2</td><td>PSNR (dB)↑</td><td>23.26</td><td>24.39</td><td>25.24</td><td>26.16</td></tr></table>
+
+In our manuscript, we adopt the typical downsample scale  $\times 4$  scale setting to validate the effectiveness of the proposed data generation method. Furthermore, our method is easily adaptable to synthesizing images at different super-resolution scales, such as  $\times 2$ , and  $\times 4$ . To further demonstrate the flexibility and superiority of our approach, we present comparison results for  $\times 4$  and  $\times 2$ . As shown in Table 11, our method achieves the best performance at both  $\times 4$  and  $\times 2$  scales.
+
+# A.9 INFERENCE TIME AND PARAMETERS COMPARISON
+
+Inference time. We test the average inference time (including IO, image processing, and generation) of producing 100 images and report the PSNR between synthetic LR and real LR on the RealSR dataset, as shown in Table 12. Using the official implementation, we measure inference time on 4 V100 GPUs. Our method adopts a sampling step of adding noise and denoising from 1 to 30, generating the highest-quality LR images and being faster than the learning-based SynDiff, although it is slower than RealESRGAN and BSRGAN. Our method and RealESRGAN can be used collaboratively and efficiently. We can use RealESRGAN for pre-training and then apply our method to create a small amount of high-quality data for fine-tuning the target scene, which does not require
+
+Table 12: Comparison of inference time with different methods.  
+
+<table><tr><td></td><td colspan="2">Hand-Crafted Physical-Based</td><td colspan="2">Learning-Based</td></tr><tr><td>Methods</td><td>BSRGAN</td><td>Real-ESRGAN</td><td>SynDiff</td><td>Ours</td></tr><tr><td>Platform</td><td>4 CPU</td><td>4 V100</td><td>4 V100</td><td>4 V100</td></tr><tr><td>Times (s)</td><td>0.4267</td><td>0.1085</td><td>0.7801</td><td>0.6086</td></tr><tr><td>PSNR (dB)</td><td>23.26</td><td>24.39</td><td>25.24</td><td>26.16</td></tr></table>
+
+Table 13: Parameters and performance comparison.  
+
+<table><tr><td>RealSR</td><td>Param (M)</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>FID↓</td><td>DISTS↓</td><td>CLIP-Score↑</td></tr><tr><td>RealSR (Syndiff)</td><td>101.121</td><td>25.24</td><td>0.7588</td><td>0.2371</td><td>119.950</td><td>0.1944</td><td>0.7984</td></tr><tr><td>RealSR (Ours)</td><td>73.676</td><td>26.16</td><td>0.7940</td><td>0.2226</td><td>107.2375</td><td>0.1865</td><td>0.8238</td></tr><tr><td>DRealSR (Syndiff)</td><td>101.121</td><td>28.0125</td><td>0.8136</td><td>0.1739</td><td>31.0855</td><td>0.1673</td><td>0.8462</td></tr><tr><td>DRealSR (Ours)</td><td>73.676</td><td>28.6629</td><td>0.8360</td><td>0.1497</td><td>30.0078</td><td>0.1567</td><td>0.8577</td></tr><tr><td>SmartPhone (Syndiff)</td><td>101.121</td><td>28.5020</td><td>0.8075</td><td>0.2423</td><td>70.6379</td><td>0.2044</td><td>0.8213</td></tr><tr><td>SmartPhone (Ours)</td><td>73.676</td><td>29.0054</td><td>0.8292</td><td>0.2121</td><td>69.1162</td><td>0.1964</td><td>0.8334</td></tr></table>
+
+high speed from our method. For example, as shown in Table 2, fine-tuning ResShift on our generated realistic data significantly improves its generalization capability in real-world scenarios. Additionally, we will explore more efficient sampling strategies and distillation methods to speed up the inference of our method further.
+
+Model parameters. Considering that BSRGAN and Real-ESRGAN are not deep-learning-based methods, following your suggestion, we compare the capacity and performance of deep learning-based Syndiff in Table 13. We see that our model not only has fewer parameters but also consistently achieves better performance on RealSR, DRealSR, and SmartPhone benchmarks.
+
+# A.10 PERFORMANCE COMPARISON WITH REAL DATA COLLECTION
+
+To further explore the performance comparison on real paired data, we further train the RRDB model using the real-collected training subset of RealSR and evaluate it on the test subset, as shown in Table 14. Since the training and test data in RealSR are captured using the same camera, they share a similar degradation distribution. Consequently, the model trained on real data achieves better performance compared to those trained on simulated data. However, collecting real-world data is often impractical, highlighting the importance of an accurate simulation method. Compared to other data synthesis approaches, our method is capable of generating realistic large-scale paired training data across diverse scenarios, enhancing the generalization capability of SR models. Furthermore, by incorporating our generated data into the real-collected training set, as shown in the last row of Table 14, the SR model's performance improves further, demonstrating the effectiveness of our approach.
+
+# A.11 MORE ABLATION STUDIES ON DEGRADATION AND CONTENT EXTRACTORS
+
+Performance comparison before and after fine-tuning degradation and content extractor. To enhance the generalization of  $E_{deg}$  and  $E_{cont}$  on unseen real distributions, as shown in the main text, only a small portion of the parameters in  $E_{deg}$  and  $E_{cont}$  are fine-tuned. We fine-tune the last layer, as shown in Figure 2. Therefore, it's nearly impossible for information to be directly passed through to the output. To further validate this, we conducted comparative experiments. As shown in Table 15, even without fine-tuning, our method can still generate realistic real LR, achieving the best performance compared to existing methods. After fine-tuning the extractors, the network's generalization ability in real scenarios is further enhanced, allowing for better generation of real LR in real-world settings.
+
+Validating the Effectiveness of Pretraining  $E_{deg}$  and  $E_{cont}$ . To Validate the effectiveness of the pretraining stage, we removed the pretrained weights of  $E_{deg}$  and  $E_{cont}$  and proceeded to train them alongside DDPM. However, without pretraining,  $E_{deg}$  and  $E_{cont}$  were unable to distinguish between
+
+Table 14: Performance comparison with existing data synthesis methods and real data collection.  
+
+<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>RRDB (Real-ESRGAN)</td><td>24.579</td><td>0.7614</td></tr><tr><td>RRDB (BSRGAN)</td><td>25.406</td><td>0.7685</td></tr><tr><td>RRDB (SynDiff)</td><td>25.488</td><td>0.7691</td></tr><tr><td>RRDB (Ours)</td><td>26.238</td><td>0.7747</td></tr><tr><td>RRDB (Real Data)</td><td>27.302</td><td>0.7934</td></tr><tr><td>RRDB (Real Data+Ours)</td><td>27.302</td><td>0.7934</td></tr></table>
+
+Table 15: Performance comparison before and after fine-tuning degradation and content extractor.  
+
+<table><tr><td>RealSR</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>Real-ESRGAN</td><td>24.39</td><td>0.679</td><td>0.388</td></tr><tr><td>BSRGAN</td><td>23.26</td><td>0.604</td><td>0.509</td></tr><tr><td>SynDiff</td><td>25.24</td><td>0.758</td><td>0.237</td></tr><tr><td>Our methods w/o pretraining</td><td>24.13</td><td>0.761</td><td>0.301</td></tr><tr><td>Our methods w/o finetuning</td><td>25.93</td><td>0.786</td><td>0.234</td></tr><tr><td>Our methods</td><td>26.16</td><td>0.794</td><td>0.222</td></tr></table>
+
+content and degradation effectively. This incapacity led to a significant decline in performance when synthesizing data, as they failed to extract degradation accurately from Real LR. Simultaneously, the content extractor struggled to capture high-fidelity content representations, resulting in generated images with lower fidelity, as shown in Table 15. We observed that without pretraining,  $E_{deg}$  and  $E_{cont}$  could not properly extract the respective degradations, making it challenging to maintain content consistency and degradation authenticity during the generation process, leading to a marked drop in network performance.
+
+# A.12 COMPARISON WITH BICUBIC
+
+We incorporate bicubic degradation as a baseline and compare the SR performance. To ensure a fair comparison, we utilize the officially released pre-trained RRDB models trained on bicubic data and evaluate them on the RealSR and Smartphone datasets, as shown in Tables 16. The results clearly demonstrate that the RRDB model trained on our data achieves significantly better SR performance across various evaluation metrics, particularly in the perceptual-oriented LPIPS metric. This highlights the limitations of a single bicubic model, which struggles to handle the high complexity of real-world degradations (Wang et al., 2021b; Zhang et al., 2021).
+
+# A.13 PERFORMANCE EVALUATION ON BLURED LR SCENARIOS
+
+To evaluate performance on blurred LR scenarios, we select blurred LR images from the popular REDS benchmark (Nah et al., 2019) to test and validate the improved generalization of our proposed method in more real-world scenarios. Specifically, we evaluate RRDB models pre-trained using synthesized data from different methods. The results, as shown in Table 17, demonstrate that our method consistently achieves the best performance on out-of-distribution data with blur degradation.
+
+# A.14 COMPARISON WITH DIFFERENT CONTRASTIVE LEARNING METHODS
+
+In (Wang et al., 2021a), the contrastive learning approach for training the degradation model constructs positive samples from different patches of the same image and negative samples from different images. However, the degradation between different patches of the same image is not always identical. For instance, in a defocused LR image, the defocused and non-defocused areas exhibit completely different degradations, and the degradation between different images may sometimes be similar. In contrast, our strategy constructs positive samples by keeping the degradation consistent while varying the content and generates negative samples by keeping the content consistent while varying the degradation, as described in Section 2.1 of the main text. This design ensures that the degradation
+
+Table 16: Performance comparison with bicubic.  
+
+<table><tr><td colspan="2">Dataset</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td rowspan="2">Smartphone</td><td>Bicubic</td><td>28.45</td><td>0.845</td><td>0.395</td></tr><tr><td>Our method</td><td>28.75</td><td>0.850</td><td>0.312</td></tr><tr><td rowspan="2">RealSR</td><td>Bicubic</td><td>25.98</td><td>0.755</td><td>0.442</td></tr><tr><td>Our method</td><td>26.23</td><td>0.774</td><td>0.297</td></tr></table>
+
+Table 17: Performance comparison on REDS.  
+
+<table><tr><td>REDS</td><td>BSRGAN</td><td>Real-ESRGAN</td><td>Syndiff</td><td>Ours</td></tr><tr><td>PSNR↑</td><td>23.4629</td><td>23.6475</td><td>23.6472</td><td>23.9834</td></tr><tr><td>SSIM↑</td><td>0.6615</td><td>0.6600</td><td>0.6443</td><td>0.6712</td></tr></table>
+
+extractor captures the complete and unique degradation distribution of the current LR image. To validate this, we replace the degradation model in (Wang et al., 2021a) and train the diffusion model, then generate real LR images on the RealSR dataset. The results, shown in Table 18, demonstrate that our proposed degradation model produces more realistic LR images compared to (Wang et al., 2021a).
+
+# A.15 MORE VISUAL COMPARISON ON INTERNET REAL LR IMAGES
+
+To evaluate real LR images from the internet, we compared existing methods on the SwinIR model and visualized the results, as shown in Figure 10 in the Appendix. It can be seen that our method better generates textures, while other methods tend to be over-smooth, validating the effectiveness of our approach.
+
+![](images/c0569a1c5c6f4eaf4d5577d71f6969fc1211d3083bbf46f93e8df24486af2a7c.jpg)  
+Real LR
+
+![](images/db8a1b78ad4acca495f3a2fa87fa3f10afa2414d3357a2ec2186cc75fbb35f4a.jpg)  
+BSRGAN
+
+![](images/ba16a2957c7a6a7505c2babd5e8eada291b0346f79c281364bbc07db639e4ff5.jpg)  
+Real-ESRGAN  
+Figure 10: Visual comparison of generated HR image from real internet LR images.
+
+![](images/5c4900762d3ab48ff96180518f8277526ddc487e66d5fac66e4a7c6c513cafd6.jpg)  
+SynDiff
+
+![](images/cff1839f927e1602efb7491b19ba47000416ba3d5d52a2d291fcf5c0e09115d9.jpg)  
+Ours
+
+# A.16 MORE VISUAL COMPARISON
+
+Here, we present additional visual results on the DRealSR benchmark to demonstrate the superiority of our method in adaptively generating accurate and realistic LR images, as shown in Figure 11 and 12. Additionally, we also present some visual results on the SmartPhone benchmark, as shown in Figure 13.
+
+# A.17 MORE RELATED WORK
+
+Recent and notable advancements in generative models and real-world image super-resolution, low-level vision tasks can be explored in these papers (Lu et al., 2024a; 2023; 2024b; Peng et al., 2025; Li et al., 2023; Lugmayr et al., 2020b; Li et al., 2022a; Peng et al., 2024b; Sun & Chen, 2024; Peng et al., 2024a; Lugmayr et al., 2019a; Peng et al., 2024c; Di et al., 2024; Wang et al., 2023; Peng et al., 2020; Chen et al., 2023a; Huang et al., 2020; Lugmayr et al., 2020b; Li et al., 2022a; Lugmayr et al., 2019a; Sun & Chen, 2024; Miao et al., 2024; Li et al., 2024b; Wang et al., 2022; 2024a;b; Han et al., 2024a;b;a; Ye et al., 2025; 2023; Zhang et al., 2025b; Gu et al., 2023; 2024; He et al., 2025b;a; 2024; Zhang et al., 2025a; 2023a; Li et al., 2024a).
+
+Table 18: Performance comparison with different contrastive learning.  
+
+<table><tr><td>RealSR</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>DASR</td><td>25.73</td><td>0.778</td><td>0.265</td></tr><tr><td>Our method</td><td>26.16</td><td>0.794</td><td>0.222</td></tr></table>
+
+![](images/349c0ef6f5416240e593efb284b8d5731b3e6249e83e0b0a2881dd7455832c6b.jpg)  
+Figure 11: Visual comparison of generated LR on DRealSR.
+
+![](images/1677c7b23b55277c6938c839d49853014626dd47a7162cc94237835af475c73b.jpg)
+
+![](images/f173d177dff112afb0bc59817e5b9a15feb083d54f0da57cc6641d2d2369d9f0.jpg)  
+Figure 12: Visual comparison of generated LR on DRealSR.  
+Figure 13: Visual comparison of generated LR images on the smartphone benchmark.

2025/Towards Realistic Data Generation for Real-World Super-Resolution/images.zip

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