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+    {
+        "type": "text",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
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+        "text": "Deyi Ji  $^{12}$  Feng Zhao  $^{1}$  Lanyun Zhu  $^{3}$  Wenwei Jin  $^{2}$  Hongtao Lu  $^{4}$  Jieping Ye  $^{2}$",
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+        "type": "text",
+        "text": "Abstract",
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+        "text": "In this paper, we address the challenge of Perspective-Invariant Learning in machine learning and computer vision, which involves enabling a network to understand images from varying perspectives to achieve consistent semantic interpretation. While standard approaches rely on the labor-intensive collection of multi-view images or limited data augmentation techniques, we propose a novel framework, Discrete Latent Perspective Learning (DLPL), for latent multi-perspective fusion learning using conventional single-view images. DLPL comprises three main modules: Perspective Discrete Decomposition (PDD), Perspective Homography Transformation (PHT), and Perspective Invariant Attention (PIA), which work together to discretize visual features, transform perspectives, and fuse multi-perspective semantic information, respectively. DLPL is a universal perspective learning framework applicable to a variety of scenarios and vision tasks. Extensive experiments demonstrate that DLPL significantly enhances the network's capacity to depict images across diverse scenarios (daily photos, UAV,自动驾驶) and tasks (detection, segmentation).",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
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+    {
+        "type": "text",
+        "text": "According to the theory of photography (Elkins, 2006; Bengio et al., 2013), image perspective/viewpoint refers to the perception of depth and spatial relationships between objects within an image. In the fields of machine learning and computer vision, current research often employs a variety of methods to enhance networks' Perspective-Invariant Learning (YU2, 2018; LO1, 1997), which involves en",
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+    {
+        "type": "text",
+        "text": "<sup>1</sup>University of Science and Technology of China <sup>2</sup>Alibaba Group <sup>3</sup>Singapore University of Technology and Design <sup>4</sup>Dept. of CSE, MOE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University. Correspondence to: Feng Zhao <fzhao956@ustc.edu.cn>, Jieping Ye <yejieping.ye@alibaba-inc.com>.",
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+    {
+        "type": "text",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
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+    {
+        "type": "text",
+        "text": "Perspective-Oriented Learning: Multi-Perspective/View Dataset Training",
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+        "img_path": "images/e6efa306705331875e4c158ebf3e0bd71fba0257bba7b1459c646f7d06d4edca.jpg",
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+        "image_caption": [
+            "Perspective-Oriented Learning: Data Augmentation Training"
+        ],
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+        "img_path": "images/4b91a20a03fcde16d56348fd9672913149fc1f4040f301087f3d068aff413907.jpg",
+        "image_caption": [
+            "Perspective-Oriented Learning: Discrete Latent Perspective Learning",
+            "Figure 1. The comparison of perspective-oriented learning methods: Perspective Augmentation, Multi-View Data Training, and the proposed Discrete Latent Perspective Learning."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "abling the network to learn scene information from different viewpoints and ensuring consistency in semantic learning across these varying perspectives, thus improving the portrayal of semantic information within images<sup>1</sup>. As shown in Figure 1, the most commonly used approaches include: (1) collecting multi-view images of a given scene to conduct multi-view training (Yu et al., 2023), and (2) utilizing data augmentation techniques (Cubuk et al., 2019), such as rotation and flipping, to increase the diversity of perspectives.",
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+        "text": "However, in the former case, the acquisition of multi-view images is significantly more challenging than that of single-view images, since the latter can rely on fixed cameras with a wide range of sources (e.g., Internet, vehicle-mounted cameras, surveillance cameras), whereas multi-view im",
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+        "type": "page_footnote",
+        "text": "<sup>1</sup>In this paper, the terms \"perspective\" and \"viewpoint\" are used interchangeably, the former is derived from the theory of photography, while the latter is more commonly employed in general discourse.",
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+        "text": "1",
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+        "image_caption": [
+            "Almost parallel to the ground (Angle: close to  $90^{\\circ}$ )"
+        ],
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+    {
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+        "img_path": "images/1a0aab368969baa2e82708880e1dd20caa8f126be0364e01108d30f54cdc65cd.jpg",
+        "image_caption": [
+            "Oblique View (Angle: about  $40^{\\circ} \\sim 60^{\\circ}$ )"
+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1154b22649af8ab9bc2d6107ad97bdf7655de91e73fdfa2b84e9b92f04210094.jpg",
+        "image_caption": [
+            "Almost perpendicular to the ground (Angle: close to  $0^{\\circ}$ )",
+            "Figure 2. The variation in image perspectives in three typical scenarios: auto-driving, UAV flight, and daily photos. UAV images exhibit more diverse and dynamic viewpoints due to variations in flight heights, angles, and other factors. However, these datasets lack multi-view images for a single scene, preventing existing networks from effective perspective-invariant learning of the images."
+        ],
+        "image_footnote": [],
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+    {
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+        "text": "ages are heavily dependent on manual collection, requiring multi-directional shooting of a scene, which is both time-consuming and labor-intensive. In practice, the costs associated with this method are difficult to justify, and the organization and annotation of multi-view images are also more expensive due to the need to ensure consistency in labeling. In the latter case, data augmentation methods often employ hand-crafted rotation operations to augment image perspectives, aiming to enhance training diversity. However, these operations yield limited performance improvements as they generate similar and rigid results, failing to capture the practical perspective changes encountered in real-world applications, such as Unmanned Aerial Vehicle (UAV) flights, auto-driving and daily photos, as shown in Figure 2.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we propose to perform latent multi-perspective fusion of Invariant Learning within the discrete feature space, based on conventional single-view images. To achieve this, we present a systematic Discrete Latent Perspective Learning (DLPL) framework, comprising Perspec",
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+    {
+        "type": "text",
+        "text": "tive Representation, Transformation, and Multi-Perspective Fusion. Firstly, since perspective lacks a clear formulation in current research, we begin with Perspective Discrete Decomposition (PDD) to discretize the visual feature $^2$  to decompose the latent perspective representation. Following this, in line with photography theory, we treat the perspective as a spatial projection of the image, and transform it to generate new descriptive perspectives with the Perspective Homography Transformation (PHT) module. These new perspectives correspond to alternatives spatial projection of the image. Finally, we integrate both the perspectives for fusion learning, with Perspective Invariant Attention (PIA), to obtain perspective-invariant semantic information of the image, ultimately enhancing the model's ability to depict the image. It is important to note that DLPL is a universal learning framework for image perspectives and applicable broadly to different scenarios (daily photos, UAVs, Autodriving) and various vision tasks (classification, detection, segmentation).",
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+    {
+        "type": "text",
+        "text": "The overall contributions can be summarized as follows:",
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+    {
+        "type": "list",
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+        "list_items": [
+            "- DLPL is a general perspective-invariant learning framework that enables multi-perspective fusion learning within latent feature space for single-view images.",
+            "- Specifically, DLPL systematically includes key processes such as Perspective Representation (PDD), Perspective Transformation (PHT), and Multi-Perspective fusion (Multi-Level PIA).",
+            "- Experiments demonstrate that DLPL achieves impressive results across a wide range of scenarios (daily photos, UAV, Auto-driving) and different vision tasks (detection and segmentation)."
+        ],
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+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1. Scene Understanding",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In existing visual datasets, changes in the perspective of images are very common and pose numerous challenges to image understanding. Among these, high-altitude scenes, such as those captured by UAVs or satellites, feature particularly rich variations in viewpoints. Initial works on core",
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+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "In this paper, we define \"visual feature\" as characteristics extracted directly from input images using conventional CNNs or ViTs, primarily containing visual information and representing the image's semantics. This aligns with the understanding in existing works (Caron et al., 2020; 2018; Jing & Tian, 2020). \"Perspective feature/representation\" refers to features related to the image's perspective, extracted from visual features using our designed Perspective Discrete Decomposition (PDD) module, mainly representing the image's perspective information. Additionally, visual features can be reconstructed from perspective representations through Visual Reconstruction (VR) module.",
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+    {
+        "type": "page_number",
+        "text": "2",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/089855c1cb1b1b81649d4b4a4e8e323efe5f970ddc6245d9dee9447fb34d88cf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b5baa157b4130e1774d7880c0c6ba31d3cb236ad8a13b44432b53bc72fcefc5.jpg",
+        "image_caption": [
+            "Discrete Latent Perspective Learning (DLPL)",
+            "Figure 3. The architecture of DLPL framework within a Vision Transformer (ViT) setup, which is divided into an encoder with a Visual Feature Extraction module and four DLPL blocks, and a task-specific decoder. The DLPL block include: (1) Perspective Discrete Decomposition (PDD) module decomposes input visual features  $\\mathbf{I}$  into a latent representation  $\\mathbf{P}$ ; (2) Visual Reconstruction (VR) module reconstructs visual features from  $\\mathbf{P}$  with a loss function  $\\mathcal{L}_{rec}$ ; (3) Perspective Embedding Space  $\\mathcal{P}$  constructed through EMA, based on  $\\mathbf{P}$ ; (4) Perspective Transformation leverages Homography for perspective variation, followed by VR to obtain reconstructed visual features  $\\mathbf{I}_{rec}^{\\prime}$  of the transformed perspective  $\\mathbf{P}^{\\prime}$ . Finally, Perspective-Invariant Learning fuses the features of the original and transformed perspectives using Perspective-Invariant Attention (PIA) to learn semantic information invariant to perspective changes."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "high-altitude vision tasks like object detection and semantic segmentation (Fu et al., 2022; Ji et al., 2023b;a; Chen et al., 2024b) often adapt network architectures from methods designed for natural scenes to establish baseline results. In contrast, the variations in perspectives within auto-driving (Cordts et al., 2016) and daily photo (Zhou et al., 2017) scenarios are not as pronounced. A plethora of methods have been proposed, including early CNN-based approaches (Chen et al., 2018a; Yuan et al., 2020; Zhu et al., 2021a; Ji et al., 2021; 2022; Hu et al., 2020; Ji et al., 2024a;b; Wang et al., 2021b;a; 2023; Feng et al., 2018; Ji et al., 2019) and recent methodologies based on Vision Transformers (ViTs) (Xie et al., 2021; Zheng et al., 2021; Yu et al., 2022; Carion et al., 2020; Zhu et al., 2023b;c; 2024a), as well as emerging studies on large vision models and multimodal models (Kirillov et al., 2023; Zhu et al., 2023a; Chen et al., 2023b; Zhu et al., 2024b). Nevertheless, economically and effectively addressing the challenges posed by perspective changes remains a difficult issue to be resolved.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2. Perspective-Oriented Learning",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Current work on Perspective Learning primarily focuses on data augmentation techniques based on single-view images (Cordts et al., 2016; Zhou et al., 2017) and joint training with multi-view datasets (Yu et al., 2023; Chen et al., 2023a).",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A substantial body of research on data augmentation techniques (Mumuni & Mumuni, 2022; Cubuk et al., 2019; Chen et al., 2024a) has been conducted in existing works, including earlier strategies based on manual rules and later studies that are based on trained learning approaches. The construction of multi-view datasets is a more recent area of research, which involves capturing scenes and targets from multiple orientations. The creation of such datasets often demands significant human effort, and in tasks requiring fine-grained annotation like segmentation, there is the challenge of maintaining consistency across multiple orientations, that is, ensuring that the fine-grained annotations for the same target from different perspectives have consistent semantic meanings.",
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+    {
+        "type": "text",
+        "text": "3. Discrete Latent Perspective Learning",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "3.1. Top-Down Overview",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DLPL is a flexible and systematic framework for enhancing perspective-invariant learning capabilities in both CNNs and ViT networks. Specifically, as shown in Figure 3, taking a general ViT architecture as an example, DLPL can be implemented in the form of Transformer Blocks. The encoder includes a basic Visual Feature Extraction module and four DLPL Blocks. The decoder, on the other hand,",
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+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/98cfb3ff33355a39c32b2b8ac9b1d21278b19026b63d4c9fe4399f3effddb947.jpg",
+        "image_caption": [
+            "Figure 4. The illustration of PDD. It takes visual features  $\\mathbf{I}$  as input and processes through three key steps: (1) Extraction of perspective prototypes from a point-ness map  $\\mathbf{S}$  using spatially structured supporting points; (2) Discretization of the continuous perspective distribution into  $M$  levels within the latent feature space; (3) Construction of a statistical perspective graph  $\\mathbf{G}$  by encoding the intensity and positional relationships of discretized levels. This process yields the latent perspective feature map  $\\mathbf{P}$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "chooses the appropriate structure based on the specific task type (classification/segmentation/detection).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Extracting Original Perspective Representation (Section 3.2). Within each DLPL Block, given the input visual feature  $\\mathbf{I}$  (which contains the original image perspective), we extract the original perspective representation from  $\\mathbf{I}$ , by the Perspective Discrete Decomposition (PDD) module.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reconstructing Visual Feature (Section 3.3). In order to ensure that the visual feature can be reconstructed from the perspective representation, we design a Visual Reconstruction (VR) module. We constrain  $\\mathbf{I}$  and the reconstructed visual feature  $\\mathbf{I}_{rec}$  to be equivalent through a loss function  $\\mathcal{L}_{rec}$ . This is a self-training process.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Perspective Space Construction (Section 3.4). Concurrently, during training, based on the perspective  $\\mathbf{P}$  of each image, we can incrementally build the Perspective Embedding Space  $\\mathcal{P}$  for the entire dataset. To maintain training stability, we update  $\\mathcal{P}$  using an Exponential Moving Average (EMA) method without backpropagating gradients.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Perspective Transformation (Section 3.5). As the VR's reconstruction ability and  $\\mathcal{P}$  become stable, we leverage the spatial Homography properties of image perspective, combined with the distribution of  $\\mathcal{P}$ , to transform  $\\mathbf{P}$  and obtain another perspective  $\\mathbf{P}'$ , of the image. Then, through VR, we obtain the corresponding visual feature  $\\mathbf{I}_{rec}^{\\prime}$  of  $\\mathbf{P}'$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Perspective-Invariant Learning (Section 3.6). We fuse the visual feature  $\\mathbf{I}$  (with original perspective  $\\mathbf{P}$ ) and",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf{I}_{rec}^{\\prime}$  (with transformed perspective  $\\mathbf{P}^{\\prime}$ ) using multi-level Perspective-Invariant Attention (PIA) to learn the network's semantic expression of image information under different perspectives.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In training, Visual Reconstruction and Perspective Space Construction accompany the entire training process. In contrast, Perspective Transformation training begins after the first two processes stabilize, at the  $30\\%$  of the entire training epoch. Perspective-Invariant Learning also occurs throughout the entire training process. During the initial  $30\\%$  training epoch, since  $\\mathbf{P}'$  is not ready, PIA degenerates to Self-Attention learning solely based on  $\\mathbf{P}$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2. Perspective Discrete Decomposition",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since perspective lacks a clear formulation in current research, we begin with the PDD module. Drawing from visual features extracted by the previous module, we construct prototypes of image perspectives based on key supporting vectors of the perspective. These are then discretized within the latent feature space and statistically analyzed to decompose the implicit perspective distribution map  $\\mathbf{P}$ , as shown in Figure 4.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Formally, PDD receives the visual feature  $\\mathbf{I}^{C\\times H\\times W}$  as input and produces a perspective feature  $\\mathbf{P}\\in \\mathbb{R}^{M\\times M\\times C'}$ , where  $C,C^{\\prime}$  are the feature channels,  $H,W$  are the spatial dimensions,  $M$  is the discretization levels. Generally, PDD exploits the explicit well-formulated distribution of structured supporting points which build up the spatial structured prototype to tell the implicit perspective information. In concrete, PDD consists of three procedures:",
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/74dc6099b2d6aef2840e0f6e7f73bb3eba66e6b87ea7e274e0e12a91c33f36b9.jpg",
+        "image_caption": [
+            "Figure 5. The illustration of Visual Reconstruction (VR). It is essentially the inverse of the PDD process, which reconstructs the visual feature from perspective representation."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            89,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Perspective Prototype Extraction. Besides I, we utilize a pretrained SuperPoint (DeTone et al., 2018) network to extract point-ness map S of the original image, which serve as the supporting structure of image perspective. The theoretical basis behind this motivation stems from related research such as multi-view visual understanding(Ma et al., 2021; Dong et al., 2019; Yaghmur & Ates, 2023), which utilizes the structured spatial information of keypoints as support points to characterize the spatial structure of the image's perspective during actual photography (Zeng et al., 2018; Liu & Li, 2023; Zhang & Hanson, 1996; Li et al., 2020). Next, we apply point-ness attention to I and S to obtain a basic perspective prototype,",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {F} = \\operatorname {S o f t m a x} (\\frac {\\mathbf {I} \\times \\mathbf {S} ^ {\\top}}{\\sqrt {C}}) \\times \\mathbf {S}, \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            517,
+            475,
+            554
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\top$  is the matrix transposition operation.  $\\mathbf{F} \\in \\mathbb{R}^{C \\times H \\times W}$  is the perspective-structured visual feature.",
+        "bbox": [
+            84,
+            561,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discretization. In order to effectively represent perspective within deep neural networks, we firstly perform Discretization on it. Inspired by (Zhu et al., 2021a; Ji et al., 2022), PDD defines  $M$  levels by equally dividing  $\\mathbf{F}$  in the spatial dimension. Specifically the  $m$ -th ( $m \\in [1, M]$ ) level  $\\mathbf{L}_m \\in \\mathbb{R}^C$  is computed as:",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {L} _ {m} = \\frac {m \\cdot \\operatorname* {m a x} (\\mathbf {F}) + (M - m) \\cdot \\operatorname* {m i n} (\\mathbf {F})}{M}. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            133,
+            710,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, based on  $\\mathbf{L}$ , we use  $M$  discretization functions to discretize  $F$  into an embedding  $\\mathbf{D} \\in \\mathbb{R}^{M \\times C \\times H \\times W}$ , where each  $\\mathbf{F}_{i,j}(i \\in [1,H], j \\in [1,W])$  is discretized into  $\\mathbf{D}_{i,j} \\in \\mathbb{R}^{M \\times C}$ . Specifically,",
+        "bbox": [
+            84,
+            752,
+            475,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {D} _ {m, i, j} = \\left\\{ \\begin{array}{l l} \\mathbf {e} - \\left| \\mathbf {L} _ {m} - \\mathbf {F} _ {i, j} \\right| \\\\ \\text {i f} \\left| \\mathbf {L} _ {m} - \\mathbf {F} _ {i, j} \\right| <   \\frac {\\mathbf {L} _ {m + 1} - \\mathbf {L} _ {m}}{2}, \\\\ \\mathbf {0} & \\text {e l s e} \\end{array} \\right. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            89,
+            824,
+            475,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{e} \\in \\mathbb{R}^C$ ,  $\\mathbf{0} \\in \\mathbb{R}^C$  denote the Unit vector and Zero vector respectively. Note a spireshaped function is used instead of a binary one to ensure that the operation is differentiable.  $\\mathbf{D}$  can be viewed as the form of a histogram for perspective distribution. According to the theory of photography, the perspective is related to spatial relationship between objects/pixels within the image. Correspondingly, we encode the spatial pixel co-occurring relationships by:",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {\\mathbf {D}} _ {i, j} = \\mathbf {D} _ {i, j} \\mathbf {D} _ {i, j} ^ {\\top}, \\quad \\hat {\\mathbf {L}} = \\left\\{\\hat {\\mathbf {L}} _ {\\hat {m}} \\right\\}, \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {m} = [ m _ {1}, m _ {2} ], m _ {1} \\in [ 1, M _ {1} ], m _ {2} \\in [ 1, M _ {2} ],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            542,
+            239,
+            839,
+            255
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $M_1 = M_2 = M$ ,  $\\hat{\\mathbf{q}} \\in \\mathbb{R}^{M \\times M \\times C \\times H \\times W}$  is spatial co-occurring perspective histogram, and  $\\hat{\\mathbf{L}}$  is the corresponding co-occurring level.  $\\top$  indicates matrix transposition.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Statistical Representation. Further, we construct the perspective graph  $\\mathbf{G} \\in \\mathbb{R}^{M \\times M \\times C'}$  by squeezing the spatial dimensions in form of counting the intensity of each level, then encoding with position corresponding relationship (Zhu et al., 2021a; Ji et al., 2022):",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {G} _ {\\hat {m}} = \\Phi \\left(\\frac {\\sum_ {i = 1} ^ {H} \\sum_ {j = 1} ^ {W} \\hat {\\mathbf {D}} _ {m , i , j}}{\\sum_ {m _ {1} = 1} ^ {M _ {1}} \\sum_ {m _ {2} = 1} ^ {M _ {2}} \\sum_ {i = 1} ^ {H} \\sum_ {j = 1} ^ {W} \\hat {\\mathbf {D}} _ {m , n , i , j}}\\right) \\tag {5} \\\\ + \\sum_ {i = 1} ^ {H} \\sum_ {j = 1} ^ {W} \\hat {\\mathbf {D}} _ {m _ {1}, m _ {2}, i, j} \\mathrm {P o s \\_ E n c} (i, j)), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            519,
+            414,
+            885,
+            503
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathrm{Pos\\_Enc}$  indicates the position encoding operation. Cat is the concatenation operation.  $\\Phi$  is MLP layer for dimension normalization. As shown in Figure 4,  $\\mathbf{G}$  can be viewed as a Perspective Distribution Graph, where the levels  $\\{L_1L_1,L_1L_2,\\dots,L_{M - 1}L_M,L ML_M\\}$  are nodes. Finally, we correlate the perspective graph over all nodes to generate the overall perspective representation  $\\mathbf{P}\\in \\mathbb{R}^{M\\times M\\times C'}$  ..",
+        "bbox": [
+            496,
+            513,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {P} = \\underset {\\hat {m}} {\\operatorname {M A}} \\left(\\mathbf {G} _ {\\hat {m}}\\right), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            635,
+            631,
+            885,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathrm{MA}(\\cdot)$  indicate performing multi-head attention over all  $\\hat{m}$  levels. By decomposing in form of discrete graph in latent embedding space,  $\\mathbf{P}$  is formulated as pointness-structure-aware thus able to represent the overall perspective information of the image.",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3. Visual Reconstruction",
+        "text_level": 1,
+        "bbox": [
+            496,
+            761,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Section 3.1 and Figure 5, we exploit the VR to reconstruct visual feature  $\\mathbf{I}_{rec}$  from the perspective  $\\mathbf{P}$ . It is essentially the inverse of the PDD process. Given the input perspective feature  $\\mathbf{P}$ , we first restore the spatial dimension using an unsqueeze operation to obtain  $\\hat{\\mathbf{D}}' \\in \\mathbb{R}^{M \\times M \\times (C \\times H \\times W)}$ . Furthermore, we perform levelwise aggregation to mutually enhance the restored spatial information in form of multi-head attention. Afterward, we",
+        "bbox": [
+            496,
+            784,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
+            266,
+            56,
+            705,
+            70
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            480,
+            922,
+            491,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/697eba22500bbde66b20700b85fb197a2ce1aa4cd39f06512f7d321cd44c2ee7.jpg",
+        "image_caption": [
+            "Mini-batch Perspective  $P_{s}$"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            84,
+            375,
+            232
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "embed the discretization levels into the visual feature using a level-wise squeeze method to obtain  $\\mathbf{D}' \\in \\mathbb{R}^{C \\times H \\times W}$ , and finally, a transformer decoder is used to reconstruct the final visual feature  $\\mathbf{I}_{rec}$ . All the unsqueeze and squeeze processes can be accomplished using simple multi-layer MLPs.",
+        "bbox": [
+            84,
+            357,
+            473,
+            434
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to enable the reconstruction ability (Ji et al., 2023b), we use a reconstruction loss to supervise the decoding process to constrain the structure affinity between  $\\mathbf{I}$  and  $\\mathbf{I}_{rec}$ ,",
+        "bbox": [
+            84,
+            440,
+            475,
+            486
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {r e c} = \\left\\| \\mathbf {I} ^ {\\top} \\mathbf {I} - \\mathbf {I} _ {r e c} ^ {\\top} \\mathbf {I} _ {r e c} \\right\\| _ {2}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4. Perspective Embedding Space",
+        "text_level": 1,
+        "bbox": [
+            84,
+            541,
+            328,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on the decomposed perspective representation  $\\mathbf{P}$ , we define a learnable latent perspective embedding space  $\\mathcal{P} \\in \\mathbb{R}^{K \\times (M \\times M \\times C')} = \\{\\mathcal{P}_k \\in \\mathbb{R}^{M \\times M \\times C'}\\}(k \\in [1, K])$ , where  $K$  is the size of the latent space. As shown in Figure 6, in training, we update  $\\mathcal{P}$  as function of exponential moving average (EMA) of the coming mini-batch perspective representations  $\\mathbf{Ps}$  from PDD:",
+        "bbox": [
+            84,
+            565,
+            475,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each  $\\mathbf{P}$  in minibatch  $\\mathbf{Ps}$ ,",
+        "bbox": [
+            106,
+            681,
+            325,
+            695
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {P} _ {k} := \\left\\{ \\begin{array}{c c} \\alpha \\mathcal {P} _ {k} + (1 - \\alpha) \\mathbf {P} & \\\\ \\text {i f} \\quad \\underset {k} {\\arg \\min } | | \\mathcal {P} _ {k} - \\mathbf {P} | | _ {2} = k, \\\\ \\mathcal {P} _ {k} & \\text {e l s e} \\end{array} , \\right. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            104,
+            700,
+            473,
+            765
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\alpha$  is the moving weight and we find  $\\alpha = 0.9$  to work well in practice.",
+        "bbox": [
+            84,
+            773,
+            473,
+            805
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then, we can obtain the overall perspective distribution of the whole dataset in form of Gaussian Mixed Model (GMM) over  $\\mathcal{P}$ , as,",
+        "bbox": [
+            84,
+            811,
+            475,
+            858
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np = \\sum_ {k = 1} ^ {K} \\pi_ {k} \\mathcal {N} \\left(\\mathcal {P} _ {k}, \\boldsymbol {\\Sigma} _ {k}\\right), \\quad \\sum_ {k = 1} ^ {K} \\pi_ {k} = 1, \\quad 0 \\leq \\pi_ {k} \\leq 1, \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            96,
+            866,
+            475,
+            909
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/00ea57055d6c20bfe4988a1d9848800eac163d3e6fb842d17b48d59973a29707.jpg",
+        "image_caption": [
+            "Figure 6. The illustration of the construction of Perspective Embedding Space  $\\mathcal{P}$ , where  $K$  is the size of  $\\mathcal{P}$  and  $\\mathcal{P}_k$  is the  $k$ -th embedding in  $\\mathcal{P}$  ( $k \\in [1, K]$ ). In training, we update  $\\mathcal{P}$  as a function of the exponential moving average (EMA) of the incoming mini-batch perspective  $\\mathbf{Ps}$  from PDD.",
+            "Figure 7. The illustration of PHT. Based on the characteristics of projection transformation, we transform the original perspective  $\\mathbf{P}$  to generate another descriptive perspective  $\\mathbf{P}'$ , which corresponds to an alternative spatial projection of the image."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            501,
+            82,
+            885,
+            290
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{N}(\\cdot)$  indicates the Gaussian Distribution, the  $k$ -th component of GMM has the center of  $\\mathcal{P}^k$  with the variance of  $\\Sigma_k$ , and  $\\pi_k$  is the mixture coefficient.",
+        "bbox": [
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+            398,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5. Perspective Homography Transformation",
+        "text_level": 1,
+        "bbox": [
+            496,
+            460,
+            821,
+            476
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In line with photography theory (Elkins, 2006), we treat the perspective  $\\mathbf{P}$  as a spatial projection of the image (Malis & Vargas Villanueva, 2007). Based on the characteristics of projection transformation, we transform the perspective to generate another descriptive perspective  $\\mathbf{P}'$ , which corresponds to an alternative spatial projection of the image. Based on Homography concept (Kriegman, 2007), the original perspective  $\\mathbf{P}$  and the transformed perspective  $\\mathbf{P}'$  form a Homography, and this transformation can be realized with a Homography matrix.",
+        "bbox": [
+            495,
+            484,
+            887,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 7, in concrete, we generate a close semantic-related transformation  $\\mathbf{P}'$  of the original perspective  $\\mathbf{P}$ , by leveraging the all the perspective distribution and performing Homography transformation:",
+        "bbox": [
+            495,
+            642,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {P} ^ {\\prime} = \\mathbf {H} (\\mathbf {P}, \\mathcal {P}) \\mathbf {P} = \\sum_ {k = 1} ^ {K} H (\\mathbf {P}, \\mathcal {P} _ {k}) p (\\mathbf {P}) \\tag {10} \\\\ = \\sum_ {k = 1} ^ {K} \\pi_ {n} \\mathcal {N} (\\mathbf {P} | \\mathcal {P} _ {k}, \\boldsymbol {\\Sigma} _ {k}) H (\\mathbf {P}, \\mathcal {P} _ {k}), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            557,
+            712,
+            885,
+            797
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{H}(\\mathbf{P},\\mathcal{P})$  is the Homography Matrix of the original perspective  $\\mathbf{P}$  over all the perspective embedding space  $\\mathcal{P}$ , and calculated by the sum of Homography Matrix of  $\\mathbf{P}$  over each  $\\mathcal{P}_k$  in the space, that is,  $\\sum_{k=1}^{K} H(\\mathbf{P},\\mathcal{P}_k)$ .  $H(\\mathbf{P},\\mathcal{P}_k)$  indicates the estimation of Homography matrix between  $\\mathbf{P}$  and  $\\mathcal{P}_k$ . Mathematically, Homography matrix can represent",
+        "bbox": [
+            495,
+            814,
+            887,
+            906
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
+            266,
+            56,
+            705,
+            70
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/937e93d354013f7ff5be23db9f08b6b211119d75028184b0c314263f9ed691b4.jpg",
+        "table_caption": [
+            "Table 1. The scenes, datasets and task settings in our experiments."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Scenes</td><td>Semantic Segmentation</td><td>Object Detection</td></tr><tr><td>Auto-Driving</td><td>Cityscapes (Cordts et al., 2016)</td><td>-</td></tr><tr><td>UAV</td><td>UDD6 (Chen et al., 2018b) iSAID (Waqas Zamir et al., 2019) UAVid (Lyu et al., 2020) Aeroscapes (Nigam et al., 2018) DroneSeg</td><td>VisDrone (Zhu et al., 2021b) UAVDT (Wu et al., 2021)</td></tr><tr><td>Daily Photos</td><td>ADE20K (Zhou et al., 2017)</td><td>-</td></tr></table>",
+        "bbox": [
+            86,
+            107,
+            470,
+            209
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/6380e8b234752610f9db807074372ad3614740a489e34001a998fdea815b7e73.jpg",
+        "table_caption": [
+            "Table 2. UAV Scene Segmentation: Comparison with state-of-the-art on UDD6, iSAID, UAVid, Aeroscapes and our proposed DroneSeg datasets."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Param.</td><td colspan=\"5\">mIoU (%)</td></tr><tr><td>UDD6</td><td>iSAID</td><td>UAVid</td><td>Aeroscapes</td><td>DroneSeg</td></tr><tr><td>Deeplab</td><td>63M</td><td>71.84</td><td>59.20</td><td>56.82</td><td>51.40</td><td>38.69</td></tr><tr><td>OCR_W48</td><td>70M</td><td>73.37</td><td>62.73</td><td>63.10</td><td>58.19</td><td>43.10</td></tr><tr><td>PSPNet</td><td>67M</td><td>72.95</td><td>60.30</td><td>58.20</td><td>57.98</td><td>37.03</td></tr><tr><td>FarSeg</td><td>-</td><td>-</td><td>63.70</td><td>-</td><td>-</td><td>-</td></tr><tr><td>PFNet</td><td>-</td><td>-</td><td>66.90</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SCO</td><td>-</td><td>-</td><td>69.10</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SETR</td><td>318.3M</td><td>68.00</td><td>62.77</td><td>58.52</td><td>50.34</td><td>48.23</td></tr><tr><td>UperNet</td><td>234.0M</td><td>73.13</td><td>66.45</td><td>61.91</td><td>64.32</td><td>53.34</td></tr><tr><td>PoolFormer</td><td>77.1M</td><td>74.54</td><td>65.55</td><td>61.73</td><td>62.27</td><td>53.94</td></tr><tr><td>SegFormer</td><td>84.7M</td><td>74.28</td><td>67.19</td><td>62.01</td><td>66.40</td><td>55.33</td></tr><tr><td>DLPL (SETR)</td><td>321.1M</td><td>72.62</td><td>65.45</td><td>60.23</td><td>55.02</td><td>52.94</td></tr><tr><td>DLPL (UperNet)</td><td>236.8M</td><td>76.00</td><td>68.81</td><td>63.27</td><td>66.25</td><td>56.10</td></tr><tr><td>DLPL (PoolFormer)</td><td>79.9M</td><td>75.98</td><td>68.57</td><td>64.02</td><td>65.03</td><td>56.44</td></tr><tr><td>DLPL (SegFormer)</td><td>87.5M</td><td>76.88</td><td>70.60</td><td>65.30</td><td>68.22</td><td>59.10</td></tr></table>",
+        "bbox": [
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+            282,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "the coordinates transformation from one perspective plane to another. In our formulation,  $\\mathbf{P}$  and  $\\mathcal{P}_k$  act as two perspective planes, thus  $H(\\mathbf{P},\\mathcal{P}_k)$  can characterize the perspective transformation from  $\\mathbf{P}$  to  $\\mathcal{P}_k$ .",
+        "bbox": [
+            84,
+            566,
+            473,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we can obtain the corresponding visual feature  $\\mathbf{I}_{rec}^{\\prime}$  with the transformed perspective  $\\mathbf{P}^{\\prime}$ , by applying the Visual Reconstruction (VR) module to  $\\mathbf{P}^{\\prime}$ .",
+        "bbox": [
+            84,
+            633,
+            473,
+            679
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.6. Perspective-Invariant Learning",
+        "text_level": 1,
+        "bbox": [
+            84,
+            695,
+            338,
+            710
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By the PHT and VR, we obtain the semantic-related perspective-transformed visual feature  $\\mathbf{I}_{rec}^{\\prime}$  for the original visual feature  $\\mathbf{I}$ . As seen that  $\\mathbf{I}$  and  $\\mathbf{I}_{rec}^{\\prime}$  contain closely identical scene context and semantic information yet differ in perspective. Next, they are aggregated by the multi-level Perspective Invariant Attention (PIA) to leverage the relationship between the  $\\mathbf{I}$  (with original perspective  $\\mathbf{P}$ ) and  $\\mathbf{I}_{rec}^{\\prime}$  (with transformed perspective  $\\mathbf{P}^{\\prime}$ ). Formally, PIA is formulated as a cross-attention manner (Chen et al., 2021):",
+        "bbox": [
+            84,
+            719,
+            475,
+            854
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {P I A} \\left(\\mathbf {I}, \\mathbf {I} _ {r e c} ^ {\\prime}\\right) = \\operatorname {S o f t m a x} \\left(\\frac {\\mathbf {I} \\times \\mathbf {I} _ {r e c} ^ {\\prime \\top}}{\\sqrt {C}}\\right) \\times \\mathbf {I} _ {r e c} ^ {\\prime}. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            122,
+            875,
+            475,
+            909
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/231eef952e5bff40aefc41ac8923727c57129e656badcc8e897c8702eda89991.jpg",
+        "table_caption": [
+            "Table 3. UAV Scene Detection: Comparison with state-of-the-arts on VisDrone and UAVDT datasets."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">VisDrone-dev</td><td colspan=\"3\">UAVDT-test</td></tr><tr><td>AP</td><td>\\(AP_{50}\\)</td><td>\\(AP_{75}\\)</td><td>AP</td><td>\\(AP_{50}\\)</td><td>\\(AP_{75}\\)</td></tr><tr><td>Dynamic R-CNN</td><td>13.70</td><td>24.70</td><td>13.50</td><td>57.60</td><td>65.30</td><td>64.6</td></tr><tr><td>RetinaNet</td><td>18.94</td><td>31.67</td><td>20.25</td><td>33.95</td><td>-</td><td>-</td></tr><tr><td>FRCNN(PVT2-B0)</td><td>19.40</td><td>31.80</td><td>20.80</td><td>60.80</td><td>71.50</td><td>69.8</td></tr><tr><td>RefineDet</td><td>19.89</td><td>37.27</td><td>20.18</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DetNet</td><td>20.07</td><td>37.54</td><td>21.26</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FPN</td><td>22.06</td><td>39.57</td><td>22.50</td><td>49.05</td><td>-</td><td>-</td></tr><tr><td>Light-RCNN</td><td>22.08</td><td>39.56</td><td>23.24</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FRCNN(ResNet-50)</td><td>22.40</td><td>37.20</td><td>23.70</td><td>55.70</td><td>69.80</td><td>66.90</td></tr><tr><td>CornerNet</td><td>23.43</td><td>41.18</td><td>25.02</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Cascade-RCNN</td><td>25.20</td><td>40.10</td><td>26.80</td><td>61.50</td><td>73.50</td><td>71.70</td></tr><tr><td>DETR</td><td>24.32</td><td>40.67</td><td>25.89</td><td>61.10</td><td>73.41</td><td>71.20</td></tr><tr><td>Deformable DETR</td><td>26.28</td><td>43.02</td><td>27.90</td><td>62.99</td><td>76.32</td><td>73.78</td></tr><tr><td>Sparse DETR</td><td>26.98</td><td>44.30</td><td>28.70</td><td>63.21</td><td>76.81</td><td>74.22</td></tr><tr><td>DLPL (DETR)</td><td>26.45</td><td>43.28</td><td>27.95</td><td>63.36</td><td>76.80</td><td>74.00</td></tr><tr><td>DLPL (Deformable DETR)</td><td>28.00</td><td>45.89</td><td>29.60</td><td>65.07</td><td>78.67</td><td>76.01</td></tr><tr><td>DLPL (Sparse DETR)</td><td>28.75</td><td>46.38</td><td>30.55</td><td>65.57</td><td>79.02</td><td>76.92</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The fundamental concept is to drive the model to jointly learn perspective-invariant semantic information in the image from multiple distinct viewpoints. Each layer of PIA generates a visual feature containing a new perspective. We cascade multiple layers of PIA and, through repeated perspective-invariant fusion, ultimately enhance the model's robustness to perspective in practical applications (e.g.自动驾驶 or UAV flights).",
+        "bbox": [
+            495,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.7. Optimization",
+        "text_level": 1,
+        "bbox": [
+            496,
+            554,
+            624,
+            569
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The overall loss  $\\mathcal{L}$  is intuitive: the combination of the main task-specific loss  $\\mathcal{L}_{task}$  and the reconstruction loss  $\\mathcal{L}_{rec}$ :",
+        "bbox": [
+            495,
+            578,
+            885,
+            609
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} = \\mathcal {L} _ {\\text {t a s k}} + \\lambda \\mathcal {L} _ {\\text {r e c}}, \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            620,
+            621,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lambda$  is the loss weight and set to 0.4.",
+        "bbox": [
+            496,
+            648,
+            767,
+            664
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            496,
+            684,
+            630,
+            700
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1. Scenes, Datasets, Tasks and Implementation Details",
+        "text_level": 1,
+        "bbox": [
+            496,
+            709,
+            885,
+            724
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DLPL is a general and applicable to a broad range of scenes and tasks. In our experiments, we focus on three types of scenarios (auto-driving, UAV, daily photos) and two fine-grained tasks (segmentation and detection). Furthermore, we concentrate on the more challenging task of segmentation, especially within the UAV scenario due to its rich variety of scene changes. Specifically, given the current lack of a large-scale UAV segmentation benchmark dataset within the field, we propose  $\\text{DroneSeg}^3$ , the largest-scale",
+        "bbox": [
+            495,
+            732,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
+            266,
+            56,
+            704,
+            70
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>3</sup>We release the proposed DroneSeg dataset at https://github. com/jankyee/DroneSeg.",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            480,
+            922,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/d29b27d705d16fa85780212c8dfc810935bd63c28cbdc55c40729628384cab6f.jpg",
+        "table_caption": [
+            "Table 4. Auto-Driving and Daily photos Segmentation: Comparison with state-of-the-arts on Cityscapes and ADE20K datasets."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Backbone</td><td rowspan=\"2\">Pram.</td><td colspan=\"2\">mIoU(%)</td></tr><tr><td>Cityscapes val</td><td>ADE20K val</td></tr><tr><td>FCN</td><td>ResNet-101</td><td>68.6M</td><td>76.6</td><td>41.4</td></tr><tr><td>EncNet</td><td>ResNet-101</td><td>55.1M</td><td>76.9</td><td>44.7</td></tr><tr><td>PSPNet</td><td>ResNet-101</td><td>68.1M</td><td>78.5</td><td>44.4</td></tr><tr><td>CCNet</td><td>ResNet-101</td><td>68.9M</td><td>80.2</td><td>45.2</td></tr><tr><td>DeeplabV3+</td><td>ResNet-101</td><td>62.7M</td><td>80.9</td><td>44.1</td></tr><tr><td>OCRNet</td><td>HRNet-W48</td><td>70.5M</td><td>81.1</td><td>45.6</td></tr><tr><td>SETR</td><td>ViT-Large</td><td>318.3M</td><td>82.2</td><td>50.2</td></tr><tr><td>Segmenter</td><td>ViT-Large</td><td>307.0M</td><td>80.7</td><td>52.2</td></tr><tr><td>SegFormer</td><td>MiT-B4</td><td>64.1M</td><td>83.8</td><td>51.1</td></tr><tr><td>SegFormer</td><td>MiT-B5</td><td>84.7M</td><td>84.0</td><td>51.8</td></tr><tr><td>DLPL (SETR)</td><td>ViT-Large</td><td>321.1M</td><td>83.9</td><td>53.1</td></tr><tr><td>DLPL (Segmenter)</td><td>ViT-Large</td><td>309.8M</td><td>82.3</td><td>53.5</td></tr><tr><td>DLPL (SegFormer)</td><td>MiT-B4</td><td>66.9M</td><td>85.0</td><td>53.5</td></tr><tr><td>DLPL (SegFormer)</td><td>MiT-B5</td><td>87.5M</td><td>85.1</td><td>53.7</td></tr></table>",
+        "bbox": [
+            86,
+            121,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "and semantically richest fine-grained annotated UAV segmentation dataset, to date.",
+        "bbox": [
+            88,
+            441,
+            473,
+            470
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In all experiments, we adopt the MMSegmentation (MMSegmentation, 2020) and MMDetection (Chen et al., 2019) toolboxes as codebases and follow the default basic configurations. In DLPL block, we employ the pretrained and SuperPoint parameters in (DeTone et al., 2018) and froze them in PDD. In our experiments, we incorporate DLPL with various representative ViT segmentors and detectors.",
+        "bbox": [
+            88,
+            479,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2. Comparison with State-of-the-Arts",
+        "text_level": 1,
+        "bbox": [
+            88,
+            602,
+            359,
+            616
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.1.UAVSCENE",
+        "text_level": 1,
+        "bbox": [
+            88,
+            625,
+            220,
+            638
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Compared to the other two types of scenarios, UAVs present more challenging perspective understanding due to the rich variety of perspective changes caused by real-time movement and deviation during flight. Therefore, segmentation and detection in UAV scenarios become the focus of our experiments. As shown in Tables 2 and 3, we apply DLPL to the currently advanced ViT networks and achieve significant performance improvements in segmentation and detection with less than 3M additional parameters. Specifically, on our newly proposed DroneSeg dataset, which presents the level of challenge to date, we achieve the highest improvement of nearly  $5\\%$  over four types of plain ViT. In the UAV detection scenario, where objects are relatively sparse, we also achieve an improvement of nearly  $3\\%$  over various plain ViT detectors. This consistent performance enhancement directly confirms the effectiveness, generality, and robustness of our proposed DLPL.",
+        "bbox": [
+            88,
+            648,
+            473,
+            904
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5. The comparison of DLPL with other perspective-oriented learning methods (data augmentation). Perspective-Vertical and Perspective-Horizontal means adjusting perspectives in vertical and horizontal directions, and are implemented by the default \"torchvisiontransforms\" interfaces in PyTorch (Paszke et al., 2019) For each naive augmentation, we try multiple magnitudes and find the best performance, for fair comparisons.",
+        "bbox": [
+            500,
+            93,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0b4d69840a229d9ed8783781eaccdc0e2d6bad09a49ac36d492acbff0b8a218c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Perspective-Oriented Learning Method</td><td>mIoU (%)</td></tr><tr><td colspan=\"2\">Baseline (SegFormer w/o data augmentation)</td><td>52.03</td></tr><tr><td rowspan=\"5\">Naive Augmentation</td><td>+ Random Rotate</td><td>52.94</td></tr><tr><td>+ Random Scale</td><td>53.09</td></tr><tr><td>+ Random Perspective-Vertical</td><td>53.56</td></tr><tr><td>+ Random Perspective-Horizontal</td><td>53.42</td></tr><tr><td>+ Random Combination</td><td>55.33</td></tr><tr><td colspan=\"2\">AutoAugmentation (Cubuk et al., 2019)</td><td>55.80</td></tr><tr><td colspan=\"2\">DLPL (SegFormer w/o data augmentation)</td><td>58.67</td></tr><tr><td colspan=\"2\">DLPL (SegFormer + Random Combination)</td><td>59.10</td></tr><tr><td colspan=\"2\">DLPL (SegFormer + AutoAugmentation)</td><td>59.09</td></tr></table>",
+        "bbox": [
+            500,
+            189,
+            885,
+            369
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.2. AUTO-DRIVING AND DAILY PHOTOS",
+        "text_level": 1,
+        "bbox": [
+            500,
+            426,
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+            440
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "These two scenes typically also encompass various shooting perspectives. To further validate the universality of DLPL, we conduct experimental verifications within these two types of scenes as well. Specifically, we focused primarily on the segmentation task whose performance is more evidently affected by perspective changes. As shown in Table 4, ViT detectors integrated with DLPL achieve highest performance increases of  $3\\%$  compared to the plain ViT detectors.",
+        "bbox": [
+            500,
+            449,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3. Ablation Study",
+        "text_level": 1,
+        "bbox": [
+            500,
+            602,
+            635,
+            616
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We perform ablation studies on semantic segmentation on DroneSeg test set, SegFormer is used as baseline network.",
+        "bbox": [
+            500,
+            626,
+            883,
+            655
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3.1. THE EFFICACY OF DLPL",
+        "text_level": 1,
+        "bbox": [
+            500,
+            671,
+            720,
+            684
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 2, as one type of perspective-oriented learning, we compare DLPL with the various ordinary perspective augmentation techniques. We notice that in all the scenes, the perspective changes are almost always related to the flying height and angles, which can be reflected on the scales, rotations, perspective-vertical and perspective-horizontal augmentations. So, we use the combination of these data augmentation methods for the comparison with DLPL. Extensively, we also include the auto-augmentation techniques (Cubuk et al., 2019) to further support our claims. As shown in Table 5, DLPL shows more potentials than both naive and auto augmentations. We also show that DLPL is still adaptive to the ordinary data augmentation to obtain further improvements.",
+        "bbox": [
+            500,
+            694,
+            885,
+            904
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
+        "bbox": [
+            267,
+            56,
+            704,
+            70
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/14cf263cf455ac89605b0144f25b49127d9f2d9cfd3e9d34764231494839603b.jpg",
+        "table_caption": [
+            "Table 6. The significance of PDD by comparing it with ordinary perspective representation."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Perspective Representation</td><td>mIoU (%)</td></tr><tr><td>Ordinary Transformer Network</td><td>57.07</td></tr><tr><td>Directly using F</td><td>55.91</td></tr><tr><td>PDD</td><td>59.10</td></tr></table>",
+        "bbox": [
+            135,
+            121,
+            419,
+            195
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c50c8a7c8975ec55797352db8ad2584a1108808a7d32a43d477c9f0292701c2.jpg",
+        "table_caption": [
+            "Table 7. The effectiveness of PHT by comparing with ordinary perspective transformation methods."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Perspective Transformation</td><td>mIoU (%)</td></tr><tr><td>KNN</td><td>56.57</td></tr><tr><td>Momentum Update</td><td>56.01</td></tr><tr><td>Naive GMM</td><td>56.94</td></tr><tr><td>PHT</td><td>59.10</td></tr></table>",
+        "bbox": [
+            148,
+            251,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3.2. THE SIGNIFICANCE OF PDD",
+        "text_level": 1,
+        "bbox": [
+            84,
+            367,
+            333,
+            382
+        ],
+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "For the first time, PDD provides heuristic inspiration for latent perspective representation. We demonstrate the significance of PDD by comparing with other ordinary perspective representation techniques: (1) directly forcing the perspective learning using an ordinary Transformer network, (2) representing the perspective by directly using  $\\mathbf{F}$ , followed by MLPs. The former lacks of in-depth characterization, and the latter may result in coarse-grained learning. As shown in Table 6, PDD shows undisputed brilliant performance, confirming its rationality and effectiveness.",
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+    {
+        "type": "text",
+        "text": "4.3.3.EFFECTIVENESS OF PHT",
+        "text_level": 1,
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+        "type": "text",
+        "text": "PHT is for perspective transformation over the original perspective  $\\mathbf{P}$  and perspective embedding space  $\\mathcal{P}$ , in the form of Homography. In order to validate the effectiveness of such choice, we also implement some other perspective transformation methods for comparisons, including K-Nearest Neighbor (KNN), Momentum Update and naive GMM. Results in Table 7 demonstrate that PHT achieves the highest performance.",
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+        "text": "4.3.4. VISUALIZATION",
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+        "type": "text",
+        "text": "To provide a clearer understanding of perspective transformation, we visualize and compare the feature maps before and after applying DLPL, as depicted in Figure 8. It shows that after passing through DLPL, the perspective reflected in visual feature maps are obviously transformed. For more in-depth insights, we select representative transformation intensities: weak (row 1), medium (row 2), and strong (row 3). These intensities accurately capture the various stages of gradual perspective changes during network training, which is conducive to network training stability and also in line with the practical scene perspective changes such as UAV",
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+        "image_caption": [
+            "(a) Image"
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+        "img_path": "images/3064ae3c9068eb3a2391af4c295a7f7515ce549d419ff44c0104af1a20cea03e.jpg",
+        "image_caption": [
+            "(b) Feature with Original Perspective"
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+    {
+        "type": "image",
+        "img_path": "images/545e887b44b1ea96824feccaf5ee222ce00fea0b6a75250a377fdc6e6ba493a4.jpg",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/a341ba8b1a30ee39d6f94ebc41949afa1df0016a42aff1825fe7a5d48c0975c9.jpg",
+        "image_caption": [
+            "(c) Feature with Transformed Perspective",
+            "Figure 8. Visualization comparison on feature maps before and after DLPL. We select three representative transformation intensities: weak (row 1), medium (row 2), and strong (row 3). We can find the overall views are obviously transformed after DLPL."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "flights. As the training deepens, the network can eventually learn rich perspective diversity.",
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+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We introduce a Discrete Latent Perspective Learning (DLPL) framework that enhances Perspective-Invariant Learning using single-view images. The framework integrates Perspective Discrete Decomposition (PDD), Perspective Homography Transformation (PHT), and Multi-Perspective Fusion via Perspective Invariant Attention (PIA), overcoming the limitations of multi-view data collection and perspective augmentations. Extensive experiments have proven that DLPL can enhance the robustness of neural networks to perspective changes, achieving notable improvements in a wide range of scenes and tasks.",
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+    {
+        "type": "text",
+        "text": "Acknowledgements",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the Anhui Provincial Natural Science Foundation under Grant 2108085UD12, the JKW Research Funds under Grant 20-163-14-LZ-001-004-01, the National Key R&D Program of China under Grant 2020AAA0103902, NSFC (No. 62176155), Shanghai Municipal Science and Technology Major Project, China (2021SHZDZX0102).",
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+    {
+        "type": "text",
+        "text": "We acknowledge the support of GPU cluster built by MCC Lab of Information Science and Technology Institution, USTC.",
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+    {
+        "type": "text",
+        "text": "We would like to express our gratitude to Qi Zhu for his brilliant suggestions regarding the writings and figures of this paper, as well as the insightful comments from the Reviewers, Area Chairs, and Program Chairs.",
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+    {
+        "type": "header",
+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
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+    {
+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.",
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+    {
+        "type": "text",
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+            "Zhu, L., Chen, T., Yin, J., See, S., and Liu, J. Addressing background context bias in few-shot segmentation through iterative modulation. In IEEE/CVF International Conference on Computer Vision, 2024a.",
+            "Zhu, L., Ji, D., Chen, T., Xu, P., Ye, J., and Liu, J. Ibid: Alleviating hallucinations in large vision-language models via image-biased decoding. arXiv preprint arXiv:2402.18476, 2024b.",
+            "Zhu, P., Wen, L., Du, D., Bian, X., Fan, H., Hu, Q., and Ling, H. Detection and tracking meet drones challenge. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(11):7380-7399, 2021b."
+        ],
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+        "text": "Discrete Latent Perspective Learning for Segmentation and Detection",
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2024/Discrete Latent Perspective Learning for Segmentation and Detection/full.md

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+# Discrete Latent Perspective Learning for Segmentation and Detection
+
+Deyi Ji  $^{12}$  Feng Zhao  $^{1}$  Lanyun Zhu  $^{3}$  Wenwei Jin  $^{2}$  Hongtao Lu  $^{4}$  Jieping Ye  $^{2}$
+
+# Abstract
+
+In this paper, we address the challenge of Perspective-Invariant Learning in machine learning and computer vision, which involves enabling a network to understand images from varying perspectives to achieve consistent semantic interpretation. While standard approaches rely on the labor-intensive collection of multi-view images or limited data augmentation techniques, we propose a novel framework, Discrete Latent Perspective Learning (DLPL), for latent multi-perspective fusion learning using conventional single-view images. DLPL comprises three main modules: Perspective Discrete Decomposition (PDD), Perspective Homography Transformation (PHT), and Perspective Invariant Attention (PIA), which work together to discretize visual features, transform perspectives, and fuse multi-perspective semantic information, respectively. DLPL is a universal perspective learning framework applicable to a variety of scenarios and vision tasks. Extensive experiments demonstrate that DLPL significantly enhances the network's capacity to depict images across diverse scenarios (daily photos, UAV,自动驾驶) and tasks (detection, segmentation).
+
+# 1. Introduction
+
+According to the theory of photography (Elkins, 2006; Bengio et al., 2013), image perspective/viewpoint refers to the perception of depth and spatial relationships between objects within an image. In the fields of machine learning and computer vision, current research often employs a variety of methods to enhance networks' Perspective-Invariant Learning (YU2, 2018; LO1, 1997), which involves en
+
+<sup>1</sup>University of Science and Technology of China <sup>2</sup>Alibaba Group <sup>3</sup>Singapore University of Technology and Design <sup>4</sup>Dept. of CSE, MOE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University. Correspondence to: Feng Zhao <fzhao956@ustc.edu.cn>, Jieping Ye <yejieping.ye@alibaba-inc.com>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+Perspective-Oriented Learning: Multi-Perspective/View Dataset Training
+
+![](images/e6efa306705331875e4c158ebf3e0bd71fba0257bba7b1459c646f7d06d4edca.jpg)
+
+![](images/35db8da87d1413012f7df611a70317cbf0abbc2c17fac0699b6de18650e9779e.jpg)  
+Perspective-Oriented Learning: Data Augmentation Training
+
+![](images/4b91a20a03fcde16d56348fd9672913149fc1f4040f301087f3d068aff413907.jpg)  
+Perspective-Oriented Learning: Discrete Latent Perspective Learning  
+Figure 1. The comparison of perspective-oriented learning methods: Perspective Augmentation, Multi-View Data Training, and the proposed Discrete Latent Perspective Learning.
+
+abling the network to learn scene information from different viewpoints and ensuring consistency in semantic learning across these varying perspectives, thus improving the portrayal of semantic information within images<sup>1</sup>. As shown in Figure 1, the most commonly used approaches include: (1) collecting multi-view images of a given scene to conduct multi-view training (Yu et al., 2023), and (2) utilizing data augmentation techniques (Cubuk et al., 2019), such as rotation and flipping, to increase the diversity of perspectives.
+
+However, in the former case, the acquisition of multi-view images is significantly more challenging than that of single-view images, since the latter can rely on fixed cameras with a wide range of sources (e.g., Internet, vehicle-mounted cameras, surveillance cameras), whereas multi-view im
+
+![](images/e08a37c08eb421cc7599ea8c8aec134fedd9bbd85f1c4e7b5c35b6c2c3fc53d3.jpg)
+
+![](images/877e84d15de2e2df5211ed654b3a19d993a8db64195a7e206d1a11a1a29a9e65.jpg)  
+Almost parallel to the ground (Angle: close to  $90^{\circ}$ )
+
+![](images/9617768af39f2f144506850bcbf7838c019703f3809fb5bc4537bb08ab47a1bf.jpg)
+
+![](images/1a0aab368969baa2e82708880e1dd20caa8f126be0364e01108d30f54cdc65cd.jpg)  
+Oblique View (Angle: about  $40^{\circ} \sim 60^{\circ}$ )
+
+![](images/1154b22649af8ab9bc2d6107ad97bdf7655de91e73fdfa2b84e9b92f04210094.jpg)  
+Almost perpendicular to the ground (Angle: close to  $0^{\circ}$ )  
+Figure 2. The variation in image perspectives in three typical scenarios: auto-driving, UAV flight, and daily photos. UAV images exhibit more diverse and dynamic viewpoints due to variations in flight heights, angles, and other factors. However, these datasets lack multi-view images for a single scene, preventing existing networks from effective perspective-invariant learning of the images.
+
+ages are heavily dependent on manual collection, requiring multi-directional shooting of a scene, which is both time-consuming and labor-intensive. In practice, the costs associated with this method are difficult to justify, and the organization and annotation of multi-view images are also more expensive due to the need to ensure consistency in labeling. In the latter case, data augmentation methods often employ hand-crafted rotation operations to augment image perspectives, aiming to enhance training diversity. However, these operations yield limited performance improvements as they generate similar and rigid results, failing to capture the practical perspective changes encountered in real-world applications, such as Unmanned Aerial Vehicle (UAV) flights, auto-driving and daily photos, as shown in Figure 2.
+
+In this paper, we propose to perform latent multi-perspective fusion of Invariant Learning within the discrete feature space, based on conventional single-view images. To achieve this, we present a systematic Discrete Latent Perspective Learning (DLPL) framework, comprising Perspec
+
+tive Representation, Transformation, and Multi-Perspective Fusion. Firstly, since perspective lacks a clear formulation in current research, we begin with Perspective Discrete Decomposition (PDD) to discretize the visual feature $^2$  to decompose the latent perspective representation. Following this, in line with photography theory, we treat the perspective as a spatial projection of the image, and transform it to generate new descriptive perspectives with the Perspective Homography Transformation (PHT) module. These new perspectives correspond to alternatives spatial projection of the image. Finally, we integrate both the perspectives for fusion learning, with Perspective Invariant Attention (PIA), to obtain perspective-invariant semantic information of the image, ultimately enhancing the model's ability to depict the image. It is important to note that DLPL is a universal learning framework for image perspectives and applicable broadly to different scenarios (daily photos, UAVs, Autodriving) and various vision tasks (classification, detection, segmentation).
+
+The overall contributions can be summarized as follows:
+
+- DLPL is a general perspective-invariant learning framework that enables multi-perspective fusion learning within latent feature space for single-view images.  
+- Specifically, DLPL systematically includes key processes such as Perspective Representation (PDD), Perspective Transformation (PHT), and Multi-Perspective fusion (Multi-Level PIA).  
+- Experiments demonstrate that DLPL achieves impressive results across a wide range of scenarios (daily photos, UAV, Auto-driving) and different vision tasks (detection and segmentation).
+
+# 2. Related Work
+
+# 2.1. Scene Understanding
+
+In existing visual datasets, changes in the perspective of images are very common and pose numerous challenges to image understanding. Among these, high-altitude scenes, such as those captured by UAVs or satellites, feature particularly rich variations in viewpoints. Initial works on core
+
+![](images/089855c1cb1b1b81649d4b4a4e8e323efe5f970ddc6245d9dee9447fb34d88cf.jpg)
+
+![](images/3b5baa157b4130e1774d7880c0c6ba31d3cb236ad8a13b44432b53bc72fcefc5.jpg)  
+Discrete Latent Perspective Learning (DLPL)  
+Figure 3. The architecture of DLPL framework within a Vision Transformer (ViT) setup, which is divided into an encoder with a Visual Feature Extraction module and four DLPL blocks, and a task-specific decoder. The DLPL block include: (1) Perspective Discrete Decomposition (PDD) module decomposes input visual features  $\mathbf{I}$  into a latent representation  $\mathbf{P}$ ; (2) Visual Reconstruction (VR) module reconstructs visual features from  $\mathbf{P}$  with a loss function  $\mathcal{L}_{rec}$ ; (3) Perspective Embedding Space  $\mathcal{P}$  constructed through EMA, based on  $\mathbf{P}$ ; (4) Perspective Transformation leverages Homography for perspective variation, followed by VR to obtain reconstructed visual features  $\mathbf{I}_{rec}^{\prime}$  of the transformed perspective  $\mathbf{P}^{\prime}$ . Finally, Perspective-Invariant Learning fuses the features of the original and transformed perspectives using Perspective-Invariant Attention (PIA) to learn semantic information invariant to perspective changes.
+
+high-altitude vision tasks like object detection and semantic segmentation (Fu et al., 2022; Ji et al., 2023b;a; Chen et al., 2024b) often adapt network architectures from methods designed for natural scenes to establish baseline results. In contrast, the variations in perspectives within auto-driving (Cordts et al., 2016) and daily photo (Zhou et al., 2017) scenarios are not as pronounced. A plethora of methods have been proposed, including early CNN-based approaches (Chen et al., 2018a; Yuan et al., 2020; Zhu et al., 2021a; Ji et al., 2021; 2022; Hu et al., 2020; Ji et al., 2024a;b; Wang et al., 2021b;a; 2023; Feng et al., 2018; Ji et al., 2019) and recent methodologies based on Vision Transformers (ViTs) (Xie et al., 2021; Zheng et al., 2021; Yu et al., 2022; Carion et al., 2020; Zhu et al., 2023b;c; 2024a), as well as emerging studies on large vision models and multimodal models (Kirillov et al., 2023; Zhu et al., 2023a; Chen et al., 2023b; Zhu et al., 2024b). Nevertheless, economically and effectively addressing the challenges posed by perspective changes remains a difficult issue to be resolved.
+
+# 2.2. Perspective-Oriented Learning
+
+Current work on Perspective Learning primarily focuses on data augmentation techniques based on single-view images (Cordts et al., 2016; Zhou et al., 2017) and joint training with multi-view datasets (Yu et al., 2023; Chen et al., 2023a).
+
+A substantial body of research on data augmentation techniques (Mumuni & Mumuni, 2022; Cubuk et al., 2019; Chen et al., 2024a) has been conducted in existing works, including earlier strategies based on manual rules and later studies that are based on trained learning approaches. The construction of multi-view datasets is a more recent area of research, which involves capturing scenes and targets from multiple orientations. The creation of such datasets often demands significant human effort, and in tasks requiring fine-grained annotation like segmentation, there is the challenge of maintaining consistency across multiple orientations, that is, ensuring that the fine-grained annotations for the same target from different perspectives have consistent semantic meanings.
+
+# 3. Discrete Latent Perspective Learning
+
+# 3.1. Top-Down Overview
+
+DLPL is a flexible and systematic framework for enhancing perspective-invariant learning capabilities in both CNNs and ViT networks. Specifically, as shown in Figure 3, taking a general ViT architecture as an example, DLPL can be implemented in the form of Transformer Blocks. The encoder includes a basic Visual Feature Extraction module and four DLPL Blocks. The decoder, on the other hand,
+
+![](images/98cfb3ff33355a39c32b2b8ac9b1d21278b19026b63d4c9fe4399f3effddb947.jpg)  
+Figure 4. The illustration of PDD. It takes visual features  $\mathbf{I}$  as input and processes through three key steps: (1) Extraction of perspective prototypes from a point-ness map  $\mathbf{S}$  using spatially structured supporting points; (2) Discretization of the continuous perspective distribution into  $M$  levels within the latent feature space; (3) Construction of a statistical perspective graph  $\mathbf{G}$  by encoding the intensity and positional relationships of discretized levels. This process yields the latent perspective feature map  $\mathbf{P}$ .
+
+chooses the appropriate structure based on the specific task type (classification/segmentation/detection).
+
+Extracting Original Perspective Representation (Section 3.2). Within each DLPL Block, given the input visual feature  $\mathbf{I}$  (which contains the original image perspective), we extract the original perspective representation from  $\mathbf{I}$ , by the Perspective Discrete Decomposition (PDD) module.
+
+Reconstructing Visual Feature (Section 3.3). In order to ensure that the visual feature can be reconstructed from the perspective representation, we design a Visual Reconstruction (VR) module. We constrain  $\mathbf{I}$  and the reconstructed visual feature  $\mathbf{I}_{rec}$  to be equivalent through a loss function  $\mathcal{L}_{rec}$ . This is a self-training process.
+
+Perspective Space Construction (Section 3.4). Concurrently, during training, based on the perspective  $\mathbf{P}$  of each image, we can incrementally build the Perspective Embedding Space  $\mathcal{P}$  for the entire dataset. To maintain training stability, we update  $\mathcal{P}$  using an Exponential Moving Average (EMA) method without backpropagating gradients.
+
+Perspective Transformation (Section 3.5). As the VR's reconstruction ability and  $\mathcal{P}$  become stable, we leverage the spatial Homography properties of image perspective, combined with the distribution of  $\mathcal{P}$ , to transform  $\mathbf{P}$  and obtain another perspective  $\mathbf{P}'$ , of the image. Then, through VR, we obtain the corresponding visual feature  $\mathbf{I}_{rec}^{\prime}$  of  $\mathbf{P}'$ .
+
+Perspective-Invariant Learning (Section 3.6). We fuse the visual feature  $\mathbf{I}$  (with original perspective  $\mathbf{P}$ ) and
+
+$\mathbf{I}_{rec}^{\prime}$  (with transformed perspective  $\mathbf{P}^{\prime}$ ) using multi-level Perspective-Invariant Attention (PIA) to learn the network's semantic expression of image information under different perspectives.
+
+In training, Visual Reconstruction and Perspective Space Construction accompany the entire training process. In contrast, Perspective Transformation training begins after the first two processes stabilize, at the  $30\%$  of the entire training epoch. Perspective-Invariant Learning also occurs throughout the entire training process. During the initial  $30\%$  training epoch, since  $\mathbf{P}'$  is not ready, PIA degenerates to Self-Attention learning solely based on  $\mathbf{P}$ .
+
+# 3.2. Perspective Discrete Decomposition
+
+Since perspective lacks a clear formulation in current research, we begin with the PDD module. Drawing from visual features extracted by the previous module, we construct prototypes of image perspectives based on key supporting vectors of the perspective. These are then discretized within the latent feature space and statistically analyzed to decompose the implicit perspective distribution map  $\mathbf{P}$ , as shown in Figure 4.
+
+Formally, PDD receives the visual feature  $\mathbf{I}^{C\times H\times W}$  as input and produces a perspective feature  $\mathbf{P}\in \mathbb{R}^{M\times M\times C'}$ , where  $C,C^{\prime}$  are the feature channels,  $H,W$  are the spatial dimensions,  $M$  is the discretization levels. Generally, PDD exploits the explicit well-formulated distribution of structured supporting points which build up the spatial structured prototype to tell the implicit perspective information. In concrete, PDD consists of three procedures:
+
+![](images/74dc6099b2d6aef2840e0f6e7f73bb3eba66e6b87ea7e274e0e12a91c33f36b9.jpg)  
+Figure 5. The illustration of Visual Reconstruction (VR). It is essentially the inverse of the PDD process, which reconstructs the visual feature from perspective representation.
+
+Perspective Prototype Extraction. Besides I, we utilize a pretrained SuperPoint (DeTone et al., 2018) network to extract point-ness map S of the original image, which serve as the supporting structure of image perspective. The theoretical basis behind this motivation stems from related research such as multi-view visual understanding(Ma et al., 2021; Dong et al., 2019; Yaghmur & Ates, 2023), which utilizes the structured spatial information of keypoints as support points to characterize the spatial structure of the image's perspective during actual photography (Zeng et al., 2018; Liu & Li, 2023; Zhang & Hanson, 1996; Li et al., 2020). Next, we apply point-ness attention to I and S to obtain a basic perspective prototype,
+
+$$
+\mathbf {F} = \operatorname {S o f t m a x} (\frac {\mathbf {I} \times \mathbf {S} ^ {\top}}{\sqrt {C}}) \times \mathbf {S}, \tag {1}
+$$
+
+where  $\top$  is the matrix transposition operation.  $\mathbf{F} \in \mathbb{R}^{C \times H \times W}$  is the perspective-structured visual feature.
+
+Discretization. In order to effectively represent perspective within deep neural networks, we firstly perform Discretization on it. Inspired by (Zhu et al., 2021a; Ji et al., 2022), PDD defines  $M$  levels by equally dividing  $\mathbf{F}$  in the spatial dimension. Specifically the  $m$ -th ( $m \in [1, M]$ ) level  $\mathbf{L}_m \in \mathbb{R}^C$  is computed as:
+
+$$
+\mathbf {L} _ {m} = \frac {m \cdot \operatorname* {m a x} (\mathbf {F}) + (M - m) \cdot \operatorname* {m i n} (\mathbf {F})}{M}. \tag {2}
+$$
+
+Next, based on  $\mathbf{L}$ , we use  $M$  discretization functions to discretize  $F$  into an embedding  $\mathbf{D} \in \mathbb{R}^{M \times C \times H \times W}$ , where each  $\mathbf{F}_{i,j}(i \in [1,H], j \in [1,W])$  is discretized into  $\mathbf{D}_{i,j} \in \mathbb{R}^{M \times C}$ . Specifically,
+
+$$
+\mathbf {D} _ {m, i, j} = \left\{ \begin{array}{l l} \mathbf {e} - \left| \mathbf {L} _ {m} - \mathbf {F} _ {i, j} \right| \\ \text {i f} \left| \mathbf {L} _ {m} - \mathbf {F} _ {i, j} \right| <   \frac {\mathbf {L} _ {m + 1} - \mathbf {L} _ {m}}{2}, \\ \mathbf {0} & \text {e l s e} \end{array} \right. \tag {3}
+$$
+
+where  $\mathbf{e} \in \mathbb{R}^C$ ,  $\mathbf{0} \in \mathbb{R}^C$  denote the Unit vector and Zero vector respectively. Note a spireshaped function is used instead of a binary one to ensure that the operation is differentiable.  $\mathbf{D}$  can be viewed as the form of a histogram for perspective distribution. According to the theory of photography, the perspective is related to spatial relationship between objects/pixels within the image. Correspondingly, we encode the spatial pixel co-occurring relationships by:
+
+$$
+\hat {\mathbf {D}} _ {i, j} = \mathbf {D} _ {i, j} \mathbf {D} _ {i, j} ^ {\top}, \quad \hat {\mathbf {L}} = \left\{\hat {\mathbf {L}} _ {\hat {m}} \right\}, \tag {4}
+$$
+
+$$
+\hat {m} = [ m _ {1}, m _ {2} ], m _ {1} \in [ 1, M _ {1} ], m _ {2} \in [ 1, M _ {2} ],
+$$
+
+where  $M_1 = M_2 = M$ ,  $\hat{\mathbf{q}} \in \mathbb{R}^{M \times M \times C \times H \times W}$  is spatial co-occurring perspective histogram, and  $\hat{\mathbf{L}}$  is the corresponding co-occurring level.  $\top$  indicates matrix transposition.
+
+Statistical Representation. Further, we construct the perspective graph  $\mathbf{G} \in \mathbb{R}^{M \times M \times C'}$  by squeezing the spatial dimensions in form of counting the intensity of each level, then encoding with position corresponding relationship (Zhu et al., 2021a; Ji et al., 2022):
+
+$$
+\begin{array}{l} \mathbf {G} _ {\hat {m}} = \Phi \left(\frac {\sum_ {i = 1} ^ {H} \sum_ {j = 1} ^ {W} \hat {\mathbf {D}} _ {m , i , j}}{\sum_ {m _ {1} = 1} ^ {M _ {1}} \sum_ {m _ {2} = 1} ^ {M _ {2}} \sum_ {i = 1} ^ {H} \sum_ {j = 1} ^ {W} \hat {\mathbf {D}} _ {m , n , i , j}}\right) \tag {5} \\ + \sum_ {i = 1} ^ {H} \sum_ {j = 1} ^ {W} \hat {\mathbf {D}} _ {m _ {1}, m _ {2}, i, j} \mathrm {P o s \_ E n c} (i, j)), \\ \end{array}
+$$
+
+where  $\mathrm{Pos\_Enc}$  indicates the position encoding operation. Cat is the concatenation operation.  $\Phi$  is MLP layer for dimension normalization. As shown in Figure 4,  $\mathbf{G}$  can be viewed as a Perspective Distribution Graph, where the levels  $\{L_1L_1,L_1L_2,\dots,L_{M - 1}L_M,L ML_M\}$  are nodes. Finally, we correlate the perspective graph over all nodes to generate the overall perspective representation  $\mathbf{P}\in \mathbb{R}^{M\times M\times C'}$  ..
+
+$$
+\mathbf {P} = \underset {\hat {m}} {\operatorname {M A}} \left(\mathbf {G} _ {\hat {m}}\right), \tag {6}
+$$
+
+where  $\mathrm{MA}(\cdot)$  indicate performing multi-head attention over all  $\hat{m}$  levels. By decomposing in form of discrete graph in latent embedding space,  $\mathbf{P}$  is formulated as pointness-structure-aware thus able to represent the overall perspective information of the image.
+
+# 3.3. Visual Reconstruction
+
+As illustrated in Section 3.1 and Figure 5, we exploit the VR to reconstruct visual feature  $\mathbf{I}_{rec}$  from the perspective  $\mathbf{P}$ . It is essentially the inverse of the PDD process. Given the input perspective feature  $\mathbf{P}$ , we first restore the spatial dimension using an unsqueeze operation to obtain  $\hat{\mathbf{D}}' \in \mathbb{R}^{M \times M \times (C \times H \times W)}$ . Furthermore, we perform levelwise aggregation to mutually enhance the restored spatial information in form of multi-head attention. Afterward, we
+
+![](images/697eba22500bbde66b20700b85fb197a2ce1aa4cd39f06512f7d321cd44c2ee7.jpg)  
+Mini-batch Perspective  $P_{s}$
+
+embed the discretization levels into the visual feature using a level-wise squeeze method to obtain  $\mathbf{D}' \in \mathbb{R}^{C \times H \times W}$ , and finally, a transformer decoder is used to reconstruct the final visual feature  $\mathbf{I}_{rec}$ . All the unsqueeze and squeeze processes can be accomplished using simple multi-layer MLPs.
+
+In order to enable the reconstruction ability (Ji et al., 2023b), we use a reconstruction loss to supervise the decoding process to constrain the structure affinity between  $\mathbf{I}$  and  $\mathbf{I}_{rec}$ ,
+
+$$
+\mathcal {L} _ {r e c} = \left\| \mathbf {I} ^ {\top} \mathbf {I} - \mathbf {I} _ {r e c} ^ {\top} \mathbf {I} _ {r e c} \right\| _ {2}. \tag {7}
+$$
+
+# 3.4. Perspective Embedding Space
+
+Based on the decomposed perspective representation  $\mathbf{P}$ , we define a learnable latent perspective embedding space  $\mathcal{P} \in \mathbb{R}^{K \times (M \times M \times C')} = \{\mathcal{P}_k \in \mathbb{R}^{M \times M \times C'}\}(k \in [1, K])$ , where  $K$  is the size of the latent space. As shown in Figure 6, in training, we update  $\mathcal{P}$  as function of exponential moving average (EMA) of the coming mini-batch perspective representations  $\mathbf{Ps}$  from PDD:
+
+For each  $\mathbf{P}$  in minibatch  $\mathbf{Ps}$ ,
+
+$$
+\mathcal {P} _ {k} := \left\{ \begin{array}{c c} \alpha \mathcal {P} _ {k} + (1 - \alpha) \mathbf {P} & \\ \text {i f} \quad \underset {k} {\arg \min } | | \mathcal {P} _ {k} - \mathbf {P} | | _ {2} = k, \\ \mathcal {P} _ {k} & \text {e l s e} \end{array} , \right. \tag {8}
+$$
+
+where  $\alpha$  is the moving weight and we find  $\alpha = 0.9$  to work well in practice.
+
+Then, we can obtain the overall perspective distribution of the whole dataset in form of Gaussian Mixed Model (GMM) over  $\mathcal{P}$ , as,
+
+$$
+p = \sum_ {k = 1} ^ {K} \pi_ {k} \mathcal {N} \left(\mathcal {P} _ {k}, \boldsymbol {\Sigma} _ {k}\right), \quad \sum_ {k = 1} ^ {K} \pi_ {k} = 1, \quad 0 \leq \pi_ {k} \leq 1, \tag {9}
+$$
+
+![](images/00ea57055d6c20bfe4988a1d9848800eac163d3e6fb842d17b48d59973a29707.jpg)  
+Figure 6. The illustration of the construction of Perspective Embedding Space  $\mathcal{P}$ , where  $K$  is the size of  $\mathcal{P}$  and  $\mathcal{P}_k$  is the  $k$ -th embedding in  $\mathcal{P}$  ( $k \in [1, K]$ ). In training, we update  $\mathcal{P}$  as a function of the exponential moving average (EMA) of the incoming mini-batch perspective  $\mathbf{Ps}$  from PDD.  
+Figure 7. The illustration of PHT. Based on the characteristics of projection transformation, we transform the original perspective  $\mathbf{P}$  to generate another descriptive perspective  $\mathbf{P}'$ , which corresponds to an alternative spatial projection of the image.
+
+where  $\mathcal{N}(\cdot)$  indicates the Gaussian Distribution, the  $k$ -th component of GMM has the center of  $\mathcal{P}^k$  with the variance of  $\Sigma_k$ , and  $\pi_k$  is the mixture coefficient.
+
+# 3.5. Perspective Homography Transformation
+
+In line with photography theory (Elkins, 2006), we treat the perspective  $\mathbf{P}$  as a spatial projection of the image (Malis & Vargas Villanueva, 2007). Based on the characteristics of projection transformation, we transform the perspective to generate another descriptive perspective  $\mathbf{P}'$ , which corresponds to an alternative spatial projection of the image. Based on Homography concept (Kriegman, 2007), the original perspective  $\mathbf{P}$  and the transformed perspective  $\mathbf{P}'$  form a Homography, and this transformation can be realized with a Homography matrix.
+
+As shown in Figure 7, in concrete, we generate a close semantic-related transformation  $\mathbf{P}'$  of the original perspective  $\mathbf{P}$ , by leveraging the all the perspective distribution and performing Homography transformation:
+
+$$
+\begin{array}{l} \mathbf {P} ^ {\prime} = \mathbf {H} (\mathbf {P}, \mathcal {P}) \mathbf {P} = \sum_ {k = 1} ^ {K} H (\mathbf {P}, \mathcal {P} _ {k}) p (\mathbf {P}) \tag {10} \\ = \sum_ {k = 1} ^ {K} \pi_ {n} \mathcal {N} (\mathbf {P} | \mathcal {P} _ {k}, \boldsymbol {\Sigma} _ {k}) H (\mathbf {P}, \mathcal {P} _ {k}), \\ \end{array}
+$$
+
+where  $\mathbf{H}(\mathbf{P},\mathcal{P})$  is the Homography Matrix of the original perspective  $\mathbf{P}$  over all the perspective embedding space  $\mathcal{P}$ , and calculated by the sum of Homography Matrix of  $\mathbf{P}$  over each  $\mathcal{P}_k$  in the space, that is,  $\sum_{k=1}^{K} H(\mathbf{P},\mathcal{P}_k)$ .  $H(\mathbf{P},\mathcal{P}_k)$  indicates the estimation of Homography matrix between  $\mathbf{P}$  and  $\mathcal{P}_k$ . Mathematically, Homography matrix can represent
+
+Table 1. The scenes, datasets and task settings in our experiments.  
+
+<table><tr><td>Scenes</td><td>Semantic Segmentation</td><td>Object Detection</td></tr><tr><td>Auto-Driving</td><td>Cityscapes (Cordts et al., 2016)</td><td>-</td></tr><tr><td>UAV</td><td>UDD6 (Chen et al., 2018b) iSAID (Waqas Zamir et al., 2019) UAVid (Lyu et al., 2020) Aeroscapes (Nigam et al., 2018) DroneSeg</td><td>VisDrone (Zhu et al., 2021b) UAVDT (Wu et al., 2021)</td></tr><tr><td>Daily Photos</td><td>ADE20K (Zhou et al., 2017)</td><td>-</td></tr></table>
+
+Table 2. UAV Scene Segmentation: Comparison with state-of-the-art on UDD6, iSAID, UAVid, Aeroscapes and our proposed DroneSeg datasets.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Param.</td><td colspan="5">mIoU (%)</td></tr><tr><td>UDD6</td><td>iSAID</td><td>UAVid</td><td>Aeroscapes</td><td>DroneSeg</td></tr><tr><td>Deeplab</td><td>63M</td><td>71.84</td><td>59.20</td><td>56.82</td><td>51.40</td><td>38.69</td></tr><tr><td>OCR_W48</td><td>70M</td><td>73.37</td><td>62.73</td><td>63.10</td><td>58.19</td><td>43.10</td></tr><tr><td>PSPNet</td><td>67M</td><td>72.95</td><td>60.30</td><td>58.20</td><td>57.98</td><td>37.03</td></tr><tr><td>FarSeg</td><td>-</td><td>-</td><td>63.70</td><td>-</td><td>-</td><td>-</td></tr><tr><td>PFNet</td><td>-</td><td>-</td><td>66.90</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SCO</td><td>-</td><td>-</td><td>69.10</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SETR</td><td>318.3M</td><td>68.00</td><td>62.77</td><td>58.52</td><td>50.34</td><td>48.23</td></tr><tr><td>UperNet</td><td>234.0M</td><td>73.13</td><td>66.45</td><td>61.91</td><td>64.32</td><td>53.34</td></tr><tr><td>PoolFormer</td><td>77.1M</td><td>74.54</td><td>65.55</td><td>61.73</td><td>62.27</td><td>53.94</td></tr><tr><td>SegFormer</td><td>84.7M</td><td>74.28</td><td>67.19</td><td>62.01</td><td>66.40</td><td>55.33</td></tr><tr><td>DLPL (SETR)</td><td>321.1M</td><td>72.62</td><td>65.45</td><td>60.23</td><td>55.02</td><td>52.94</td></tr><tr><td>DLPL (UperNet)</td><td>236.8M</td><td>76.00</td><td>68.81</td><td>63.27</td><td>66.25</td><td>56.10</td></tr><tr><td>DLPL (PoolFormer)</td><td>79.9M</td><td>75.98</td><td>68.57</td><td>64.02</td><td>65.03</td><td>56.44</td></tr><tr><td>DLPL (SegFormer)</td><td>87.5M</td><td>76.88</td><td>70.60</td><td>65.30</td><td>68.22</td><td>59.10</td></tr></table>
+
+the coordinates transformation from one perspective plane to another. In our formulation,  $\mathbf{P}$  and  $\mathcal{P}_k$  act as two perspective planes, thus  $H(\mathbf{P},\mathcal{P}_k)$  can characterize the perspective transformation from  $\mathbf{P}$  to  $\mathcal{P}_k$ .
+
+Finally, we can obtain the corresponding visual feature  $\mathbf{I}_{rec}^{\prime}$  with the transformed perspective  $\mathbf{P}^{\prime}$ , by applying the Visual Reconstruction (VR) module to  $\mathbf{P}^{\prime}$ .
+
+# 3.6. Perspective-Invariant Learning
+
+By the PHT and VR, we obtain the semantic-related perspective-transformed visual feature  $\mathbf{I}_{rec}^{\prime}$  for the original visual feature  $\mathbf{I}$ . As seen that  $\mathbf{I}$  and  $\mathbf{I}_{rec}^{\prime}$  contain closely identical scene context and semantic information yet differ in perspective. Next, they are aggregated by the multi-level Perspective Invariant Attention (PIA) to leverage the relationship between the  $\mathbf{I}$  (with original perspective  $\mathbf{P}$ ) and  $\mathbf{I}_{rec}^{\prime}$  (with transformed perspective  $\mathbf{P}^{\prime}$ ). Formally, PIA is formulated as a cross-attention manner (Chen et al., 2021):
+
+$$
+\operatorname {P I A} \left(\mathbf {I}, \mathbf {I} _ {r e c} ^ {\prime}\right) = \operatorname {S o f t m a x} \left(\frac {\mathbf {I} \times \mathbf {I} _ {r e c} ^ {\prime \top}}{\sqrt {C}}\right) \times \mathbf {I} _ {r e c} ^ {\prime}. \tag {11}
+$$
+
+Table 3. UAV Scene Detection: Comparison with state-of-the-arts on VisDrone and UAVDT datasets.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">VisDrone-dev</td><td colspan="3">UAVDT-test</td></tr><tr><td>AP</td><td>\(AP_{50}\)</td><td>\(AP_{75}\)</td><td>AP</td><td>\(AP_{50}\)</td><td>\(AP_{75}\)</td></tr><tr><td>Dynamic R-CNN</td><td>13.70</td><td>24.70</td><td>13.50</td><td>57.60</td><td>65.30</td><td>64.6</td></tr><tr><td>RetinaNet</td><td>18.94</td><td>31.67</td><td>20.25</td><td>33.95</td><td>-</td><td>-</td></tr><tr><td>FRCNN(PVT2-B0)</td><td>19.40</td><td>31.80</td><td>20.80</td><td>60.80</td><td>71.50</td><td>69.8</td></tr><tr><td>RefineDet</td><td>19.89</td><td>37.27</td><td>20.18</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DetNet</td><td>20.07</td><td>37.54</td><td>21.26</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FPN</td><td>22.06</td><td>39.57</td><td>22.50</td><td>49.05</td><td>-</td><td>-</td></tr><tr><td>Light-RCNN</td><td>22.08</td><td>39.56</td><td>23.24</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FRCNN(ResNet-50)</td><td>22.40</td><td>37.20</td><td>23.70</td><td>55.70</td><td>69.80</td><td>66.90</td></tr><tr><td>CornerNet</td><td>23.43</td><td>41.18</td><td>25.02</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Cascade-RCNN</td><td>25.20</td><td>40.10</td><td>26.80</td><td>61.50</td><td>73.50</td><td>71.70</td></tr><tr><td>DETR</td><td>24.32</td><td>40.67</td><td>25.89</td><td>61.10</td><td>73.41</td><td>71.20</td></tr><tr><td>Deformable DETR</td><td>26.28</td><td>43.02</td><td>27.90</td><td>62.99</td><td>76.32</td><td>73.78</td></tr><tr><td>Sparse DETR</td><td>26.98</td><td>44.30</td><td>28.70</td><td>63.21</td><td>76.81</td><td>74.22</td></tr><tr><td>DLPL (DETR)</td><td>26.45</td><td>43.28</td><td>27.95</td><td>63.36</td><td>76.80</td><td>74.00</td></tr><tr><td>DLPL (Deformable DETR)</td><td>28.00</td><td>45.89</td><td>29.60</td><td>65.07</td><td>78.67</td><td>76.01</td></tr><tr><td>DLPL (Sparse DETR)</td><td>28.75</td><td>46.38</td><td>30.55</td><td>65.57</td><td>79.02</td><td>76.92</td></tr></table>
+
+The fundamental concept is to drive the model to jointly learn perspective-invariant semantic information in the image from multiple distinct viewpoints. Each layer of PIA generates a visual feature containing a new perspective. We cascade multiple layers of PIA and, through repeated perspective-invariant fusion, ultimately enhance the model's robustness to perspective in practical applications (e.g.自动驾驶 or UAV flights).
+
+# 3.7. Optimization
+
+The overall loss  $\mathcal{L}$  is intuitive: the combination of the main task-specific loss  $\mathcal{L}_{task}$  and the reconstruction loss  $\mathcal{L}_{rec}$ :
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {t a s k}} + \lambda \mathcal {L} _ {\text {r e c}}, \tag {12}
+$$
+
+where  $\lambda$  is the loss weight and set to 0.4.
+
+# 4. Experiments
+
+# 4.1. Scenes, Datasets, Tasks and Implementation Details
+
+DLPL is a general and applicable to a broad range of scenes and tasks. In our experiments, we focus on three types of scenarios (auto-driving, UAV, daily photos) and two fine-grained tasks (segmentation and detection). Furthermore, we concentrate on the more challenging task of segmentation, especially within the UAV scenario due to its rich variety of scene changes. Specifically, given the current lack of a large-scale UAV segmentation benchmark dataset within the field, we propose  $\text{DroneSeg}^3$ , the largest-scale
+
+Table 4. Auto-Driving and Daily photos Segmentation: Comparison with state-of-the-arts on Cityscapes and ADE20K datasets.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td rowspan="2">Pram.</td><td colspan="2">mIoU(%)</td></tr><tr><td>Cityscapes val</td><td>ADE20K val</td></tr><tr><td>FCN</td><td>ResNet-101</td><td>68.6M</td><td>76.6</td><td>41.4</td></tr><tr><td>EncNet</td><td>ResNet-101</td><td>55.1M</td><td>76.9</td><td>44.7</td></tr><tr><td>PSPNet</td><td>ResNet-101</td><td>68.1M</td><td>78.5</td><td>44.4</td></tr><tr><td>CCNet</td><td>ResNet-101</td><td>68.9M</td><td>80.2</td><td>45.2</td></tr><tr><td>DeeplabV3+</td><td>ResNet-101</td><td>62.7M</td><td>80.9</td><td>44.1</td></tr><tr><td>OCRNet</td><td>HRNet-W48</td><td>70.5M</td><td>81.1</td><td>45.6</td></tr><tr><td>SETR</td><td>ViT-Large</td><td>318.3M</td><td>82.2</td><td>50.2</td></tr><tr><td>Segmenter</td><td>ViT-Large</td><td>307.0M</td><td>80.7</td><td>52.2</td></tr><tr><td>SegFormer</td><td>MiT-B4</td><td>64.1M</td><td>83.8</td><td>51.1</td></tr><tr><td>SegFormer</td><td>MiT-B5</td><td>84.7M</td><td>84.0</td><td>51.8</td></tr><tr><td>DLPL (SETR)</td><td>ViT-Large</td><td>321.1M</td><td>83.9</td><td>53.1</td></tr><tr><td>DLPL (Segmenter)</td><td>ViT-Large</td><td>309.8M</td><td>82.3</td><td>53.5</td></tr><tr><td>DLPL (SegFormer)</td><td>MiT-B4</td><td>66.9M</td><td>85.0</td><td>53.5</td></tr><tr><td>DLPL (SegFormer)</td><td>MiT-B5</td><td>87.5M</td><td>85.1</td><td>53.7</td></tr></table>
+
+and semantically richest fine-grained annotated UAV segmentation dataset, to date.
+
+In all experiments, we adopt the MMSegmentation (MMSegmentation, 2020) and MMDetection (Chen et al., 2019) toolboxes as codebases and follow the default basic configurations. In DLPL block, we employ the pretrained and SuperPoint parameters in (DeTone et al., 2018) and froze them in PDD. In our experiments, we incorporate DLPL with various representative ViT segmentors and detectors.
+
+# 4.2. Comparison with State-of-the-Arts
+
+# 4.2.1.UAVSCENE
+
+Compared to the other two types of scenarios, UAVs present more challenging perspective understanding due to the rich variety of perspective changes caused by real-time movement and deviation during flight. Therefore, segmentation and detection in UAV scenarios become the focus of our experiments. As shown in Tables 2 and 3, we apply DLPL to the currently advanced ViT networks and achieve significant performance improvements in segmentation and detection with less than 3M additional parameters. Specifically, on our newly proposed DroneSeg dataset, which presents the level of challenge to date, we achieve the highest improvement of nearly  $5\%$  over four types of plain ViT. In the UAV detection scenario, where objects are relatively sparse, we also achieve an improvement of nearly  $3\%$  over various plain ViT detectors. This consistent performance enhancement directly confirms the effectiveness, generality, and robustness of our proposed DLPL.
+
+Table 5. The comparison of DLPL with other perspective-oriented learning methods (data augmentation). Perspective-Vertical and Perspective-Horizontal means adjusting perspectives in vertical and horizontal directions, and are implemented by the default "torchvisiontransforms" interfaces in PyTorch (Paszke et al., 2019) For each naive augmentation, we try multiple magnitudes and find the best performance, for fair comparisons.
+
+<table><tr><td colspan="2">Perspective-Oriented Learning Method</td><td>mIoU (%)</td></tr><tr><td colspan="2">Baseline (SegFormer w/o data augmentation)</td><td>52.03</td></tr><tr><td rowspan="5">Naive Augmentation</td><td>+ Random Rotate</td><td>52.94</td></tr><tr><td>+ Random Scale</td><td>53.09</td></tr><tr><td>+ Random Perspective-Vertical</td><td>53.56</td></tr><tr><td>+ Random Perspective-Horizontal</td><td>53.42</td></tr><tr><td>+ Random Combination</td><td>55.33</td></tr><tr><td colspan="2">AutoAugmentation (Cubuk et al., 2019)</td><td>55.80</td></tr><tr><td colspan="2">DLPL (SegFormer w/o data augmentation)</td><td>58.67</td></tr><tr><td colspan="2">DLPL (SegFormer + Random Combination)</td><td>59.10</td></tr><tr><td colspan="2">DLPL (SegFormer + AutoAugmentation)</td><td>59.09</td></tr></table>
+
+# 4.2.2. AUTO-DRIVING AND DAILY PHOTOS
+
+These two scenes typically also encompass various shooting perspectives. To further validate the universality of DLPL, we conduct experimental verifications within these two types of scenes as well. Specifically, we focused primarily on the segmentation task whose performance is more evidently affected by perspective changes. As shown in Table 4, ViT detectors integrated with DLPL achieve highest performance increases of  $3\%$  compared to the plain ViT detectors.
+
+# 4.3. Ablation Study
+
+We perform ablation studies on semantic segmentation on DroneSeg test set, SegFormer is used as baseline network.
+
+# 4.3.1. THE EFFICACY OF DLPL
+
+As discussed in Section 2, as one type of perspective-oriented learning, we compare DLPL with the various ordinary perspective augmentation techniques. We notice that in all the scenes, the perspective changes are almost always related to the flying height and angles, which can be reflected on the scales, rotations, perspective-vertical and perspective-horizontal augmentations. So, we use the combination of these data augmentation methods for the comparison with DLPL. Extensively, we also include the auto-augmentation techniques (Cubuk et al., 2019) to further support our claims. As shown in Table 5, DLPL shows more potentials than both naive and auto augmentations. We also show that DLPL is still adaptive to the ordinary data augmentation to obtain further improvements.
+
+Table 6. The significance of PDD by comparing it with ordinary perspective representation.  
+
+<table><tr><td>Perspective Representation</td><td>mIoU (%)</td></tr><tr><td>Ordinary Transformer Network</td><td>57.07</td></tr><tr><td>Directly using F</td><td>55.91</td></tr><tr><td>PDD</td><td>59.10</td></tr></table>
+
+Table 7. The effectiveness of PHT by comparing with ordinary perspective transformation methods.  
+
+<table><tr><td>Perspective Transformation</td><td>mIoU (%)</td></tr><tr><td>KNN</td><td>56.57</td></tr><tr><td>Momentum Update</td><td>56.01</td></tr><tr><td>Naive GMM</td><td>56.94</td></tr><tr><td>PHT</td><td>59.10</td></tr></table>
+
+# 4.3.2. THE SIGNIFICANCE OF PDD
+
+For the first time, PDD provides heuristic inspiration for latent perspective representation. We demonstrate the significance of PDD by comparing with other ordinary perspective representation techniques: (1) directly forcing the perspective learning using an ordinary Transformer network, (2) representing the perspective by directly using  $\mathbf{F}$ , followed by MLPs. The former lacks of in-depth characterization, and the latter may result in coarse-grained learning. As shown in Table 6, PDD shows undisputed brilliant performance, confirming its rationality and effectiveness.
+
+# 4.3.3.EFFECTIVENESS OF PHT
+
+PHT is for perspective transformation over the original perspective  $\mathbf{P}$  and perspective embedding space  $\mathcal{P}$ , in the form of Homography. In order to validate the effectiveness of such choice, we also implement some other perspective transformation methods for comparisons, including K-Nearest Neighbor (KNN), Momentum Update and naive GMM. Results in Table 7 demonstrate that PHT achieves the highest performance.
+
+# 4.3.4. VISUALIZATION
+
+To provide a clearer understanding of perspective transformation, we visualize and compare the feature maps before and after applying DLPL, as depicted in Figure 8. It shows that after passing through DLPL, the perspective reflected in visual feature maps are obviously transformed. For more in-depth insights, we select representative transformation intensities: weak (row 1), medium (row 2), and strong (row 3). These intensities accurately capture the various stages of gradual perspective changes during network training, which is conducive to network training stability and also in line with the practical scene perspective changes such as UAV
+
+![](images/f7459983d863a0616d6e6ea3988f990e85a720c2f7439c21c9704f3cef61979c.jpg)
+
+![](images/70d780e0d694c1082a177eaa7ec28640fe1905ea9d53003ea215215e56eff06e.jpg)
+
+![](images/160a9567c190763b2c6652d708c595cb9167ce8ccba8f8ff3a2d40d78b106b5e.jpg)  
+(a) Image
+
+![](images/2eef43c81d1ace8720484d744d4c5bf0900cd9197e0d593f57f10c7e17689fae.jpg)
+
+![](images/dafd57358d9073b28b2d55ca542ac98bc57422b7dfc6623bf16f00e9eb0be962.jpg)
+
+![](images/3064ae3c9068eb3a2391af4c295a7f7515ce549d419ff44c0104af1a20cea03e.jpg)  
+(b) Feature with Original Perspective
+
+![](images/4514cd9c7fb3f11e17a28c9c1d23beea6f22c5507bbd7a7366c255a2ef3f356a.jpg)
+
+![](images/545e887b44b1ea96824feccaf5ee222ce00fea0b6a75250a377fdc6e6ba493a4.jpg)
+
+![](images/a341ba8b1a30ee39d6f94ebc41949afa1df0016a42aff1825fe7a5d48c0975c9.jpg)  
+(c) Feature with Transformed Perspective  
+Figure 8. Visualization comparison on feature maps before and after DLPL. We select three representative transformation intensities: weak (row 1), medium (row 2), and strong (row 3). We can find the overall views are obviously transformed after DLPL.
+
+flights. As the training deepens, the network can eventually learn rich perspective diversity.
+
+# 5. Conclusion
+
+We introduce a Discrete Latent Perspective Learning (DLPL) framework that enhances Perspective-Invariant Learning using single-view images. The framework integrates Perspective Discrete Decomposition (PDD), Perspective Homography Transformation (PHT), and Multi-Perspective Fusion via Perspective Invariant Attention (PIA), overcoming the limitations of multi-view data collection and perspective augmentations. Extensive experiments have proven that DLPL can enhance the robustness of neural networks to perspective changes, achieving notable improvements in a wide range of scenes and tasks.
+
+# Acknowledgements
+
+This work was supported by the Anhui Provincial Natural Science Foundation under Grant 2108085UD12, the JKW Research Funds under Grant 20-163-14-LZ-001-004-01, the National Key R&D Program of China under Grant 2020AAA0103902, NSFC (No. 62176155), Shanghai Municipal Science and Technology Major Project, China (2021SHZDZX0102).
+
+We acknowledge the support of GPU cluster built by MCC Lab of Information Science and Technology Institution, USTC.
+
+We would like to express our gratitude to Qi Zhu for his brilliant suggestions regarding the writings and figures of this paper, as well as the insightful comments from the Reviewers, Area Chairs, and Program Chairs.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+# Disentangled 3D Scene Generation with Layout Learning
+
+Dave Epstein  $^{1,2}$  Ben Poole  $^{2}$  Ben Mildenhall  $^{2}$  Alexei A. Efros  $^{1}$  Aleksander Holynski  $^{1,2}$
+
+# Abstract
+
+We introduce a method to generate 3D scenes that are disentangled into their component objects. This disentanglement is unsupervised, relying only on the knowledge of a large pretrained text-to-image model. Our key insight is that objects can be discovered by finding parts of a 3D scene that, when rearranged spatially, still produce valid configurations of the same scene. Concretely, our method jointly optimizes multiple NeRFs from scratch—each representing its own object—along with a set of layouts that composite these objects into scenes. We then encourage these composited scenes to be in-distribution according to the image generator. We show that despite its simplicity, our approach successfully generates 3D scenes decomposed into individual objects, enabling new capabilities in text-to-3D content creation. See our project page for results and an interactive demo: https://dave.ml/layoutlearning/
+
+# 1. Introduction
+
+A remarkable ability of many seeing organisms is object individuation (Piaget et al., 1952), the ability to discern separate objects from light projected onto the retina (Wertheimer, 1938). Indeed, from a very young age, humans and other creatures are able to organize the physical world they perceive into the three-dimensional entities that comprise it (Spelke, 1990; Wilcox, 1999; Hoffmann et al., 2011). The analogous task of object discovery has captured the attention of the artificial intelligence community from its very inception (Roberts, 1963; Ohta et al., 1978), since agents that can autonomously parse 3D scenes into their component objects are better able to navigate and interact with their surroundings.
+
+Fifty years later, generative models of images are advancing at a frenzied pace (Nichol et al., 2021; Ramesh et al., 2022; Sahara et al., 2022; Yu et al., 2022; Chang et al., 2023).
+
+While these models can generate high-quality samples, their internal workings are hard to interpret, and they do not explicitly represent the distinct 3D entities that make up the images they create. Nevertheless, the priors learned by these models have proven incredibly useful across various tasks involving 3D reasoning (Hedlin et al., 2023; Ke et al., 2023; Liu et al., 2023; Luo et al., 2023; Wu et al., 2023), suggesting that they may indeed be capable of decomposing generated content into the underlying 3D objects depicted.
+
+One particularly exciting application of these text-to-image networks is 3D generation, leveraging the rich distribution learned by a diffusion model to optimize a 3D representation, e.g. a neural radiance field (NeRF, Mildenhall et al., 2020), such that rendered views resemble samples from the prior. This technique allows for text-to-3D generation without any 3D supervision (Poole et al., 2022; Wang et al., 2023b), but most results focus on simple prompts depicting just one or two isolated objects (Lin et al., 2023; Wang et al., 2023c).
+
+Our method builds on this work to generate complex scenes that are automatically disentangled into the objects they contain. To do so, we instantiate and render multiple NeRFs for a given scene instead of just one, encouraging the model to use each NeRF to represent a separate 3D entity. At the crux of our approach is an intuitive definition of objects as parts of a scene that can be manipulated independently of others while keeping the scene "well-formed" (Biederman, 1981). We implement this by learning a set of different layouts—3D affine transformations of every NeRF—which must yield composited scenes that render into in-distribution 2D images given a text prompt (Poole et al., 2022).
+
+We find that this lightweight inductive bias, which we term layout learning, results in surprisingly effective object disentanglement in generated 3D scenes (Figure 1), enabling object-level scene manipulation in the text-to-3D pipeline. We demonstrate the utility of layout learning on several tasks, such as building a scene around a 3D asset of interest, sampling different plausible arrangements for a given set of assets, and even parsing a provided NeRF into the objects it contains, all without any supervision beyond just a text prompt. We further quantitatively verify that, despite requiring no auxiliary models or per-example human annotation, the object-level decomposition that emerges through layout learning is meaningful and outperforms baselines.
+
+![](images/18fde10526409dd770de872e3e7256d62e97f6b6abe859cca452bd56e9c97529.jpg)  
+(a) Generated scene  
+"achenk hunting for easter eggs
+
+![](images/ba69e5160b66aa52b257a622156393fe2fc510126b7fc7cc965502ea810e67c9.jpg)
+
+![](images/94da6dc75b669a16c59c42ddab57c7649c5edb72e425f16de2fab895b718120e.jpg)  
+(b) Disentangled objects  
+#
+
+![](images/b53593cac260b2ed7ba128105e99393a4234870a9832c8d046ab8b2fe62555c6.jpg)
+
+![](images/9e1cefa9f745b734dc97ed6ede074d8a23c8e702273c9eb44b6f8144d67b768f.jpg)
+
+![](images/9d67eb80658f9a90c71ed6d431980d9e242bb726e728fe6b9daf101dde735952.jpg)  
+"a chef rat standing on a tiny stool and cooking a stew"
+
+![](images/149f59fe1b661efd2694b50c7d54846422d9c60a5973a45cfb50be73b3b5edbd.jpg)
+
+![](images/212dc2becb4ff92424c5756fb8a8ff89755f5159be186ded8c82468cf49f9572.jpg)
+
+![](images/68d8c59fb1552489ab740efad0914d43638fd905391d923eae58d74f908461a4.jpg)
+
+![](images/c2731f3015113bb8a349dc3fd5143bef8e6fa37cebeb555a9bce67c70cf8a305.jpg)
+
+![](images/e079df9cf6dcb356e4082521dafe4651bdde52b3f213e41a7c82f2f8fe74e46b.jpg)  
+"a pigeon having some coffee and a bagel, reading the newspaper"
+
+![](images/730c949a6dc39b7622c974ee8248ead16bbe1178af0dfd902c0cb91cfed6375f.jpg)
+
+![](images/d1c8df83640b1ece2b97dd0e08e2cec89c837de0d16108606e39b85d654e410a.jpg)
+
+![](images/e7d65280d842cd2442ccc3a7dd74f3a81f6a6ed3926f44a70f9c67045bc701d2.jpg)
+
+![](images/ab19bbfb18d0798d62cf69c994bd99a00dfb7fdc1eace2523fe5f6a6fb90dcb1.jpg)
+
+![](images/2e16d7fd3171ae4a2c396095ae4ff1d9edb9d23ae2041d91c31f63c67703dcec.jpg)  
+"two dogs in matching outfits paddling a kayak"
+
+![](images/cbc3ca77d2654bb0e676e7f8ac44c2a90e68da2917adf3d1b7c7f11358415159.jpg)
+
+![](images/01ec656c03266db1bcb82abadf5a6f63c270672bc8219db9137b1ec9a63a5f1a.jpg)
+
+![](images/1ddad831a71bc82da022fd84d3a97fcb2da4f1a7f7caf993ae8ec66d54cbc9e4.jpg)
+
+![](images/3f715036d5f1e721c025cfb273dff4e8c278fb47ae61902a21b3ba041f93d84a.jpg)
+
+![](images/b3e66eec7b594708bb46d2b6eb52b14177d11757a185e0cc5bf4da3925e9fadc.jpg)  
+"a sloth sitting on a beanbag with popcorn and a remote control"
+
+![](images/09b3ff44e9e7e807fd9f549a653be7b84818c6b879297aa5c52b488dbb283e2d.jpg)
+
+![](images/c4ac7ea04020dd0a340fd0565001c051d8e461fef1ed965d0ec6c0adde5a8a18.jpg)
+
+![](images/4d9f9330567af26c90e0259bbf5f03b645f510f99cd55a413f6448457d7416da.jpg)
+
+![](images/9094c4d6a14be4f2547b2c9614f4d0557bfcd47ee342978f46558e512d805efa.jpg)
+
+![](images/6460dae474eee0de2c976b860c1b3a87007eb3168259ad0194c1109bb5997eaa.jpg)  
+"a bald eagle having a burger and a drink at the park"
+
+![](images/ab19c50c7182b07a9d379c6d46eec7b3e385d84bb1530d774e23877a8b20f2d8.jpg)
+
+![](images/d79abb6fce4d14a964f2b856d71b115ae041aae4645f64b2219b19b4fa95eb4c.jpg)
+
+![](images/e71a3e25b9bb54fa881eccc09511d1a9b4705674b4068eb494d962a5f0dda23d.jpg)
+
+![](images/0a4e3f2a65acde318336f9f7d7ea110f8cf711bc954864935f63b7003910e45e.jpg)
+
+![](images/067ea86fc3854e1210c73a805d5e6c2dfb12069cb7d39016610f1390baaca251.jpg)  
+"a bear wearing a flannel camping and reading a book by the fire"
+
+![](images/60e26cafdf5ad3de4e21885ea1fe1a461549e44df09849beddbdaf1c9881aad0.jpg)
+
+![](images/8f6e7ff0db7232ecc620eaa3890e606b5822a42b96fe0a97eb5de57892421359.jpg)
+
+![](images/354192bb0c37026a4ebd2d74bd57e97dc7b91edf021cc421fdf2c8b3a3b796a7.jpg)
+
+![](images/106197cb4cf2bacde1892db50bca183feb25f96508ce52c082542fb0238c1f77.jpg)  
+Figure 1: Layout learning generates disentangled 3D scenes given a text prompt and a pretrained text-to-image diffusion model. We learn an entire 3D scene (left, shown from two views along with surface normals and a textureless render) that is composed of multiple NeRFs (right) representing different objects and arranged according to a learned layout.
+
+Our key contributions are as follows:
+
+- We introduce a simple, tractable definition of objects as portions of a scene that can be manipulated independently of each other and still produce valid scenes.  
+- We incorporate this notion into the architecture of a neural network, enabling the compositional generation of 3D scenes by optimizing a set of NeRFs as well as a set of layouts for these NeRFs.  
+- We apply layout learning to a range of novel 3D scene generation and editing tasks, demonstrating its ability to disentangle complex data despite requiring no object labels, bounding boxes, fine-tuning, external models, or any other form of additional supervision.
+
+# 2. Background
+
+# 2.1. Neural 3D representations
+
+To output three-dimensional scenes, we must use an architecture capable of modeling 3D data, such as a neural radiance field (NeRF, Mildenhall et al., 2020). We build on MLP-based NeRFs (Barron et al., 2021), that represent a volume using an MLP  $f$  that maps from a point in 3D space  $\mu$  to a density  $\tau$  and albedo  $\rho$ :
+
+$$
+(\tau , \boldsymbol {\rho}) = f (\boldsymbol {\mu}; \theta).
+$$
+
+We can differentiably render this volume by casting a ray  $\mathbf{r}$  into the scene, and then alpha-compositing the densities and colors at sampled points along the ray to produce a color and accumulated alpha value. For 3D reconstruction, we would optimize the colors for the rendered rays to match a known pixel value at an observed image and camera pose, but for 3D generation we sample a random camera pose, render the corresponding rays, and score the resulting image using a generative model.
+
+# 2.2. Text-to-3D using 2D diffusion models
+
+Our work builds on text-to-3D generation using 2D diffusion priors (Poole et al., 2022). These methods turn a diffusion model into a loss function that can be used to optimize the parameters of a 3D representation. Given an initially random set of parameters  $\theta$ , at each iteration we randomly sample a camera  $c$  and render the 3D model to get an image  $x = g(\theta, c)$ . We can then score the quality of this rendered image given some conditioning text  $y$  by evaluating the score function of a noised version of the image  $z_t = \alpha_t x + \sigma_t \epsilon$  using the pretrained diffusion model  $\hat{\epsilon}(z_t; y, t)$ . We update the parameters of the 3D representation using score distillation:
+
+$$
+\nabla_ {\theta} \mathcal {L} _ {\mathrm {S D S}} (\theta) = \mathbb {E} _ {t, \epsilon , c} \left[ w (t) \left(\hat {\epsilon} \left(z _ {t}; y, t\right) - \epsilon\right) \frac {\partial x}{\partial \theta} \right] \tag {1}
+$$
+
+where  $w(t)$  is a noise-level dependent weighting.
+
+SDS and related methods enable the use of rich 2D priors obtained from large text-image datasets to inform the structure of 3D representations. However, they often require careful tuning of initialization and hyperparameters to yield high quality 3D models, and past work has optimized these towards object generation. The NeRF is initialized with a Gaussian blob of density at the origin, biasing the optimization process to favor an object at the center instead of placing density in a skybox-like environment in the periphery of the 3D representation. Additionally, bounding spheres are used to prevent creation of density in the background. The resulting 3D models can produce high-quality individual objects, but often fail to generate interesting scenes, and the resulting 3D models are a single representation that cannot be easily split apart into constituent entities.
+
+# 3. Method
+
+To bridge the gap from monolithic 3D representations to scenes with multiple objects, we introduce a more expressive 3D representation. Here, we learn multiple NeRFs along with a set of layouts, i.e. valid ways to arrange these NeRFs in 3D space. We transform the NeRFs according to these layouts and composite them, training them to form high-quality scenes as evaluated by the SDS loss with a text-to-image prior. This structure causes each individual NeRF to represent a different object while ensuring that the composite NeRF represents a high-quality scene. See Figure 2 for an overview of our approach.
+
+# 3.1. Compositing multiple volumes
+
+We begin by considering perhaps the most naive approach to generating 3D scenes disentangled into separate entities. We simply declare  $K$  NeRFs  $\{f_k\}$  each one intended to house its own object—and jointly accumulate densities from all NeRFs along a ray, proceeding with training as normal by rendering the composite volume. This can be seen as an analogy to set-latent representations (Locatello et al., 2020; Jaegle et al., 2021a,b; Jabri et al., 2023), which have been widely explored in other contexts. In this case, rather than arriving at the final albedo  $\pmb{\rho}$  and density  $\tau$  of a point  $\pmb{\mu}$  by querying one 3D representation, we query  $K$  such representations, obtaining a set  $\{\pmb{\rho}_k, \tau_k\}_{k=1}^K$ . The final density at  $\pmb{\mu}$  is then  $\tau' = \sum \tau_k$  and the final albedo is the density-weighted average  $\pmb{\rho}' = \sum \frac{\tau_k}{\tau'} \pmb{\rho}_k$ .
+
+This formulation provides several potential benefits. First, it may be easier to optimize this representation to generate a larger set of objects, since there are  $K$  distinct 3D Gaussian density spheres to deform at initialization, not just one. Second, many representations implicitly contain a local smoothness bias (Tancik et al., 2020) which is helpful for
+
+![](images/6bdd79fe8c34219b9c8645088f8628ccd2450ebdf94a3a67bc967ea2fda448d2.jpg)  
+Figure 2: Method. Layout learning works by optimizing  $K$  NeRFs  $f_{k}$  and learning  $N$  different layouts  $\mathbf{L}_n$  for them, each consisting of per-NeRF affine transforms  $\mathbf{T}_k$ . Every iteration, a random layout is sampled and used to transform all NeRFs into a shared coordinate space. The resultant volume is rendered and optimized with score distillation sampling (Poole et al., 2022) as well as per-NeRF regularizations to prevent degenerate decompositions and geometries (Barron et al., 2022). This simple structure causes object disentanglement to emerge in generated 3D scenes.
+
+generating objects but not spatially discontinuous scenes. Thus, our representation might be inclined toward allocating each representation toward a spatially smooth entity, i.e. an object.
+
+However, just as unregularized sets of latents are often highly uninterpretable, simply spawning  $K$  instances of a NeRF does not produce meaningful decompositions. In practice, we find each NeRF often represents a random point-cloud-like subset of 3D space (Fig. 3).
+
+To produce scenes with disentangled objects, we need a method to encourage each 3D instance to represent a coherent object, not just a different part of 3D space.
+
+# 3.2. Layout learning
+
+We are inspired by other unsupervised definitions of objects that operate by imposing a simple inductive bias or regularization in the structure of a model's latent space, e.g. query-axis softmax attention (Locatello et al., 2020), spatial ellipsoid feature maps (Epstein et al., 2022), and diagonal Hessian matrices (Peebles et al., 2020). In particular, Niemeyer & Geiger (2021) learn a 3D-aware GAN that composites multiple NeRF volumes in the forward pass, where the latent code contains a random affine transform for each NeRF's output. Through this structure, each NeRF learns to associate itself with a different object, facilitating the kind of disentanglement we are after. However, their approach relies on pre-specified independent distributions of each object's location, pose, and size, preventing scaling
+
+beyond narrow datasets of images with one or two objects and minimal variation in layout.
+
+In our setting, not only does the desired output comprise numerous open-vocabulary, arbitrary objects, but these objects must be arranged in a particular way for the resultant scene to be valid or "well-formed" (Biederman et al., 1982). Why not simply learn this arrangement?
+
+To do this, we equip each individual NeRF  $f_{k}$  with its own learnable affine transform  $\mathbf{T}_k$ , and denote the set of transforms across all volumes a layout  $\mathbf{L} \equiv \{\mathbf{T}_k\}_{k=1}^K$ . Each  $\mathbf{T}_k$  has a rotation  $\mathbf{R}_k \in \mathbb{R}^{3 \times 3}$  (in practice expressed via a quaternion  $\mathbf{q} \in \mathbb{R}^4$  for ease of optimization), translation  $\mathbf{t}_k \in \mathbb{R}^3$ , and scale  $s_k \in \mathbb{R}$ . We apply this affine transform to the camera-to-world rays  $\mathbf{r}$  before sampling the points used to query  $f_{k}$ . This implementation is simple, makes no assumptions about the underlying form of  $f$ , and updates parameters with standard backpropagation, as sampling and embedding points along the ray is fully differentiable (Lin et al., 2021). Concretely, a ray  $\mathbf{r}$  with origin  $\mathbf{o}$  and direction  $\mathbf{d}$  is transformed into an instance-specific ray  $\mathbf{r}_k$  via the following transformations:
+
+$$
+\mathbf {o} _ {k} = s _ {k} \left(\mathbf {R} _ {k} \mathbf {o} - \mathbf {t} _ {k}\right) \tag {2}
+$$
+
+$$
+\mathbf {d} _ {k} = s _ {k} \mathbf {R} _ {k} \mathbf {d} \tag {3}
+$$
+
+$$
+\mathbf {r} _ {k} (t) = \mathbf {o} _ {k} + t \mathbf {d} _ {k} \tag {4}
+$$
+
+Though we input a different  $H \times W$  grid of rays to each  $f_{k}$ , we composite their outputs as if they all sit in the same
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Average Score ↑</td></tr><tr><td>CLIP Color</td><td>B/16 Geo</td><td>CLIP Color</td><td>L/14 Geo</td></tr><tr><td>Random objects</td><td>23.4</td><td>22.4</td><td>17.2</td><td>18.3</td></tr><tr><td>Per-object SDS</td><td>32.3</td><td>30.5</td><td>27.2</td><td>25.9</td></tr><tr><td>K NeRFs</td><td>26.7</td><td>25.4</td><td>21.0</td><td>21.2</td></tr><tr><td>+ Per-NeRF losses</td><td>27.3</td><td>26.1</td><td>21.6</td><td>22.6</td></tr><tr><td>+ Empty NeRF loss</td><td>27.7</td><td>26.2</td><td>22.8</td><td>23.2</td></tr><tr><td>+ Learn layout</td><td>29.9</td><td>28.8</td><td>24.9</td><td>23.5</td></tr><tr><td>+ Learn N layouts</td><td>31.3</td><td>29.9</td><td>27.1</td><td>24.8</td></tr><tr><td>Relative layouts</td><td>30.4</td><td>29.2</td><td>25.7</td><td>24.0</td></tr><tr><td>View dep. prompt</td><td>31.0</td><td>29.1</td><td>25.6</td><td>23.6</td></tr></table>
+
+Figure 3: Evaluating disentanglement and quality. We optimize a model with  $K = 3$  NeRFs on a list of 30 prompts, each containing three objects. We then automatically pair each NeRF with a description of one of the objects in the prompt and report average NeRF-object CLIP score (see text for details). We also generate each of the  $30 \times 3 = 90$  objects from the prompt list individually and compute its score with both the corresponding prompt and a random other one, providing upper and lower bounds for performance on this task. Training  $K$  NeRFs provides some decomposition, but most objects are scattered across 2 or 3 models. Learning one layout alleviates some of these issues, but only with multiple layouts do we see strong disentanglement. We show two representative examples of emergent objects to visualize these differences.
+
+coordinate space—for example, the final density at  $\pmb{\mu} = \mathbf{r}(t)$  is the sum of densities output by every  $f_{k}$  at  $\pmb{\mu}_{k} = \mathbf{r}_{k}(t)$ .
+
+Compared to the naive formulation that instantiates  $K$  models with identical initial densities, learning the size, orientation, and position of each model makes it easier to place density in different parts of 3D space. In addition, the inherent stochasticity of optimization may further dissuade degenerate solutions.
+
+While introducing layout learning significantly increases the quality of object disentanglement (Tbl. 3b), the model is still able to adjoin and utilize individual NeRFs in undesirable ways. For example, it can still place object parts next to each other in the same way as K NeRFs without layout learning.
+
+Learning multiple layouts. We return to our statement that objects must be "arranged in a particular way" to form scenes that render to in-distribution images. While we already enable this with layout learning in its current form, we are not taking advantage of one key fact: there are many "particular ways" to arrange a set of objects, each of which gives an equally valid composition. Rather than only learning one layout, we instead learn a distribution over layouts  $P(\mathbf{L})$  or a set of  $N$  randomly initialized layouts  $\{\mathbf{L}_n\}_{n=1}^N$ . We opt for the latter, and sample one of the  $N$  layouts from the set at each training step to yield transformed rays  $\mathbf{r}_k$ .
+
+With this in place, we have arrived at our final definition of objectness (Figure 2): objects are parts of a scene that can be arranged in different ways to form valid compo
+
+sitions. We have "parts" by incorporating multiple volumes, and "arranging in different ways" through multiple-Layout learning. This simple approach is easy to implement (Fig. 9), adds very few parameters ( $8NK$  to be exact), requires no fine-tuning or manual annotation, and is agnostic to choices of text-to-image and 3D model. In Section 4, we verify that layout learning enables the generation and disentanglement of complex 3D scenes.
+
+Regularization. We build on Mip-NeRF 360 (Barron et al., 2022) as our 3D backbone, inheriting their orientation, distortion, and accumulation losses to improve visual quality of renderings and minimize artifacts. However, rather than computing these losses on the final composited scene, we apply them on a per-NeRF basis. Importantly, we add a loss penalizing degenerate empty NeRFs by regularizing the soft-binarized version of each NeRF's accumulated density,  $\alpha_{\mathrm{bin}}$ , to occupy at least  $10\%$  of the canvas:
+
+$$
+\mathcal {L} _ {\text {e m p t y}} = \max  (0. 1 - \bar {\alpha} _ {\text {b i n}}, 0) \tag {5}
+$$
+
+We initialize parameters  $s \sim \mathcal{N}(1,0.3)$ ,  $\mathbf{t}^{(i)} \sim \mathcal{N}(0,0.3)$ , and  $\mathbf{q}^{(i)} \sim \mathcal{N}(\mu_i,0.1)$  where  $\mu_i$  is 1 for the last element and 0 for all others. We use a  $10 \times$  higher learning rate to train layout parameters. See Appendix A.1 for more details.
+
+# 4. Experiments
+
+We examine the ability of layout learning to generate and disentangle 3D scenes across a wide range of text prompts. We
+
+![](images/000e4d1e109532ef76c813eff1bb5ed213b49bff6af8b39f5d3fff4caa419e10.jpg)
+
+(a) Frozen object  
+![](images/9a3811712ecde0f3b46d6f1ace0b75d23e36bf426d1111c9651794d41b2af6b6.jpg)  
+"a cat wearing a hawaiian shirt and sunglasses, having a drink on a beach towel"
+
+![](images/3a40ea5a529c608718ca1ecf4017848b3e95e3d28d8ae47945abef35cbc1c7f6.jpg)  
+(b) Disentangled objects
+
+![](images/170d74c818e8a03c23806a6a034272e4e7be3b852d4ee54a20c671d7580fdd52.jpg)
+
+![](images/81ddf182b89c314271ab1aa32acaa6a364665b60679400282b594b3193489dc9.jpg)  
+(c) Generated scene
+
+![](images/b65ce177bfb22742e16a21a674f5f854cbf4dac14bbd7285603366e7ff700237.jpg)
+
+![](images/873e4a85e9766a50a195229a67327117186b2e0eab73d5526030b6fc874f64fa.jpg)
+
+![](images/85060b84d6f0003635533688005d82a47363ecdd90dafe98d7118c809fc52d60.jpg)
+
+![](images/da69970c404305d0ea28799d98983cff6c219d84cb9b8c3b9c917536a6d632be.jpg)
+
+![](images/1094b35018222efbc1ceb464f3644c0d1ae5c20707e54ee6955131a47af3532f.jpg)  
+"a cat wearing a Santa costume holding a present next to a miniature Christmas tree"
+
+![](images/5ea164c395cd05534dd2ba8c720ab1606ef0eedb662182b3fc9062769a85c575.jpg)
+
+![](images/ca08898d7edef385f70553103e380cca6073b12d74ff41943a3dc6e1429a524f.jpg)
+
+![](images/cfdcd99daa04615bd97109bc38ed3b86b47311af0139bc7175fddc73a73fa757.jpg)
+
+![](images/b9ad5f7c04b740c44635742db2cc362f0b286d1712380eab7b310074ddde909b.jpg)
+
+![](images/1057586089df864dd29bc0d3e94bb8bf7c1be14f1c1cfbf0b097fcc60393700f.jpg)
+
+![](images/65d2bb5be784971328b926895b61717a1041b615e35d351817f49adbad41579b.jpg)  
+Figure 4: Conditional optimization. We can take advantage of our structured representation to learn a scene given a 3D asset in addition to a text prompt, such as a specific cat or motorcycle (a). By freezing the NeRF weights but not the layout weights, the model learns to arrange the provided asset in the context of the other objects it discovers (b). We show the entire composite scenes the model creates in (c) from two views, along with surface normals and a textureless render.
+
+![](images/d8ab98fa3db681168e9998b0534f9b9b1012c7cc978c045c2cc1e06bbfb5bf7d.jpg)
+
+![](images/8d35ce9615e39181cb088ee27d6e4c9595530e855c97463aef4110c3c28186ea.jpg)
+
+![](images/669f17df7b990a31d0636d6a8aa3fee661906854ad00da3088bc4992b5d6d43b.jpg)
+
+![](images/f17362338e0f26b21f51310f29c77f21f81d2fe5c905fd21e6fafa0fc2ba8fc5.jpg)
+
+![](images/cc4dc67d5c837f1ef65953841fa277c98ca9862d0e62927667f2cbd75bc66db8.jpg)
+
+![](images/80706340a6ce4acfe05dfb8f2be84f012b9d44d423265e19f605cc9e664ac13d.jpg)  
+"alion in a leather jacket riding a motorcycle between two differently colored cones"
+
+![](images/e8ba3b8bd9f2b93a2a0dbbda5aaf2af6398d4e16d8dd5345978c895c94eab5ff.jpg)
+
+![](images/123e572a55c1e1aadd6fafba8a80359efc6cf4b0a451d285cc1cd1e496f845e3.jpg)
+
+![](images/3c945f56e7f9c91ab969bbf7048c3badc0f1f75900bd99996477b2b7c10a93c7.jpg)  
+"a modern nightstand with a lamp and a miniature motorcycle model on it, on top of a small rug"
+
+![](images/473cf8413bc34abe1e1b52a7d5b84af9ee51d1cf45a5991d3327b76250392974.jpg)
+
+![](images/aa812c562487dd89edbf3950b5d28fb63f3f379231b6ed111bd2248e371c7d95.jpg)
+
+![](images/595e6e6e79d13d874addbdc4ee19138dd27adb53c6bc7af4f1edb92e50fa675d.jpg)
+
+![](images/55f9f36d487789b5bc99e8fdcd508c582e535788db1be6c6dfbd59eb54babf02.jpg)
+
+![](images/87b66a52ba4a78be0827bad558b28ae1089f98ebfe31a8a13f2751526ac24c7e.jpg)
+
+first verify our method's effectiveness through an ablation study and comparison to baselines, and then demonstrate various applications enabled by layout learning.
+
+# 4.1. Qualitative evaluation
+
+In Figure 1, we demonstrate several examples of our full system with layout learning. In each scene, we find that the composited 3D generation is high-quality and matches the text prompt, while the individual NeRFs learn to correspond to objects within the scene. Interestingly, since our approach does not directly rely on the input prompt, we can disentangle entities not mentioned in the text, such as a basket filled with easter eggs, a chef's hat, and a picnic table.
+
+# 4.2. Quantitative evaluation
+
+Measuring the quality of text-to-3D generation remains an open problem due to a lack of ground truth data—there is
+
+no "true" scene corresponding to a given prompt. Similarly, there is no true disentanglement for a certain text description. Following Park et al. (2021); Jain et al. (2022); Poole et al. (2022), we attempt to capture both of these aspects using scores from a pretrained CLIP model (Radford et al., 2021; Li et al., 2017). Specifically, we create a diverse list of 30 prompts, each containing 3 objects, and optimize a model with  $K = 3$  NeRFs on each prompt. We compute the  $3 \times 3$  matrix of CLIP scores  $(100 \times$  cosine similarity) for each NeRF with descriptions "a DSLR photo of [object  $1/2/3$ ]", finding the optimal NeRF-to-object matching and reporting the average score across all 3 objects.
+
+We also run SDS on the  $30 \times 3 = 90$  per-object prompts individually and compute scores, representing a maximum attainable CLIP score under perfect disentanglement (we equalize parameter counts across all models for fairness).
+
+As a low-water mark, we compute scores between per-object NeRFs and a random other prompt from the pool of 90.
+
+The results in Table 3b show these CLIP scores, computed both on textured ("Color") and textureless, geometry-only ("Geo") renders. The final variant of layout learning achieves competitive performance, only 0.1 points away from supervised per-object rendering when using the largest CLIP model as an oracle, indicating high quality of both object disentanglement and appearance. Please see Appendix A.3 for a complete list of prompts and more details.
+
+Ablation. We justify the sequence of design decisions presented in Section 3 by evaluating different variants of layout learning, starting from a simple collection of  $K$  NeRFs and building up to our final architecture. The simple setting leads to some non-trivial separation (Figure 3a) but parts of objects are randomly distributed across NeRFs—CLIP scores are significantly above random, but far below the upper bound. Adding regularization losses improve scores somewhat, but the biggest gains come from introducing layout learning and then co-learning  $N$  different arrangements, validating our approach.
+
+# 4.3. Applications of layout learning
+
+To highlight the utility of the disentanglement given by layout learning beyond generation, we apply it to various 3D editing tasks. First, we show further results on object disentanglement in Figure 4, but in a scenario where one NeRF is frozen to contain an object of interest, and the rest of the scene must be constructed around it. This object's layout parameters can also be frozen, for example, if a specific position or size is desired. We examine the more challenging setting where layout parameters must also be learned, and show results incorporating a grumpy cat and green motorbike into different contexts. Our model learns plausible transformations to incorporate provided assets into scenes, while still discovering the other objects necessary to complete the prompt.
+
+In Figure 5, we visualize the different layouts learned in a single training run. The variation in discovered layouts is significant, indicating that our formulation can find various meaningful arrangements of objects in a scene. This allows users of our method to explore different permutations of the same content in the scenes they generate.
+
+Inspired by this, and to test gradient flow into layout parameters, we also examine whether our method can be used to arrange off-the-shelf, frozen 3D assets into semantically valid configurations (Figure 6). Starting from random positions, sizes, and orientations, layouts are updated using signal backpropagated from the image model. This learns reasonable transformations, such as a rubber duck shrinking and moving inside a tub, and a shower head moving
+
+![](images/8c757548bf73c8c169c191aa2ab0113cf3c48bc1ef23601cef058d6494475969.jpg)
+
+![](images/6253224270a04648b284a113f458e0989350158ea1d72978c145d1bf77b33390.jpg)
+
+![](images/fde23bd098997cd7ffe8630bd9bad9c818582da0743dacdfe51dbb356954ce09.jpg)  
+Figure 5: Layout diversity. Our method discovers different plausible arrangements for objects. Here, we optimize each example over  $N = 4$  layouts and show differences in composited scenes, e.g. flamingos wading inside vs. beside the pond, and cats in different poses around the snooker table.
+
+upwards and pointing so its stream is going into the tub.
+
+Finally, we use layout learning to disentangle a pre-existing NeRF containing multiple entities, without any per-object supervision (Fig. 8). We do this by randomly initializing a new model and training it with a caption describing the target NeRF. We require the first layout  $\mathbf{L}_1$  to create a scene that faithfully reconstructs the target NeRF in RGB space, allowing all other layouts to vary freely. We find that layout learning arrives at reasonable decompositions of the scenes it is tasked with reconstructing.
+
+# 5. Related work
+
+Object recognition and discovery. The predominant way to identify the objects present in a scene is to segment two-dimensional images using extensive manual annotation (Kirillov et al., 2023; Li et al., 2022; Wang et al., 2023a), but relying on human supervision introduces challenges and scales poorly to 3D data. As an alternative, an extensive line of work on unsupervised object discovery (Russell et al., 2006; Rubinstein et al., 2013; Oktay et al., 2018; Henaff et al., 2022; Smith et al., 2022; Ye et al., 2022; Monnier et al., 2023) proposes different inductive biases (Locatello et al., 2019) that encourage awareness of objects in a scene. However, these approaches are largely restricted to either 2D images or constrained 3D data (Yu et al., 2021; Sajjadi et al., 2022), limiting their applicability to complex 3D scenes. At the same time, large text-to-image models have been shown to implicitly encode an understanding of entities in their
+
+![](images/3d4c0235aad2c13bed794a740a78f26187327c6c77728554539e09d658d69957.jpg)  
+(a) Input objects
+
+![](images/497919bb2772e20847a0dd3105a257ee63957173b5b7650f8c4e483a337c66fc.jpg)  
+"abowl of pasta, a table, and a chair
+
+![](images/2aa894e9fcee91aa2aa14979af0cfaff5560bdf6f1b431eb00569f08a10047b1.jpg)
+
+![](images/f6c38afbb41ecc0dbf9efb5299af8821e2d079532e202ab330549bc43945b9a3.jpg)  
+(b) Learned layout
+
+![](images/8bc8367601238af53ef8cfc58fb1875fdc089ffe5dfddcc38608c5710fddb5f9.jpg)
+
+![](images/0bd5c9dae3a10a7f0492f2ef873beced9b9b3376c331d8064ea38ca77f14f66d.jpg)
+
+![](images/4adeb7b574e5530347d5cb31cf91e4b6d46bcc06cd371248b8ed85d4ebadab6f.jpg)  
+"a monitor, keyboard, and mouse"
+
+![](images/1824c5d177774a749e7f06b0d07e7ff5d27141b85dd16645f5975ec4eb5cb9ce.jpg)
+
+![](images/274620926877b829768186289700704705e83a11edbd92e3494f7fbda328dbd4.jpg)
+
+![](images/b3a32d46bcdadeabb1ee4f93b3f8ec418af01578bd60effe035ca4e0c650a8b8.jpg)
+
+![](images/bda705bc35b5ae6f4dc67c81be3e13d66a8ccf5b4aa3fade07b8c115f82be2b3.jpg)  
+Figure 6: Optimizing layout. Allowing gradients to flow only into layout parameters while freezing a set of provided 3D assets results in reasonable object configurations, such as a chair tucked into a table with spaghetti on it, despite no such guidance being provided in the text conditioning.
+
+![](images/dd6bfd12a7ca78c72b6bcebe3b7ad2cf288510df5a099f5c63bfcd57203e9922.jpg)  
+"a rubber duck, a bathtub, and a shower head"
+
+![](images/7c61b99c98826363d96f358c33c845e87f436834ecba725428fe873635ca1b61.jpg)
+
+![](images/de7edbeb991590f0d68edb6ffa4f5fdc54a3b755495ad94df521d60252f18d2e.jpg)
+
+![](images/69e4fcd6071119324e67904bce017762045446e119b900d7ff05b1599160a2ae.jpg)
+
+internals (Epstein et al., 2023), motivating their use for the difficult problem of explicit object disentanglement.
+
+Compositional 3D generation. There are many benefits to generating 3D scenes separated into objects beyond just better control. For example, generating objects one at a time and compositing them manually provides no guarantees about compatibility in appearance or pose, such as "dogs in matching outfits" in Figure 1 or a lion holding the handlebars of a motorcycle in Figure 4. Previous and concurrent work explores this area, but either requires users to painstakingly annotate 3D bounding boxes and per-object labels (Cohen-Bar et al., 2023; Po & Wetzstein, 2023) or uses external supervision such as LLMs to propose objects and layouts (Yang et al., 2023; Zhang et al., 2023), significantly slowing down the generation process and hindering quality. We show that this entire process can be solved without any additional models or labels, simply using the signal provided by a pretrained image generator.
+
+# 6. Discussion
+
+We present layout learning, a simple method for generating disentangled 3D scenes given a text prompt. By optimizing multiple NeRFs to form valid scenes across multiple layouts, we encourage each NeRF to contain its own object. This approach requires no additional supervision or auxiliary models, yet performs quite well. By generating scenes that are decomposed into objects, we provide users of text-to-3D systems with more granular, local control over the complex creations output by a black-box neural network.
+
+Disentangled objects  
+![](images/ba0ab9f29ea87c30acbafbbba59638c75de69d761269eac70cf2edb1ca963d09.jpg)  
+(a) Bad geometry: "a moose staring down a snowman by a cabin"
+
+![](images/280c65c1d8d7543d614baa0c2a6a5fa4da43cfb93a4416d65b93e4fd39d9b4de.jpg)
+
+![](images/c520b2dfd433ce65248a7df8ac8b2fb1ef24d06217f58f611c1ee0e78edc915e.jpg)
+
+![](images/6503b40a7191733c79308c2806e89456a07e7ddaebd6038c5690db76e367ece8.jpg)  
+Entire scene
+
+![](images/a6d69860e9a1d97a0f6e0b1a7e77fbeda557a1b470ed1a1b8e2b721cafe25c48.jpg)
+
+![](images/01871ec636238c1b7bf03a59ff3634a8855951f1a6547a812d4c6d698e724ec7.jpg)  
+(b) Undersegmentation: "two astronauts riding a horse together"
+
+![](images/543aa7b31d9bddb398edb4eb328a01e3807b9f1d9fdc79719edb0d4ab26d0763.jpg)
+
+![](images/1375654e88a2be43ae97c203b6b2e0c69b0d5ef1b1ee8e2d4ae1956c68ab75ef.jpg)
+
+![](images/d255018681e55886f9dff053710dfe55580409b48895c6dee5ccf1075bb81c2c.jpg)
+
+![](images/62344f64ac1ccde075a6904ea14e5a6d33a08873a3a61485109bc4088a16fab2.jpg)
+
+![](images/8951f7f7ed520cb1450b88648c0d3ee3ee5b71d03243df06694ac75e244a494e.jpg)
+
+![](images/735370f0f8611bdb698812c96e09ad55b40e095115e0c5dc4f11ae4c1bfe7e4e.jpg)
+
+![](images/8d059664571dc7ec87d86d8ae6506f51c723d02ab916bea4f144611d67ae3090.jpg)
+
+![](images/87f823267a1b8e57ad4ddd8ef399d4e78ba0cd47fd08f764000d3004b145246f.jpg)  
+(c) Clutter  $(K = 5)$ : "two fancy llamas enjoying a tea party"
+
+![](images/3d25b6b946493aa246284f0484960458c3b1a46d7abef31ecbc48540a161a41c.jpg)
+
+![](images/a42ab320de96040a336ec9f3eb8762a7b8d25af3c6ceec71ab1a1f6c8010fd37.jpg)  
+(d) Overly similar layouts: "a monkey having a whiskey and a cigar, using a typewriter"
+
+![](images/af2b14052788330a29be449c681e3e901429a61416a3be9ed93006bc888f5944.jpg)  
+Learned layouts
+
+![](images/366beba4e294eee4c4d31cf11c123b3cc724956e44e5c341e37b5d7ae7fdd0cb.jpg)
+
+![](images/0d0c11d2be24df19d4907b1fa5bd261a2177bc8bb4d94592c3273c2dc0de7fd3.jpg)
+
+Figure 7: Limitations. Layout learning inherits failure modes from SDS, such as bad geometry of a cabin with oddly intersecting exterior walls (a). It also may undesirably group objects that always move together (b) such as a horse and its rider, and (c) for certain prompts that generate many small objects, choosing  $K$  correctly is challenging, hurting disentanglement. In some cases (d), despite different initial values, layouts converge to very similar final configurations.
+
+Though layout learning is surprisingly effective on a wide variety of text prompts, the problem of object disentangle-ment in 3D is inherently ill-posed, and our definition of objects is simple. As a result, many undesirable solutions exist that satisfy the constraints we pose.
+
+Despite our best efforts, the compositional scenes output by our model do occasionally suffer from failures (Fig. 7) such as over- or under-segmentation and the "Janus problem" (where objects are depicted so that salient features appear from all views, e.g. an animal with a face on the back of its head) as well as other undesirable geometries. Further, though layouts are initialized with high standard deviation and trained with an increased learning rate, they occasionally converge to near-identical values, minimizing the effectiveness of our method. In general, we find that failures to disentangle are accompanied by an overall decrease in visual quality.
+
+# Acknowledgements
+
+We thank Dor Verbin, Ruiqi Gao, Lucy Chai, and Minyoung Huh for their helpful comments, and Arthur Brussee for help with an NGP implementation. DE was partly supported by the PD Soros Fellowship. DE conducted part of this research at Google, with additional funding from an ONR MURI grant.
+
+# Impact statement
+
+Generative models present many ethical concerns over data attribution, nefarious applications, and longer-term societal effects. Though we build on a text-to-image model trained on data that has been filtered to remove concerning imagery and captions, recent work has shown that popular datasets contain dangerous depictions of undesirable content<sup>1</sup> which may leak into model weights.
+
+Further, since we distill the distribution learned by an image generator, we inherit the potential negative use-cases enabled by the original model. By facilitating the creation of more complex, compositional 3D scenes, we perhaps expand the scope of potential issues associated with text-to-3D technologies. Taking care to minimize potential harmful deployment of our generative models through using ethically-sourced and well-curated data is of the utmost importance as our field continues to grow in size and influence.
+
+Further, by introducing an unsupervised method to disentangle 3D scenes into objects, we possibly contribute to the displacement of creative workers such as video game asset designers via increased automation. However, at the same time, methods like the one we propose have the potential to become valuable tools at the artist's disposal, providing much more control over outputs and helping create new, more engaging forms of content.
+
+# References
+
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+Barron, J. T., Mildenhall, B., Verbin, D., Srinivasan, P. P., and Hedman, P. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5470-5479, June 2022.  
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+
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+Zhang, Q., Wang, C., Siarohin, A., Zhuang, P., Xu, Y., Yang, C., Lin, D., Zhou, B., Tulyakov, S., and Lee, H.-Y. Scenewiz3d: Towards text-guided 3d scene composition. arXiv preprint arXiv:2312.08885, 2023.
+
+![](images/bf2e31b596b706c4bdd39a37779ca3fcaf5ef3209be88d651c69a95faccd05db.jpg)  
+(a) Input NeRF
+
+![](images/1abeb775f2013784cd8c1f77627844dd5ac6ee87dccc8e20e0ab2b7cf4fa8628.jpg)
+
+![](images/abd2e706fccf8f2fbdbaec0d4be989b2aae148544ffd7198d7e4cd47b21a83a5.jpg)
+
+![](images/f4ef12e3afd099bccce9e272a14d6fc72f5e70412719d574a21e9268f2bab366.jpg)  
+(b) Discovered objects
+
+![](images/2269c419f9d6a6cb5a493f3980a67749ce8470fefe90afdfaf9436f11ade2f00.jpg)
+
+![](images/1b0ec75c7215e4162b8f2ab1c585c2d3d511a9cd0397b15501e93eb5004820e7.jpg)  
+(c) Reconstruction
+
+![](images/4c2caa29f2c691051b6c0ef062815f1ffa5d52154604a7e6c1298b0a5ed871de.jpg)
+
+![](images/08ed3ef6944b590b4e9e0006bf59fa9e8216dc3183cdf5ec82e0128f1d3326be.jpg)
+
+![](images/f27b86195badf4dd60cce272371e618c171b5a5ae621667abd136759d3ad4ec6.jpg)
+
+![](images/d71de7af438fece3673be724f0521e58a06387651cee89b95846af258bded228.jpg)
+
+![](images/fb6bfe7711efd01bb93a69a12dc46f391896fc608b768960359bf25f5c6b870e.jpg)  
+"a giant husk of corn grilling hot dogs by a pool with an inner tube"
+
+![](images/7d183c368bb4561c31bd71ec13bc846a2f07834cde91ff86a426e892e2f74f03.jpg)
+
+![](images/aa358a793c5707f2dc5985d7b8d7080f74bc814148d7055524bb2d5341c09b34.jpg)
+
+![](images/d879608f3fb47530b2defc6340b3f68c5faf5ab1436dac9bd04af1d7f347b297.jpg)  
+"two cute cats wearing baseball uniforms playing catch"
+
+![](images/bb7e4389413f9fa243b7bc55496ffaaf0d521b112c67b351bda6585643b28384.jpg)
+
+![](images/046f08c1ff368509d47be37f056df25a04c9b201d9cb8fd0057e67aed1ac9a9d.jpg)
+
+![](images/6f36a01046e5ce717e8d60e0169af73a1666d90697a21602a56f74b41bad7a6c.jpg)  
+Figure 8: Decomposing NeRFs of scenes. Given a NeRF representing a scene (a) and a caption, layout learning is able to parse the scene into the objects it contains without any per-object supervision (b). We accomplish this by requiring renders of one of the  $N$  learned layouts to match the same view rendered from the target NeRF (c), using a simple  $L_{2}$  reconstruction loss with  $\lambda = 0.05$ .
+
+![](images/b03f4f88de2764f3ba1afb11269f1eac21dc0f9dccabdb1b065f77b7a94c6974.jpg)
+
+![](images/99bc98e66cacdf5d2a7ff7566711c89054e1083658a4ae1a5a11e081a54cd122.jpg)  
+"abird having some sushi and sake
+
+![](images/bb7677213c1a7afea01c1123c6c2a45700390dfe6a8c0fe685edb9480a32243c.jpg)
+
+![](images/6df57021d0290cfc523438fc62b5c89238b687bc5de877f803acaca497d0244e.jpg)
+
+![](images/e73993d3f42d94c6cca12592271d531c2c66cfd66716c670b0edf356dd3154b4.jpg)
+
+![](images/3b21a0fd427466fefcc1e0cfde371f1ed71ad9ea7cd095717ec9d07449a7c9ea.jpg)
+
+![](images/ff23e26b884ad88119f7f8ab931a28acaabbcbbe12718aeede9d0db2215e215e.jpg)
+
+# A. Appendix
+
+# A.1. Implementation details
+
+We use Mip-NeRF 360 as the 3D backbone (Barron et al., 2022) and Imagen (Saharia et al., 2022), a 128px pixel-space diffusion model, for most experiments, rendering at 512px. To composite multiple representations, we merge the output albedos and densities at each point, taking the final albedo as a weighted average given by per-NeRF density. We apply this operation to the outputs of the proposal MLPs as well as the final RGB-outputting NeRFs. We use  $\lambda_{\mathrm{dist}} = 0.001$ ,  $\lambda_{\mathrm{acc}} = 0.01$ ,  $\lambda_{\mathrm{ori}} = 0.01$  as well as  $\lambda_{\mathrm{empty}} = 0.05$ . The empty loss examines the mean of the per-pixel accumulated density along rays in a rendered view,  $\alpha$ , for each NeRF. It penalizes these mean  $\bar{\alpha}$  values if they are under a certain fraction of the image canvas (we use  $10\%$ ). For more robustness to noise, we pass  $\alpha$  through a scaled sigmoid to binarize it (Fig. 10), yielding the  $\bar{\alpha}_{\mathrm{bin}}$  used in Eq. 5. We sample camera azimuth in  $[0^{\circ}, 360^{\circ}]$  and elevation in  $[-90^{\circ}, 0^{\circ}]$  except in rare cases where we sample azimuth in a 90-degree range to minimize Janus-problem artifacts or generate indoor scenes with a diorama-like effect.
+
+We use a classifier-free guidance strength of 200 and textureless shading probability of 0.1 for SDS (Poole et al., 2022), disabling view-dependent prompting as it does not aid in the generation of compositional scenes (Table 3b).
+
+We otherwise inherit all other details, such as covariance annealing and random background rendering, from SDS. We optimize our model with Shampoo (Gupta et al., 2018) with a batch size of 1 for 15000 steps with an annealed learning rate, starting from  $10^{-9}$ , peaking at  $10^{-4}$  after 3000 steps, and decaying to  $10^{-6}$ .
+
+Optimizing NGPs. To verify the robustness of our approach to different underlying 3D representations, we also experiment with a re-implementation of Instant NGPs (Müller et al., 2022), and find that our method generalizes to that setting. Importantly, we implement an aggressive coarse-to-fine training regime in the form of slowly unlocking grid settings at resolution higher than  $64 \times 64$  only after 2000 steps. Without this constraint on the initial smoothness of geometry, the representation "optimizes too fast" and is prone to placing all density in one NGP.
+
+# A.2. Pseudo-code for layout learning
+
+In Figs. 9 and 10, we provide NumPy-like pseudocode snippets of the core logic necessary to implement layout learning, from transforming camera rays to compositing multiple 3D volumes to regularizing them.
+
+# A.3. CLIP evaluation
+
+To evaluate our approach, we use similarity scores output
+
+```julia
+Initialize variables  
+quad = normal((N, K, 4), mean=[0,0,0,1], std=0.1)  
+trans = normal((N, K, 3), mean=0., std=0.3)  
+scale = normal((N, K, 1), mean=1., std=0.3)  
+nerfs = [init_nerf() for i in range(K)]
+```
+
+```python
+Transform rays for NeRF k using layout n  
+def transform(rays, k, n):  
+    rot = quaternion_to_matrix(quat)  
+    rays['orig'] = rot[n,k] @ rays['orig'] - trans[n,k]  
+    rays['orig'] *= scale[n,k]  
+    rays['dir'] = scale[n,k] * rot[n,k] @ rays['dir']  
+return rays
+```
+
+```python
+Composite K NeRFs into one volume
+def composite_nerfs(per_nerf_rays):
+    per_nerf_out = [nerf(rays) for nerf, rays in zip(nerfs, per_nerf_rays]
+densities = [out['density'] for out in per_nerf_out]
+out = {'density': sum(densities)}
+wts = [d/sum(densities) for d in densities]
+rgbs = [out['rgb'] for out in per_nerf_out]
+out['rgb'] = sum(w*rgb for w,rgb in zip(wts, rgbs))
+return out, per_nerf_out
+```
+
+```javascript
+Train   
+optim  $=$  shampoo.params  $\equiv$  [nerfs,quat,trans, scale])   
+for step in range(num_steps): rays  $=$  sample_camera Rays() n  $=$  random.uniform(N) per_nerf Rays  $=$  [ transform(rays,k,n) for k in range(K) ] vol,per_nerf_vols  $=$  composite_nerfs(per_nerf Rays) image  $=$  render(vol,rays) loss  $=$  SDS(image,prompt,diffusion_model) loss  $+ =$  regularize(per_nerf_vols) loss.backup( optim_step_and_zero_grad(
+```
+
+Figure 9: Pseudocode for layout learning, with segments inherited from previous work abstracted into functions.
+
+by a pretrained contrastive text-image model (Radford et al., 2021), which have been shown to correlate with human judgments on the quality of compositional generation (Park et al., 2021). However, rather than compute a retrieval-based metric such as precision or recall, we report the raw  $(100 \times$  upscaled, as is common practice) cosine similarities. In addition to being a more granular metric, this avoids the dependency of retrieval on the size and difficulty of the test set (typically only a few hundred text prompts).
+
+We devise a list of 30 prompts (Fig. 11), each of which lists three objects, spanning a wide range of data, from animals to food to sports equipment to musical instruments. As described in Section 4, we then train models with  $K = 3$  NeRFs and layout learning and test whether each NeRF contains a different object mentioned in the prompt. We compute CLIP scores for each NeRF with a query prompt "a DSLR photo of [A/B/C]", yielding a  $3 \times 3$  score matrix.
+
+```python
+def soft_bin(x, t=0.01, eps=1e-7):
+    # x has shape (..., H, W)
+        bin = sigmoid((x - 0.5)/t)
+        min = bin.min(axis=-1, -2), keepdims=True)
+        max = bin.max(axis=-1, -2), keepdims=True)
+        return (bin - min) / (max - min + eps)
+soft_bin_acc = soft_bin(acc).mean((-1, -2))
+empty_loss = empty_loss_margin - soft_bin_acc
+empty_loss = max(empty_loss, 0.)
+```
+
+Figure 10: Pseudocode for empty NeRF regularization, where soft_bin_acc computes  $\bar{\alpha}_{\mathrm{bin}}$  in Equation 5.
+
+```csv
+'a cup of coffee, a croissant, and a closed book',  
+'a pair of slippers, a robe, and a candle',  
+'a basket of berries, a carton of whipped cream, and an orange',  
+'a guitar, a drum set, and an amp',  
+'a campfire, a bag of marshmallows, and a warm blanket',  
+'a pencil, an eraser, and a protractor',  
+'a fork, a knife, and a spoon',  
+'a baseball, a baseball bat, and a baseball glove',  
+'a paintbrush, an empty easel, and a palette',  
+'a teapot, a teacup, and a cucumber sandwich',  
+'a wallet, keys, and a smartphone',  
+'a backpack, a water bottle, and a bag of chips',  
+'a diamond, a ruby, and an emerald',  
+'a pool table, a dartboard, and a stool',  
+'a tennis racket, a tennis ball, and a net',  
+'sunglasses, sunscreen, and a beach towel',  
+'a ball of yarn, a pillow, and a fluffy cat',  
+'an old-fashioned typewriter, a cigar, and a glass of whiskey',  
+'a shovel, a pail, and a sandcastle',  
+'a microscope, a flask, and a laptop',  
+'a sunny side up egg, a piece of toast, and some strips of bacon',  
+'a vase of roses, a slice of chocolate cake, and a bottle of red wine',  
+'three playing cards, a stack of poker chips, and a flute of champagne',  
+'a tomato, a stalk of celery, and an onion',  
+'a coffee machine, a jar of milk, and a pile of coffee beans',  
+'a bag of flour, a bowl of eggs, and a stick of butter',  
+'a hot dog, a bottle of soda, and a picnic table',  
+'a pothos houseplant, an armchair, and a floor lamp',  
+'an alarm clock, a banana, and a calendar',  
+'a wrench, a hammer, and a measuring tape',  
+'a backpack, a bicycle helmet, and a watermelon'
+```
+
+Figure 11: Prompts used for CLIP evaluation. Each prompt is injected into the template "a DSLR photo of {prompt}, plain solid color background". To generate individual objects, the three objects in each prompt are separated into three new prompts and optimized independently.
+
+To compute NeRF-prompt CLIP scores, we average text-image similarity across 12 uniformly sampled views, each 30 degrees apart, at  $-30^{\circ}$  elevation. We then select the best NeRF-prompt assignment (using brute force, as there are only  $3! = 6$  possible choices), and run this process across 3 different seeds, choosing the one with the highest mean NeRF-prompt score.

2024/Disentangled 3D Scene Generation with Layout Learning/images.zip

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2024/Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet/3bbbf988-9032-4e0c-ade6-e44600da0c3c_origin.pdf

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2024/Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet/full.md

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+# Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet
+
+Zeyang Zhang $^{1}$  Xin Wang $^{1}$  Yijian Qin $^{1}$  Hong Chen $^{1}$  Ziwei Zhang $^{1}$  Xu Chu $^{1}$  Wenwu Zhu $^{1}$
+
+# Abstract
+
+The existing graph neural architecture search (GNAS) methods assume that the graph tasks are static during the search process, ignoring the ubiquitous scenarios where sequential graph tasks come in a continual fashion. Moreover, existing GNAS works resort to entangled graph factors during the architecture search process, resulting in the catastrophic forgetting problems. In this paper, we study the problem of continual GNAS that is expected to continually search the architecture to learn new graph tasks without forgetting the past, which remains largely unexplored in the literature. However, this problem poses the challenge of architecture conflicts, i.e., the optimal architecture for the new graph task may have performance deterioration and thus sub-optimal for past tasks. To address the challenge, we propose a novel Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet (GASIM) method, which is able to continually search the optimal architectures without forgetting past knowledge. Specifically, we first design a modular graph architecture super-network incorporating multiple modules to enable searching architecture with factor expertise. Second, we propose a factor-based task-module router that discovers the latent graph factors and routes the incoming task to the best suitable architecture module to alleviate the forgetting problem induced by architecture conflicts. Finally, we propose an invariant architecture search mechanism to capture the shared knowledge among tasks. Extensive experiments demonstrate that our method achieves state-of-the-art performance against baselines in continual graph neural architecture search.
+
+$^{1}$ Department of Computer Science and Technology, BN-RIST, Tsinghua University, Beijing, China. Correspondence to: Xin Wang <xin_wang@tsinghua.edu.cn>, Wenwu Zhu <wwzhu@tsinghua.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+# 1. Introduction
+
+Graph neural architecture search (GNAS), aiming at automating the discovery of optimal architectures for Graph neural networks (GNNs) when confronted with graph-structured data and specific tasks, has exhibited noteworthy advancements, demonstrating its efficacy in enhancing predictive capabilities and alleviating the need for extensive human involvement across a spectrum of graph applications (Zhang et al., 2021b). Most existing GNAS methods treat the entire graph as a holistic entity and search the architectures via optimizing for a fixed task applied to the entire graph, which is based on the underlying assumption that the graph tasks remain static throughout the search process.
+
+However, graph tasks come continually in many real-world scenarios. For example, in citation networks, novel types of papers and their corresponding citations may continually surface, necessitating a continually updated classifier to distinguish documents belonging to newly emerging research fields (Zhang et al., 2022d; Zhou & Cao, 2021). In the domain of drug design, the encounter with new categories or properties of molecules is a recurrent phenomenon, and thus, the predictor should be consistently updated to assimilate knowledge regarding these new molecule categories or properties (Kirkpatrick et al., 2017). As the existing GNAS methods assume the graph tasks are fixed, they will inevitably fail in the ubiquitous settings where graph tasks are continuously generated.
+
+In this paper, we study the problem of continual graph neural architecture search that is expected to continually search the graph architecture to learn new graph tasks without forgetting the past, which remains largely unexplored in the literature. However, we discover the problem is highly nontrivial with the critical challenge of architecture conflicts, i.e., graph tasks may have different optimal architectures, while the optimal architecture for the new graph task may perform worse on past tasks, and thus it could be challenging to search the architecture to learn the new graph task without harming the performance on previous tasks. We further utilize a mainstream class of differentiable GNAS methods for an example, and demonstrate with theoretical analyses that they fail to revolve the problem of architecture conflicts due to their entangled search process, resulting in
+
+catastrophic forgetting problems in continual graph architecture search process.
+
+To address these challenges, we propose a Disentangled Continual Graph Neural Architecture Search with Invariant Modular (GASIM) Supernet method, which is able to continually search the optimal graph architectures without forgetting the past graph tasks. The key idea is to design a modular graph super-network, wherein each module undertakes the search for optimal architectures tailored to the graph tasks with similar factors so that the task-architecture relationship is disentangled to foster the sharing of mutual knowledge and mitigating conflicts throughout the continuous search process. Specifically, we first design a modular graph architecture super-network incorporating multiple modules to enable searching architecture with factor expertise and resolve architecture conflicts. Each module is designed to be activated by the incoming graph before searching. Second, we propose a factor-based task-module router that discovers the latent graph factors and routes the incoming task to the best suitable architecture module to alleviate the forgetting problem induced by architecture conflicts. We adopt a distribution matching method to predict the latent graph factors, and then maintain a task-module relation graph to route the new graph task to suitable modules. Finally, we propose an invariant architecture search mechanism to capture the shared knowledge among tasks. Based on the task-module graph, we extract the relevant graph factors, and augment the search process by designing an invariance loss to discover architectures harmonizing all graph tasks that have ever routed to this specific module. Extensive experiments on real-world datasets demonstrate that the proposed method achieves state-of-the-art performance against baselines in continual graph neural architecture search.
+
+The contributions of this paper are summarized as follows:
+
+- We study the largely unexplored problem of continual graph neural architecture search and propose a novel Disentangled Continua Graph Neural Architecture Search with Invariant Modular Supernet (GASIM) method which is able to discover the optimal graph neural architectures in a continual fashion.  
+- We discover the problem of architecture conflicts in continual graph architecture search, and demonstrate with theoretical analyses that a mainstream classic of differentiable search methods can not resolve the problem due to the entangled search process.  
+- We introduce three novel modules, i) modular graph architecture super-network, ii) factor-based task-module router and iii) invariant architecture search mechanism, which can continually search the optimal architectures to learn the new graph tasks while mitigating the problem of forgetting past knowledge.
+
+- Extensive experiments on real-world datasets demonstrate that the proposed method achieves state-of-the-art performance against baselines in continual graph neural architecture search.
+
+# 2. Problem Formulation
+
+Graph Neural Architecture Search Let  $\mathcal{G}$  denote the graph space and  $\mathcal{V}$  the label space. A graph neural network is represented as a function  $a_{\alpha, w}: \mathcal{G} \to \mathcal{V}$ , with architecture parameters  $\alpha \in \mathcal{A}$  and learnable weights  $w \in \mathcal{W}$ , where  $\mathcal{A}$  is the architecture space and  $\mathcal{W}$  is the weight space. Graph Neural Architecture Search (GNAS) automates the design of graph neural architectures by searching for the optimal  $\alpha$ . As  $\alpha$  typically represents the selection of GNN operations (e.g., GCN (Kipf & Welling, 2016), GAT (Velicković et al., 2018), GIN (Xu et al., 2018)), it is referred to as operation choices for brevity. GNAS addresses the bi-level optimization problem (Elsken et al., 2019):
+
+$$
+\alpha^ {*} = \underset {\alpha \in \mathcal {A}} {\arg \min } \mathcal {L} \left(a _ {\alpha , w ^ {*} (\alpha)}, \mathcal {G}, \mathcal {Y}\right), \tag {1}
+$$
+
+$$
+\text {s . t .} w ^ {*} (\alpha) = \underset {w \in \mathcal {W} (\alpha)} {\arg \min } \mathcal {L} (a _ {\alpha , w}, \mathcal {G}, \mathcal {Y}), \tag {2}
+$$
+
+where  $\mathcal{L}$  represent the loss of predictions by the architecture  $a_{\alpha ,w}(\cdot)$  on the graph,  $\alpha^{*}$  and  $w^{*}$  is the best architecture and weight for the given task  $\mathcal{T} = (\mathcal{G},\mathcal{Y})$
+
+Continual Graph Neural Architecture Search In real-world scenarios, graphs are generated continuously. Consider a stream of graphs  $\mathcal{G} = \{\mathcal{G}_1,\mathcal{G}_2,\dots ,\mathcal{G}_t\}$ , where each graph  $\mathcal{G}_t = \{\mathcal{V}_t,\mathcal{E}_t\}$  has its own set of nodes and edges. Accompanying the graphs is a sequence of tasks  $\mathcal{T} = \{\mathcal{T}_1,\mathcal{T}_2,\ldots ,\mathcal{T}_t\}$  and their corresponding labels  $\mathcal{V} = \{\mathcal{V}_1,\mathcal{V}_2,\ldots ,\mathcal{V}_t\}$ . For example, in node classification tasks, each node  $v_{t}^{i}\in \mathcal{V}_{t}$  is expected to be classified to its node category  $y_{t}^{i}\in \mathcal{V}_{t}$ , while the incoming tasks could be distinct, e.g.,  $\mathcal{V}_{t - 1}\bigcup \mathcal{V}_t = \emptyset$ . An ideal continual graph neural architecture search is expected to continually modify the architecture to learn the new task without forgetting the past tasks. The objective can be formulated as
+
+$$
+\alpha_ {t} ^ {*}, w _ {t} ^ {*} = \underset {\alpha \in \mathcal {A}, w \in \mathcal {W}} {\arg \min } \sum_ {k = 1} ^ {t} \mathcal {L} \left(a _ {\alpha , w}, \mathcal {G} _ {k}, \mathcal {Y} _ {k}\right), \tag {3}
+$$
+
+where  $\alpha_{t}^{*}, w_{t}^{*}$  are the best architecture and weights at task  $\mathcal{T}_t$ , which can achieve the best average performance on all the tasks that have been seen so far. Since it is impractical to access all past graphs in the continual learning process, we consider the scenarios that only the current graph is available while the past graphs are not accessible in the stage of training or searching, following the continual learning literature (Wang et al., 2023a; Zhang et al., 2022c). The model should modify the architecture to learn the new task while maximally reducing the harm to the past tasks.
+
+![](images/e6d676532c0b3dc0c49868ea882609a0279b9c156b92aff12eba938b058501fa.jpg)  
+Figure 1: A showcase heatmap demonstrating that new graph tasks may have different optimal architectures on CoraFull dataset. Lighter colors denote higher performance of the architecture on the according task. Red and blue boxes denote the best performed and worst performed architectures on each task respectively. (Best viewed in color).
+
+# 3. Preliminary Studies: Architecture Conflicts
+
+In the settings of continual graph neural architecture search, the models may face the dilemma of architecture conflicts, where the optimal architecture of the new task may have poor performance for the past tasks. In such situations, directly searching architectures for the new task will bring catastrophic forgetting of the past tasks. To elaborate on this, we first conduct a preliminary experiment. We train and test six different GNN architectures for each task on CoraFull dataset, and report the performance for each task and architecture by a heatmap. From Figure 1, we have the following observations:
+
+Obs. 1. Different tasks have their own optimal architectures, and no architecture is always the winner. This phenomenon indicates that the performance is destined to be suboptimal if the architecture is fixed during graph continual learning, and also demonstrates the necessity of continual graph neural architecture search.
+
+Obs. 2. Some tasks have conflicting optimal architectures. For example, GIN performs the worst on the third task while the best on the fourth task, and in contrast, GCN performs the best on the third task while the worst on fourth task. This phenomenon brings great challenges for continual graph neural architecture search, since searching the optimal architecture for the new task may greatly deteriorate the performance of previous tasks if there exist architecture conflicts between these tasks.
+
+We further provide theoretical analyses of graph neural architecture search in the situations of architecture conflicts as follows. Consider two operations  $f_{1}(\cdot), f_{2}(\cdot)$  for two graph regression tasks, we adopt DARTS (Liu et al., 2018) as the search algorithm that searches the mixed operation  $F(\mathcal{G}) = o_{1}f_{1}(\mathcal{G}) + o_{2}f_{2}(\mathcal{G})$  with gradient methods to minimize the MSE loss  $\mathcal{L}_1(F)$  and  $\mathcal{L}_2(F)$  for each task respectively, where  $o_1 = \frac{\exp(\alpha_1)}{\exp(\alpha_1) + \exp(\alpha_2)}$  and  $o_2 = \frac{\exp(\alpha_2)}{\exp(\alpha_1) + \exp(\alpha_2)}$  and  $\alpha_{1},\alpha_{2}$  are learnable parameters. Suppose the operations are distinguishable, i.e.,  $\exists \mathcal{G}_k,f_1(\mathcal{G}_k)\neq f_2(\mathcal{G}_k)$ . Suppose  $y = f_{1}(\mathcal{G})$  and  $y = f_{2}(\mathcal{G})$  are the ground-truth labeling
+
+function for each task respectively, which means that the operations are conflicting for two tasks. We have the following proposition with proof in Appendix.
+
+Proposition 3.1. For optimizing  $\mathcal{L}_2(F)$ , after one step of gradient descent w.r.t architecture parameters, the changes of operation weights  $o_1, o_2$  will increase the loss  $\mathcal{L}_1(F)$ .
+
+This proposition shows that DARTS can not handle the architecture conflict problems, and it will inevitably fail due to forgetting the past when searching for the new task. The problem is that for both tasks, the super-network adopts the same mixed operation, i.e.,  $o_1, o_2$ , which is entangled together in the continual searching process.
+
+To this end, our key idea is to design a modular graph supernetwork, wherein each module undertakes the search for optimal architectures tailored to graph tasks with similar factors so that the task-architecture relationship is disentangled to foster the sharing of mutual knowledge and mitigating conflicts throughout the continuous search process.
+
+# 4. Methodology
+
+In this section, we introduce Disentangled Continua Graph Neural Architecture Search with Invariant Modular Supernet (GASIM) to continually search graph architectures, by proposing three key components, modular graph architecture super-network, factor-based task-module router, and invariant architecture search mechanism. The overall framework is illustrated in Figure 2.
+
+# 4.1. Modular Graph Architecture Super-network
+
+To alleviate architecture conflicts and share mutual knowledge, we propose a modular graph architecture supernetwork to concurrently estimate and search for  $K$  different modules, where each time one module is updated to accommodate graph tasks with similar factors and the other modules are kept unchanged to reduce conflicts.
+
+![](images/05ecad3795338fe5e08cf010dd7a147ae71bf7054a17ed7ff83031b0caca844c.jpg)  
+Figure 2: The framework of Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet (GASIM), including the following three key components: 1) Modular graph architecture super-network incorporating multiple modules to enable searching architectures with factor expertise, 2) Factor-based task-module router that discovers the latent graph factors and routes the incoming task to the best suitable architecture module to alleviate the forgetting problem induced by architecture conflicts, 3) Invariant architecture search mechanism to capture the shared knowledge among tasks in the routed module. (Best viewed in color)
+
+Modular Super-network Layer Each super-network layer employs  $K$  mixed operations parameterized by different vectors  $\alpha$  to learn  $K$ -chunk graph representations:
+
+$$
+\mathbf {H} _ {k} \leftarrow \overline {{\mathrm {G N N}}} _ {\alpha_ {k}} (\mathbf {H}, \mathbf {A}), \tag {4}
+$$
+
+where  $\mathbf{A}$  denotes the adjacency matrix of the graph,  $\mathbf{H}$  represents the input graph representations, and  $\overline{\mathrm{GNN}}_{\alpha_k}(\cdot)$  denotes the mixed GNN operations parameterized by  $\alpha_{k}$ . To facilitate differentiable optimization, we adopt a continuous parameterization and weight-sharing mechanism (Liu et al., 2018) for the mixed operations:
+
+$$
+\overline {{\mathrm {G N N}}} _ {\alpha_ {k}} (\mathbf {H}, \mathbf {A}) = \sum_ {i = 1} ^ {| \mathcal {O} |} \alpha_ {k, i} \mathrm {G N N} _ {i} (\mathbf {H}, \mathbf {A}), \tag {5}
+$$
+
+where  $|\mathcal{O}|$  is the number of GNN operation choices,  $\alpha_{k,i} = \frac{\exp(\theta_{\alpha_{k,i}})}{\sum_j\exp(\theta_{\alpha_{k,j}})}$  denotes the probability of the  $i$ -th operation for the  $k$ -th architecture  $\alpha_{k}$ , and  $\theta$  represents learnable parameters. The overall super-network is constructed in the form of a directed acyclic graph (DAG) with an ordered sequence of modular super-network layers.
+
+Flexible Modular Path An incoming graph can select a flexible path in the modular super-network, i.e.,
+
+$$
+\pi (\mathcal {G} _ {t}) = \left(a _ {\alpha_ {i}, w _ {i}} (\cdot), \dots , a _ {\alpha_ {j}, w _ {j}} (\cdot)\right), \tag {6}
+$$
+
+where each  $\alpha$  in the path is the chosen operations to be activated to calculate the representations of the graph. This
+
+design mitigates the conflicts of architectures by offering a broader array of flexible paths in the super-network (Guo et al., 2020). To illustrate, the 'mean' operation is adept at capturing structural properties, while the 'max' operation excels at capturing representative elements (Xu et al., 2018). When the consecutive tasks have operation conflicts, our method can simultaneously capture both aspects by selecting operations to learn the respective representations.
+
+Route then Search Directly optimizing the modular path for each graph as Eq. (6) could be difficult for its large discrete solution space. We transform the probability of the path into the expectation of multiple paths under different modules by the Bayesian formula, i.e.,
+
+$$
+p \left(\pi \left(\mathcal {G} _ {t}\right) \mid \mathcal {G} _ {t}\right) = \sum_ {k = 1} ^ {K} \underbrace {p (k \mid \mathcal {G} _ {t})} _ {\text {R o u t e}} \overbrace {p \left(\pi \left(\mathcal {G} _ {t}\right) \mid \mathcal {G} _ {t} , k\right)} ^ {\text {S e a r c h}}, \tag {7}
+$$
+
+where the probability of the graph  $\mathcal{G}_t$  to activate the  $k$ -th module  $p(k \mid \mathcal{G}_t)$  can be modeled by the relationship between modules and tasks, the probability of module-specific architectures  $p(\pi(\mathcal{G}_t) \mid \mathcal{G}_t, k)$  can be optimized by a specific super-network. To further alleviate the forgetting problem, we adopt a hard routing mechanism that only one module can be selected to be searched while others are kept unchanged, so that learning the new task will not affect the results of other past tasks using other modules.
+
+# 4.2. Factor-based Task-Module Router
+
+Graph factors are shown important in graph formation, representing various inherent graph properties (Fan et al., 2019; Ma et al., 2019). We propose to leverage the graph factors to maintain a task-module graph to implement the routing as Eq. (7). Specifically, we learn the latent factors with distribution matching following (Liu et al., 2023a), and measure the task similarity by the factors to conduct routing.
+
+Latent Factor Prediction with Distribution Matching For task  $\mathcal{T}_t$  with graph  $\mathcal{G}_t = (\mathbf{A}_t,\mathbf{X}_t)$  and labels  $\mathcal{V}_t$ , we learn the latent factors  $\mathbf{F}_t$  as a sketch of the graph task. The factors should faithfully reflect the characteristics and distributions of the original graph, so that the module can be routed through the factor similarity. To this end, we adopt Maximum Mean Discrepancy (MMD) to learn the factors.
+
+$$
+\mathbf {F} _ {t} ^ {*} = \underset {\mathbf {F}} {\arg \min } \ell_ {\mathrm {M M D}} \left(\operatorname {G N N} \left(\mathbf {X} _ {t}, \mathbf {A} _ {t}\right), \operatorname {G N N} ^ {\prime} \left(\mathbf {F}, \mathbf {A} _ {f}\right)\right), \tag {8}
+$$
+
+where  $\mathbf{A}_f$  is the adjacency matrix of the relationship between the factors and is set to a self-loop matrix for simplicity.  $\mathrm{GNN}(\cdot)$  and  $\mathrm{GNN}'(\cdot)$  are shared random encoders, which encode the original graph and the learned graph factors into a shared latent space. From the perspective of self-supervised learning, the objective aims to reconstruct the distribution characteristics in the latent space (Hou et al., 2022; Xie et al., 2022). We random sample several node features as the initialization of the factors to ease the optimization process. We learn the latent factors for each class, and denote the classes of the factors as  $\mathcal{Y}_t^f$ .
+
+Architecture Route with Task-Module Graph We maintain the task-module graph, and keep updating it to assign new tasks to modules. The task-module graph at task  $t$  is defined by a relation matrix  $\mathbf{R}_t^m \in [0,1]^{t \times K}$ , where the  $t$ -th row  $\mathbf{r}_t^m \in [0,1]^K$  denotes the probability of  $\mathcal{T}_t$  being assigning to each module. Denote the super-network at task  $t$  as  $g_t(\cdot)$  and the classifier as task  $t$  as  $f_t(\cdot)$ . We calculate the similarity between the  $t$ -th task and previous tasks by leveraging the factors and the previous classifier,
+
+$$
+\tilde {\mathbf {R}} _ {t} ^ {f} = f _ {t - 1} \circ g _ {t - 1} \left(\mathbf {F} _ {t}\right), \tag {9}
+$$
+
+where  $\tilde{\mathbf{R}}_t^f\in \mathbb{R}^{N\times C_{t - 1}}$  refers to the probabilities of factors belonging to the classes,  $N$  is the number of factors, and  $C_{t - 1}$  is the number of classes till  $(t - 1)$ -th task. We calculate the factor-task relation matrix  $\mathbf{R}_t^f\in \mathbb{R}^{N\times (t - 1)}$  by adding up the probabilities of the tasks' corresponding classes, i.e.,  $\mathbf{R}_t^f = \tilde{\mathbf{R}}_t^f\cdot \mathbf{R}_t^c$ , where  $\mathbf{R}_t^c\in \{0,1\}^{C_{t - 1}\times (t - 1)}$  is a matrix representing the relations between the classes and the tasks. Then we calculate the probability of the new task  $\mathcal{T}_t$  being assigned to the modules as
+
+$$
+\mathbf {r} _ {t} ^ {m} = \operatorname {S o f t m a x} \left(\operatorname {M e a n} \left(\mathbf {R} _ {t} ^ {f} \cdot \mathbf {R} _ {t - 1} ^ {m}\right)\right), \tag {10}
+$$
+
+where  $\operatorname{Mean}(\cdot)$  takes the average along the first dimension, and  $\mathbf{r}_t^m$  is the assigning vector for task  $\mathcal{T}_t$ . We adopt the module with maximum probability in  $\mathbf{r}_t^m$  for searching, and add the probability vector into  $\mathbf{R}_t^m$  for subsequent task routing. For the initialization, we assign one-hot vectors for the early tasks  $\mathcal{T}_t, t < K$ . By leveraging the relationship between factors, tasks and modules, we continually update the task-module graph for routing new tasks to modules with similar properties to avoid architecture conflicts while sharing mutual knowledge.
+
+# 4.3. Invariant Architecture Search Mechanism
+
+In the continual GNAS process, each module searches architectures for its routed new task, which may have distribution shifts from the past tasks that have routed to the module, even though they have similar factors. This shift between tasks may cause the module to overfit the new task while forgetting the past tasks. To this end, we propose an invariant architecture search mechanism to better share mutual knowledge via capturing the features invariant to tasks.
+
+Routing Loss Suggest the router chooses  $k$ -th module for the task  $\mathcal{T}_t$ , and denote  $g_t^k(\cdot)$  as the super-network sliced with  $k$ -th module, and  $f_t^k(\cdot)$  as the classifier. Then loss is calculated as
+
+$$
+\mathcal {L} _ {t} ^ {r} = \ell \left(f _ {t} ^ {k} \circ g _ {t} ^ {k} (\mathcal {G} _ {t}), \mathcal {Y} _ {t}\right), \tag {11}
+$$
+
+where  $\ell(\cdot)$  is the cross-entropy loss function, and  $\mathcal{G}_t$  and  $\mathcal{V}_t$  are the graph and labels at the current task. Note that only the routed module will be updated while other modules are kept unchanged to alleviate the forgetting problems.
+
+Invariance Loss Based on the task-module graph  $\mathbf{R}_t^m$ , we select the factors that have the common module choice of the current task. Suggest the router chooses  $k$ -th module for the task  $\mathcal{T}_t$ , we obtain the factor indices  $\mathcal{I}_t$  by
+
+$$
+\mathcal {I} _ {t} = \left\{j \mid \underset {i} {\arg \max } \mathbf {R} _ {t} ^ {m} (j, i) = k \right\}. \tag {12}
+$$
+
+Then we obtain a set of factors and their classes  $\{\mathbf{F}_i, \mathcal{Y}_i^f \mid \in \mathcal{I}_t\}$ , and calculate the invariance loss as
+
+$$
+\mathcal {L} _ {t} ^ {i n v} = \operatorname {V a r} \left(\ell \left(f _ {t} ^ {k} \circ g _ {t} ^ {k} (\mathbf {F} _ {i}), \mathcal {Y} _ {i} ^ {f}\right) \mid i \in \mathcal {I} _ {t}\right), \tag {13}
+$$
+
+where  $\operatorname{Var}(\cdot)$  calculates the variance. This loss constrains the searched architecture from losing the capability of handling the previous tasks routed to the same module. The final loss for searching the architectures is
+
+$$
+\mathcal {L} _ {t} = \mathcal {L} _ {t} ^ {r} + \lambda \mathcal {L} _ {t} ^ {\text {i n v}}, \tag {14}
+$$
+
+where  $\lambda$  is a hyper-parameter to make a trade-off between the performance of the new task and past tasks. Similar to
+
+(Liu et al., 2018), the architecture parameters at task  $\mathcal{T}_t$  are updated differentiably with
+
+$$
+\alpha = \alpha - \eta_ {\alpha} \nabla_ {\alpha} \mathcal {L} _ {t}, w = w - \eta_ {w} \nabla_ {w} \mathcal {L} _ {t}. \tag {15}
+$$
+
+After searching the architectures, we fix the architectures and finetune the weights with factors for better classification. The overall algorithm is summarized in Algo. 1
+
+# Algorithm 1 The pipeline for GASIM
+
+Require: The number of tasks  $T$ , hyperparameter  $\lambda, K$ .
+
+1: Construct the modular super-network in Sec. 4.1.  
+2: for  $l = 1, \dots, T$  do  
+3: Predict the latent factors as Eq. (8)  
+4: Route the task to the module as Eq. (10)  
+5: Calculate routing loss as Eq. (11)  
+6: Calculate invariance loss as Eq. (13)  
+7: Calculate the final loss as Eq. (14)  
+8: Search the architecture according to Eq. (15)  
+9: Fix the architecture and finetune the weights  
+10: end for
+
+Discussions about the framework and MoE The router of our framework aims to learn several groups of similar tasks with expert architecture modules. While the router and Mixture-of-Experts (MoE) (Shazeer et al., 2017; Fedus et al., 2022) have something in common that they both route the data to different parameterized networks for forwarding, they have different objectives. The objective of most MoE methods is to promote the performance by significantly enlarging the model capacity, while controlling the model sparsity to maintain the inference efficiency. In contrast, in our framework, the objective of the routing is to alleviate the architecture conflicts in the continual search process by routing dissimilar tasks to different modules, while routing similar tasks to the same module to boost shared knowledge. Therefore, differing from the MoE objective, the main focus of the routing is to facilitate modularization and continually search graph architectures to adapt to new tasks while not forgetting the previous knowledge.
+
+# 5. Experiments
+
+In this section, we conduct experiments on real-world continual graph datasets to verify the design of our method. We also include detailed ablation studies to analyze the effectiveness of each component.
+
+# 5.1. Experimental Setups
+
+Baselines We compare our method with 10 baselines from the following two different categories.
+
+- Manually designed GNNs. We include six representative GNNs as our baselines, i.e., GCN (Kipf &
+
+Welling, 2016), GAT (Veličković et al., 2018), GIN (Xu et al., 2018), GraphSage (Hamilton et al., 2017), GraphConv (Morris et al., 2019), SGC (Wu et al., 2019) which also constitute our search space.
+
+- Graph neural architecture search. We include two representative GNAS baselines GraphNAS (Gao et al., 2020) and GASSO (Qin et al., 2021) where GASSO is specially designed for node-level classification tasks by searching the graph structures simultaneously. We also include two classical NAS baselines, random search and DARTS (Liu et al., 2018). Since they are not specially designed for graphs, we adopt our search space for them.
+
+Since these methods are not designed for continual graph learning, we adopt the most recent and competitive continual graph learning method (Liu et al., 2023a;b) to train them to suit to the continual settings.
+
+Super-network Construction The super-network generally consists of two parts, the operation pool and the directed acyclic graph (DAG) that wires the operations. For the operation pool, we include the six node aggregation operations mentioned in the manually designed GNN baselines. For brevity, we denote 'Agg' as node aggregation operations. Following (Qin et al., 2021), the DAG for each module is a straightforward path, i.e.,  $\mathbf{H}^{l + 1} = \mathrm{Agg}^l (\mathbf{H}^l,\mathbf{A})$ , and the embeddings of the last layer are utilized for downstream tasks, where  $\mathbf{H}^l$  denotes the hidden embeddings output by the  $l$ -th layer, and  $\mathbf{A}$  denotes the graph adjacency matrix. For searching the routed module, the embedding is fed to the module's classifier  $f(\cdot)$ , and the prediction is  $f(\mathbf{H})$ . For finetuning and testing the whole model, the embeddings of each module are concatenated together, i.e.,  $\mathbf{H} = [\mathbf{H}_1||\mathbf{H}_2||\dots ||\mathbf{H}_K]$ , which is subsequently fed to the whole model's classifier for prediction.
+
+Datasets We adopt three node classification tasks in experiments, including CoraFull (McCallum et al., 2000), Arxiv (Hu et al., 2020), and Reddit (Hamilton et al., 2017). CoraFull and Arxiv are citation networks, and Reddit is a post-to-post graph. Table 1 shows the statistics of these datasets. All datasets are partitioned into a set of tasks, each focusing on the node classification problem, where each task involves nodes from two distinct classes within an incoming graph. For each task,  $60\%$  of the nodes are allocated for training,  $20\%$  for validation, and  $20\%$  for testing.
+
+Evaluation We adopt the challenging setting that no task indicator is provided in the testing stage, also known as class-IL setting (Wang et al., 2023a), and the model continues to make classifications for all categories that have been seen so far. During the continual update phase, the model can only access the newly incoming graph and all the past graphs are
+
+Table 1: A summary of dataset statistics.  
+
+<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Features</td><td>Classes</td><td>Tasks</td></tr><tr><td>CoraFull</td><td>19,793</td><td>130,622</td><td>8,710</td><td>70</td><td>35</td></tr><tr><td>Arxiv</td><td>169,343</td><td>1,166,243</td><td>128</td><td>40</td><td>20</td></tr><tr><td>Reddit</td><td>227,853</td><td>114,615,892</td><td>602</td><td>40</td><td>20</td></tr></table>
+
+not available. Denote  $m_{k,i}$  as the performance on  $i$ -th task after learning  $k$ -th task. We adopt three metrics to measure the model performance: 1) Average performance (AP), measuring the average model performance after learning from Task  $\mathcal{T}_k$ , i.e.,  $\mathrm{AP}_k = \frac{1}{k}\sum_{i=1}^k m_{k,i}$ , 2) Mean of average performance (AP), measuring the average performance of the continual learning process, i.e.,  $\overline{\mathrm{AP}} = \frac{1}{k}\sum_{i=1}^k \mathrm{AP}_i$ . 3) Average Forgetting (AF), measuring the influence of the training process for the current task on the performance of previous tasks, i.e.,  $\mathrm{AF}_k = \frac{1}{k-1}\sum_{i=1}^{k-1}(m_{k,i} - m_{i,i})$ . A higher value indicates that training the current task has a more substantial impact on historical tasks. A negative or positive number signifies a detrimental or beneficial impact.
+
+More experimental details are provided in the Appendix, including configurations and implementation details.
+
+# 5.2. Main Results
+
+From the results summarized in Table 2, we have the following observations:
+
+- No hand-designed GNN is always the best for all datasets, which verifies the demand for automated graph architecture design in a continual fashion.  
+- The existing GNAS methods do not always beat the hand-designed GNNs, showing their limitations in handling continual graph data since their underlying assumption of static structural distribution and task may not hold in real continual graph data.  
+- The existing GNAS methods have lower average forgetting in most cases. In particular, DARTS achieves  $61\%$  average performance while having  $-16.6\%$  AF on Arxiv dataset. This phenomenon indicates that DARTS can search a better architecture for each task, but the architecture may perform poorly for the previous tasks, which further verifies our discovered problem of architecture conflicts in the continual search process. Neglecting the problem will let the model overfit the newly-coming task while harming the past knowledge.  
+- Our method achieves significant improvements over the baselines. In particular, our method improves  $8\%$  in terms of average performance over the second-best baseline on CoraFull dataset. The results verify the effectiveness of our method that is able to continually search the
+
+![](images/57a061e9a24ad43c8073ca29fd8448bc41fb359ed30dce715f8f7a05c5455342.jpg)  
+Figure 3: Comparisons of different ablated variants of GASIM on real-world datasets in terms of mean average performance. 'Full' denotes the full version of the method (Best viewed in color).
+
+architectures to learn the newly-coming tasks without forgetting past knowledge.
+
+# 5.3. Additional Experiments
+
+Ablation studies To verify the effectiveness of each designed component, we compare different ablated versions on each dataset: 1) v1 removes the invariance loss in Eq. (13), 2) v2 replaces the similar factor selection in Eq. (12) with random factor selection, 3) v3 replaces the router in Eq. (10) with a random router, which routes the task to a random module, 4) v4 replaces the router in Eq. (10) with a fixed router, which always routes the task to the first module, 5) v5 replaces the router in Eq. (10) with a sequential router, which always routes the task to the next module, i.e.,  $t\% K$ , 6) v6 replaces the modular super-network with a vanilla super-network. From Fig. 3, we have the following observations:
+
+- Our proposed GASIM outperforms all the ablated variants on all datasets, demonstrating the effectiveness of each component of our proposed framework in continually searching graph architectures.  
+- The ablated versions v1 and v2 have a huge performance drop on Arxiv, showing the importance of leveraging the invariance to share mutual knowledge.  
+- The hand-designed routers v3, v4 and v5 have more suboptimal and unstable results across datasets, showing the effectiveness of our factor-based task-module router.  
+- The ablated version v6 fails drastically on CoraFull dataset, showing that the modular super-network is critical for resolving architecture conflicts to boost the continual searching process.
+
+Visualized showcases We illustrate the performance matrices of three methods, GCN, DARTS and ours in Fig. 4. The performance matrix shows the test results for all past tasks after learning the new task. We find that our method
+
+Table 2: The results of all the methods on the real-world datasets. Numbers after the  $\pm$  signs represent standard deviations. The best results are in bold.  
+
+<table><tr><td>Dataset</td><td colspan="3">CoraFull</td><td colspan="3">Arxiv</td><td colspan="3">Reddit</td></tr><tr><td>Metric</td><td>AP</td><td>AF</td><td>\( \overline{AP} \)</td><td>AP</td><td>AF</td><td>\( \overline{AP} \)</td><td>AP</td><td>AF</td><td>\( \overline{AP} \)</td></tr><tr><td>GCN</td><td>61.94±1.45</td><td>-10.17±1.01</td><td>75.54±0.76</td><td>62.64±0.91</td><td>-13.78±0.47</td><td>72.16±0.78</td><td>96.65±0.13</td><td>-0.78±0.13</td><td>98.09±0.06</td></tr><tr><td>GIN</td><td>67.70±0.52</td><td>-3.62±1.73</td><td>76.73±0.79</td><td>59.68±3.90</td><td>-16.93±3.14</td><td>72.21±0.94</td><td>97.77±0.05</td><td>0.04±0.02</td><td>98.68±0.04</td></tr><tr><td>GAT</td><td>40.26±5.70</td><td>-4.57±4.12</td><td>50.72±1.42</td><td>61.87±0.51</td><td>-9.70±0.48</td><td>70.03±1.07</td><td>78.14±1.82</td><td>-5.90±4.06</td><td>88.54±2.08</td></tr><tr><td>GraphSage</td><td>61.29±0.97</td><td>-9.17±0.76</td><td>74.25±1.02</td><td>59.56±0.92</td><td>-13.44±1.32</td><td>69.30±0.56</td><td>91.56±0.48</td><td>-1.95±0.42</td><td>94.57±0.20</td></tr><tr><td>SGC</td><td>60.87±1.68</td><td>-10.93±1.07</td><td>75.13±1.18</td><td>56.33±1.42</td><td>-10.96±0.42</td><td>66.94±0.75</td><td>96.36±0.13</td><td>-0.67±0.03</td><td>97.87±0.05</td></tr><tr><td>GraphConv</td><td>67.15±1.12</td><td>-6.21±0.93</td><td>77.42±1.11</td><td>53.48±7.30</td><td>-19.45±5.00</td><td>69.40±1.70</td><td>97.61±0.06</td><td>-0.22±0.15</td><td>98.72±0.03</td></tr><tr><td>Random</td><td>63.57±1.06</td><td>-12.69±2.86</td><td>77.03±0.60</td><td>57.94±10.48</td><td>-19.76±11.33</td><td>71.74±0.97</td><td>97.89±0.08</td><td>12.27±5.76</td><td>87.55±2.70</td></tr><tr><td>GraphNAS</td><td>60.01±8.50</td><td>-18.85±5.71</td><td>63.81±11.05</td><td>52.94±4.12</td><td>-9.58±1.28</td><td>62.94±4.34</td><td>96.98±0.48</td><td>-1.01±0.18</td><td>98.34±0.26</td></tr><tr><td>DARTS</td><td>66.02±0.52</td><td>-6.89±0.90</td><td>77.40±0.38</td><td>61.45±2.09</td><td>-16.59±1.65</td><td>71.74±1.42</td><td>97.55±0.09</td><td>-0.25±0.11</td><td>98.75±0.02</td></tr><tr><td>GASSO</td><td>61.94±1.08</td><td>-10.07±0.61</td><td>75.11±0.93</td><td>61.21±0.84</td><td>-15.02±0.50</td><td>70.93±0.80</td><td>96.43±0.19</td><td>-0.94±0.11</td><td>97.98±0.08</td></tr><tr><td>Ours</td><td>76.03±0.77</td><td>-5.22±0.80</td><td>78.95±0.71</td><td>65.03±1.40</td><td>-12.19±1.64</td><td>73.43±0.84</td><td>98.40±0.01</td><td>-0.17±0.06</td><td>99.10±0.02</td></tr></table>
+
+![](images/bf15c1a14d2871ba921f8ceee289771708c977bc86627f8b41a68a41fa03d3f7.jpg)  
+Figure 4: Performance matrices of different methods on Arxiv dataset. Lighter colors denote higher performance. The value  $P_{i,j}, j \leq i$  denotes the performance on  $j$ -th task after learning the  $i$ -th task. (Best viewed in color).
+
+![](images/5039f1d3c176926642a9c36fe0f0857e552299c794107d8d80c58dd58fac2d1b.jpg)  
+Figure 5: A showcase heatmap demonstrating that the router continually selects different modules on Arxiv dataset. Lighter colors denote higher probability of the module on the according task. Red boxes denotes the module choice on each task respectively. (Best viewed in color).
+
+can continually assimilate new knowledge by searching the architectures for the new task. For example, the performance of the 8-th task even grows after learning the 10-th task. To make deeper analyses, we illustrate the router choices in Fig. 5. For the 8-th and the 10-th tasks, the router both chooses the 0-th module, which shows that the model may share the knowledge in the same module to boost the performance of both tasks in the continual searching process.
+
+# 6. Related works
+
+Graph Neural Architecture Search Neural architecture search has gained prominence as it endeavors to automatically discover optimal architectures for specific tasks. This trend has been particularly notable in computer vision (Ren et al., 2021; Elsken et al., 2019), natural language processing (Chitty-Venkata et al., 2022). Graph neural architecture search (GNAS) methods (Guan et al., 2021; Qin et al., 2022b), confronting the unique challenge of tackling the relationship of architectures and complex structures on graphs, can be broadly categorized into reinforcement-learning-based methods (Gao et al., 2020; Zhou et al., 2019; Cai et al., 2022; Gao et al., 2022; 2023), evolutionary-based methods (Nunes & Pappa, 2020; Li & King, 2020; Shi et al., 2022; Zhang et al., 2022a;b), and differentiable methods (Ding et al., 2021; Huan et al., 2021; Li et al., 2021c; Cai et al., 2021; Qin et al., 2022a; Wei et al., 2021; Zhang et al., 2023f; e; Li et al., 2024c; Zheng et al., 2023). However, a prevailing limitation of existing GNAS methods is their dependency on the underlying assumption of static distribution and tasks, and thus limiting their application in real-world scenarios. In this paper, we study the problem of continual graph neural architecture search in the setting where graphs are continuously generated.
+
+Continual Graph Learning Continua graph learning aims to handle streaming graph data other than static graphs. Distinct from continual learning in computer vision (Kirkpatrick et al., 2017; Aljundi et al., 2018; Lopez-Paz & Ranzato, 2017; Li & Hoiem, 2017; Wang et al., 2022), continual graph learning methods should take into consideration the changes of structures as well as the tasks applied to these structures. The existing methods can be roughly classified into two categories, regularisation-based methods (Liu et al., 2021) and replay-based methods (Zhou & Cao, 2021; Zhang et al., 2022e; Liu et al., 2023a). (Liu et al., 2021) employs
+
+regularisation to preserve topological information from historical graphs. (Zhou & Cao, 2021) incorporates memory-replay mechanisms by storing representative nodes, while (Zhang et al., 2022e) employs memory banks to store sparsified subgraphs, thereby preserving structural information. (Liu et al., 2023a) condenses the graphs into memory with consideration of class imbalance, which are then adopted for replaying in the learning process. To assess and benchmark CGL methods, two recent benchmarks (Zhang et al., 2022c; Ko et al., 2022) have been developed, contributing to the evolving landscape of graph continual learning research. However, these methods adopt a fixed architecture with the assumption that the optimal architecture is static in the continual learning process. Recently, there has been increasing interest in utilizing the in-context learning capabilities of large language models for graph-related tasks (Jin et al., 2023; Zhang et al., 2023a; 2024; Yao et al., 2024; Li et al., 2024b; Chen et al., 2024b). However, it remains underexplored whether these methods can effectively adapt to continuously emerging graph domains and tasks. In this paper, we continually modify the architecture to suit for the newly-coming graph tasks in the continual learning process.
+
+Disentangled Graph Learning Disentangled representation learning aims to delineate and elucidate the distinct latent factors that influence observable data, representing each factor as a unique vector representation (Bengio et al., 2013; Wang et al., 2023b), and the obtained factors could be helpful for explaining the inherent mechanism driving the data formation as well as producing robust representations for downstream tasks. This methodology has proven valuable across diverse domains, spanning computer vision (Hsieh et al., 2018; Ma et al., 2018; Chen et al., 2016; Wang et al., 2023c; Chen et al., 2023; 2024a), and graph representation learning (Ma et al., 2019; Liu et al., 2020; Yang et al., 2020; Chen et al., 2021; Li et al., 2021a;b; 2022c). In the context most relevant to our work, (Qin et al., 2022a) characterize latent factors within graph data by employing a self-supervised disentangled graph encoder, and then conducts graph neural architecture search for each graph to address distribution shifts, while the graphs and tasks are still fixed in the optimization process. In contrast, our focus in this paper centers on the automation of continual graph neural architecture designing via disentangling the relationship between graphs and architectures with the invariant modular super-network and routing mechanism.
+
+Out-of-Distribution Generalization Many current machine learning techniques are based on the premise that training and testing data are independent and identically distributed. However, this assumption may not hold in complex real-world settings (Shen et al., 2021; Yao et al., 2022; Xu et al., 2022; Wen et al., 2024), particularly during the continual learning process where data distributions can
+
+progressively shift across sequential tasks. One classic of Out-of-Distribution (OOD) generalization methods is invariant learning that aims to obtain invariant features are robust across distributions (Arjovsky et al., 2019; Sagawa et al., 2019; Krueger et al., 2021). Recently, there is a growing interest in achieving OOD generalization on graphs (Li et al., 2024a; 2022a;b;d; 2023a;b; Wu et al., 2022a;b; Chen et al., 2022; Zhang et al., 2021a; 2022g; Fan et al., 2021) as well as dynamic graphs that come sequentially (Zhang et al., 2023b;c;d; 2022f). In our work, we leverage insights from the invariant learning literature and propose an invariance loss with regard to graph architectures and tasks to enhance knowledge sharing across different tasks and modules, thereby mitigating the distribution shifts during the continual learning process.
+
+# 7. Conclusion
+
+Existing graph neural architecture search (GNAS) methodologies fall short in addressing the dynamic nature of continually evolving graph data. In this paper, we propose Disentangled Continual Graph Neural Architecture Search with Invariant Modular Supernet (GASIM) method, specifically designed to overcome the challenges of continual GNAS. By proposing a modular graph architecture super-network and a factor-based task-module router, our approach efficiently navigates the entangled nature of graph factors and resolves conflicts arising from evolving relationships between the entire graph and optimal architectures. The proposed invariant architecture search method further ensures the retention of shared knowledge across tasks, enabling continual learning without forgetting past knowledge. Extensive experiments demonstrate that our method achieves state-of-the-art performance in continual graph neural architecture search. One limitation of this paper is that we mainly focus on homogeneous graphs, and we leave extending our method to heterogeneous graphs for further explorations.
+
+# Acknowledgements
+
+This work is supported by the National Key Research and Development Program of China No.2023YFF1205001, National Natural Science Foundation of China (No. 62222209, 62250008, 62102222, 62206149), Beijing National Research Center for Information Science and Technology under Grant No. BNR2023RC01003, BNR2023TD03006, and Beijing Key Lab of Networked Multimedia.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Appendix
+
+# A.1. Proof of Proposition 1
+
+To prove the proposition, we have to first give a Lemma A.1 inspired by (Qin et al., 2021).
+
+Lemma A.1. The operation weights are calculated by a softmax function. E.g.  $o_1 = \frac{\exp\{\alpha_1\}}{\exp\{\alpha_1\} + \exp\{\alpha_2\}}$ . Let  $g_1$  and  $g_2$  are the gradients w.r.t.  $\alpha_1$  and  $\alpha_2$ . The operation weights after a gradient descent step are  $o_1'$  and  $o_2'$ , e.g.,  $o_1' = \frac{\exp\{\alpha_1 - g_1\}}{\exp\{\alpha_1 - g_1\} + \exp\{\alpha_2 - g_2\}}$ . If  $g_1 < g_2$ , then  $o_1' > o_1$  and  $o_2' < o_2$ , and vice versa.
+
+Proof. If  $g_{1} < g_{2}$ , we should improve  $o_1' > o_1$ , that is
+
+$$
+\frac {e x p \left\{\alpha_ {1} - g _ {1} \right\}}{e x p \left\{\alpha_ {1} - g _ {1} \right\} + e x p \left\{\alpha_ {2} - g _ {2} \right\}} > \frac {e x p \left\{\alpha_ {1} \right\}}{e x p \left\{\alpha_ {1} \right\} + e x p \left\{\alpha_ {2} \right\}}
+$$
+
+$$
+\exp \left\{\alpha_ {1} - g _ {1} \right\} \left(\exp \left\{\alpha_ {1} \right\} + \exp \left\{\alpha_ {2} \right\}\right) > \exp \left\{\alpha_ {1} \right\} \left(\exp \left\{\alpha_ {1} - g _ {1} \right\} + \exp \left\{\alpha_ {2} - g _ {2} \right\}\right)
+$$
+
+$$
+\exp \left\{2 \alpha_ {1} - g _ {1} \right\} + \exp \left\{\alpha_ {1} + \alpha_ {2} - g _ {1} \right\} > \exp \left\{2 \alpha_ {1} - g _ {1} \right\} + \exp \left\{\alpha_ {1} + \alpha_ {2} - g _ {2} \right\} \tag {16}
+$$
+
+$$
+e x p \left\{\alpha_ {1} + \alpha_ {2} - g _ {1} \right\} > e x p \left\{\alpha_ {1} + \alpha_ {2} - g _ {2} \right\}
+$$
+
+$$
+\alpha_ {1} + \alpha_ {2} - g _ {1} > \alpha_ {1} + \alpha_ {2} - g _ {2}
+$$
+
+$$
+g _ {1} <   g _ {2}
+$$
+
+We already have  $g_{1} < g_{2}$ , thus  $o_1' > o_1$ . Since  $o_1 + o_2 = o_1' + o_2' = 1$ , we have  $o_2' < o_2$ .
+
+Proposition A.2. For optimizing  $\mathcal{L}_2(F)$ , after one step of gradient descent w.r.t architecture parameters, the changes of operation weights  $o_1, o_2$  will increase the loss  $\mathcal{L}_1(F)$ .
+
+Proof. The MSE loss of  $\mathcal{L}_2(F)$  is calculated as
+
+$$
+\mathcal {L} _ {2} (F) = \sum_ {k} \left(F \left(\mathcal {G} _ {k}\right) - y _ {k}\right) ^ {2}. \tag {17}
+$$
+
+Then we have
+
+$$
+\begin{array}{l} \frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {i}} = 2 \sum_ {k} \left(F (\mathcal {G} _ {k}) - y _ {k}\right) \frac {\partial F (\mathcal {G} _ {k})}{\partial o _ {i}} \\ = 2 \sum_ {k} \left(o _ {1} f _ {1} (\mathcal {G} _ {k}) + o _ {2} f _ {2} (\mathcal {G} _ {k}) - y _ {k}\right) f _ {i} (\mathcal {G} _ {k}) \\ = 2 \sum_ {k} \left(o _ {1} f _ {1} \left(\mathcal {G} _ {k}\right) + \left(1 - o _ {1}\right) f _ {2} \left(\mathcal {G} _ {k}\right) - y _ {k}\right) f _ {i} \left(\mathcal {G} _ {k}\right) \tag {18} \\ = 2 \sum_ {k} \left(o _ {1} f _ {1} (\mathcal {G} _ {k}) + (1 - o _ {1}) f _ {2} (\mathcal {G} _ {k}) - f _ {2} (\mathcal {G} _ {k})\right) f _ {i} (\mathcal {G} _ {k}) \\ = 2 o _ {1} \sum_ {k} \left(f _ {1} (\mathcal {G} _ {k}) - f _ {2} (\mathcal {G} _ {k})\right) f _ {i} (\mathcal {G} _ {k}). \\ \end{array}
+$$
+
+The second equation expands  $F(x)$  by  $F(x) = o_1 f(x) + o_2 f(x)$ . The third equation utilizes relationship of  $o_1$  and  $o_2$ , since  $o_1 = \frac{\exp\{\alpha_1\}}{\exp\{\alpha_1\} + \exp\{\alpha_2\}}$ ,  $o_2 = \frac{\exp\{\alpha_2\}}{\exp\{\alpha_1\} + \exp\{\alpha_2\}}$  and thus  $o_1 + o_2 = 1$ . The fourth equation utilizes the supposed ground-truth labeling function for the 2-rd task, i.e.,  $y_k = f_2(\mathcal{G}_k)$ . Then we have
+
+$$
+\begin{array}{l} \frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {1}} - \frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {2}} = 2 o _ {1} \sum_ {k} \left(f _ {1} (\mathcal {G} _ {k}) - f _ {2} (\mathcal {G} _ {k})\right) \left(f _ {1} (\mathcal {G} _ {k}) - f _ {2} (\mathcal {G} _ {k})\right) \\ = 2 o _ {1} \sum_ {k} \left(f _ {1} \left(\mathcal {G} _ {k}\right) - f _ {2} \left(\mathcal {G} _ {k}\right)\right) ^ {2} \tag {19} \\ > 0. \\ \end{array}
+$$
+
+The inequality utilizes that  $o_1 = \frac{\exp\{\alpha_1\}}{exp\{\alpha_1\} + exp\{\alpha_2\}} > 0$ , and the assumption that  $\exists \mathcal{G}_k, f_1(\mathcal{G}_k) \neq f_2(\mathcal{G}_k)$  so that  $\sum_k \left( f_1(\mathcal{G}_k) - f_2(\mathcal{G}_k) \right)^2 > 0$ . When  $\frac{\partial \mathcal{L}_2(F)}{\partial o_1} - \frac{\partial \mathcal{L}_2(F)}{\partial o_2} = 0$ , then  $f_1(\mathcal{G}_k) = f_2(\mathcal{G}_k) \forall \mathcal{G}_k$  and  $\frac{\partial \mathcal{L}_2(F)}{\partial o_1} = \frac{\partial \mathcal{L}_2(F)}{\partial o_2} = 0$ , which is meaningless, since in this case the operations  $f_1(\cdot), f_2(\cdot)$  are not distinguishable with the data points  $\{ (\mathcal{G}_k, y_k) \}_{k=1}^N$  for optimization.
+
+Besides, since  $o_1 = \frac{exp\{\alpha_1\}}{exp\{\alpha_1\} + exp\{\alpha_2\}}$ , we have
+
+$$
+\frac {\partial o _ {1}}{\partial \alpha_ {1}} = \frac {\exp \left\{\alpha_ {1} \right\} \left(\exp \left\{\alpha_ {1} \right\} + \exp \left\{\alpha_ {2} \right\}\right) - \exp \left\{\alpha_ {1} \right\} \cdot \exp \left\{\alpha_ {1} \right\}}{\left(\exp \left\{\alpha_ {1} \right\} + \exp \left\{\alpha_ {2} \right\}\right) ^ {2}} = \frac {\exp \left\{\alpha_ {1} + \alpha_ {2} \right\}}{\left(\exp \left\{\alpha_ {1} \right\} + \exp \left\{\alpha_ {2} \right\}\right) ^ {2}} \tag {20}
+$$
+
+Similarly,
+
+$$
+\frac {\partial o _ {2}}{\partial \alpha_ {2}} = \frac {\exp \left\{\alpha_ {1} + \alpha_ {2} \right\}}{\left(\exp \left\{\alpha_ {1} \right\} + \exp \left\{\alpha_ {2} \right\}\right) ^ {2}}. \tag {21}
+$$
+
+Thus  $\frac{\partial o_1}{\partial\alpha_1} = \frac{\partial o_2}{\partial\alpha_2}$ . Then we have
+
+$$
+\frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {1}} > \frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {2}}
+$$
+
+$$
+\frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {1}} \frac {\partial o _ {1}}{\partial \alpha_ {1}} > \frac {\partial \mathcal {L} _ {2} (F)}{\partial o _ {2}} \frac {\partial o _ {2}}{\partial \alpha_ {2}} \tag {22}
+$$
+
+$$
+\nabla_ {\alpha_ {1}} \mathcal {L} _ {2} (F) > \nabla_ {\alpha_ {2}} \mathcal {L} _ {2} (F).
+$$
+
+By Lemma A.1,  $o_1$  decreases and  $o_2$  increases. Then for the first task, we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {1} (F) = \sum_ {k} \left(F (\mathcal {G} _ {k}) - y _ {k}\right) ^ {2} \\ = \sum_ {k} \left(o _ {1} f _ {1} (\mathcal {G} _ {k}) + o _ {2} f _ {2} (\mathcal {G} _ {k}) - y _ {k}\right) ^ {2} \\ = \sum_ {k} \left(o _ {1} f _ {1} \left(\mathcal {G} _ {k}\right) + o _ {2} f _ {2} \left(\mathcal {G} _ {k}\right) - f _ {1} \left(\mathcal {G} _ {k}\right)\right) ^ {2} \tag {23} \\ = \sum_ {k} \left(- o _ {2} f _ {1} \left(\mathcal {G} _ {k}\right) + o _ {2} f _ {2} \left(\mathcal {G} _ {k}\right)\right) ^ {2} \\ = o _ {2} ^ {2} \sum_ {k} \left(f _ {2} (\mathcal {G} _ {k}) - f _ {1} (\mathcal {G} _ {k})\right) ^ {2}. \\ \end{array}
+$$
+
+The third equation utilizes the supposed ground-truth labeling function for the 1-st task, i.e.,  $y_{k} = f_{1}(\mathcal{G}_{k})$ . Since  $\sum_{k}\left(f_{2}(\mathcal{G}_{k}) - f_{1}(\mathcal{G}_{k})\right)^{2} > 0$  and  $o_2$  increases, the loss of the first task  $\mathcal{L}_1(F)$  increases.
+
+# B. Implementation Details
+
+# B.1. Hyperparameters
+
+For fair comparisons, all methods adopt the same dimensionality  $d$  as 512, number of layers as 2. Adam optimizer (Kingma & Ba, 2014) is adopted to optimize the model weights with a learning rate 1e-3 and another SGD optimizer with a learning rate 1e-2 is adopted to optimize architecture parameters for NAS methods. For our method, we adopt  $K = 3$  for all datasets, and the hyperparameter  $\lambda \in \{0.01, 0.1, 1, 10, 100\}$ . We constrain the parameter size of the learned factors same to (Liu et al., 2023a)(≤ 1% features), which is a replay-based method adopted by all the baselines for continual training. We adopt the dimensionality as  $d_k = d / K$  for each module to keep the parameter size similar with the vanilla supernet for fair comparisons. We run the all experiments with 3 random seeds, and report the average performance and standard deviations.
+
+# B.2. Configurations
+
+All experiments are conducted with:
+
+- Operating System: Ubuntu 20.04.5 LTS  
+- CPU: Intel(R) Xeon(R) Gold 5218R CPU @ 2.10GHz  
+- GPU: NVIDIA GeForce RTX 4090 with 24 GB of memory  
+- Software: Python 3.8.18, Cuda 12.2, PyTorch (Paszke et al., 2019) 2.1.2, PyTorch Geometric (Fey & Lenssen, 2019) 2.4.0.

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2024/Disentangled Graph Self-supervised Learning for Out-of-Distribution Generalization/full.md

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+# Disentangled Graph Self-supervised Learning for Out-of-Distribution Generalization
+
+Haoyang Li $^{1}$  Xin Wang $^{1}$  Zeyang Zhang $^{1}$  Haibo Chen $^{1}$  Ziwei Zhang $^{1}$  Wenwu Zhu $^{1}$
+
+# Abstract
+
+Graph out-of-distribution (OOD) generalization, aiming to generalize graph neural networks (GNNs) under distribution shifts between training and testing environments, has attracted ever-increasing attention recently. However, existing literature heavily relies on sufficient task-dependent graph labels, which are often scarce or even unavailable, limiting their applications in real-world scenarios. In this paper, we study the self-supervised graph OOD generalization problem, i.e., learning GNNs capable of achieving relatively stable performances under distribution shifts without graph labels. However, the problem remains largely unexplored, with the critical challenge that the invariant and variant information are highly entangled in graphs. To solve this problem, we propose an OOD generalized disentangled graph contrastive learning model (OODGCL), which is capable of learning disentangled graph-level representations with self-supervision that can handle distribution shifts between training and testing graph data. Specifically, we first introduce a disentangled graph encoder to map each input graph into the factorized graph representation. Then we propose a tailored disentangled invariant self-supervised learning module to maximize predictive ability of the representations and make sure the representations other than from one specific channel are invariant to the environments partitioned by this latent factor for excluding the information corresponding to this latent factor for disentanglement. Finally, the disentangled graph representations are fed into a linear predictor and finetuned for the downstream tasks. We provide comprehensive theoretical analyses
+
+to show that our model can learn disentangled graph representations and achieve OOD generalization. Extensive experiments on real-world datasets demonstrate the superiority of our model against state-of-the-art baselines under distribution shifts for graph classification tasks.
+
+# 1. Introduction
+
+Graph structured data is ubiquitous in the real world, such as social networks (Qiu et al., 2018), traffic networks (Yu et al., 2017), financial networks (Yang et al., 2021), chemical molecules (Hu et al., 2020), etc. In the last decade, graph neural networks (GNNs) (Kipf & Welling, 2017; Velicković et al., 2018; Xu et al., 2019) have been a central topic in graph machine learning and made great progress in both academia and industry. Most of the existing literature is built upon the in-distribution (I.D.) hypothesis that assumes the testing and training graph data are from the identical distribution. However, in the real world, distribution shifts between testing and training graphs widely and inevitably exist, where the performance of existing GNNs drops significantly due to lacking out-of-distribution (OOD) generalization capacity (Li et al., 2022b).
+
+Despite the notable success of graph OOD generalization methods, the existing literature relies on sufficient task-dependent annotated labels to learn OOD generalized graph representations, which could be extremely scarce, or even unavailable in practice. For example, in the context of drug discovery, it necessitates high costs and substantial human labor in clinical tests to obtain labeled data for training OOD generalized graph models in a supervised manner, if we want to predict the invariant properties of molecules under distribution shifts (Paul et al., 2021). Since the existing approaches heavily rely on supervised labels, they will fail to learn invariant representations and model truly predictive relations with labels under distribution shifts for OOD generalization, when graph labels are scarce or not available, leading to severe performance degeneration.
+
+In this paper, we study the self-supervised graph OOD generalization problem, i.e., learning GNNs capable of achieving relatively stable performances under distribution shifts
+
+without graph labels. The key goal is to pretrain the graph encoder with much more available unlabeled graph data in a self-supervised manner to produce disentangled graph representations that can achieve promising OOD generalization when evaluated on testing graphs under distribution shifts. However, self-supervised graph OOD generalization is non-trivial due to the following challenges. First, it is challenging to get rid of spurious correlations between invariant and variant information in graph representations. The existing graph self-supervised methods generally adopt a holistic view meaning that the learned representations describe graphs as a perceptual whole (Li et al., 2021a), so that the invariant and variant information are highly entangled in the graph representations. Second, the existing methods tend to conduct pretraining tasks with spurious variant information (Li et al., 2022e), which fail to capture disentangled factors that reflect the semantic information which can be the labels in downstream tasks, leading to poor generalization performances.
+
+To tackle these challenges, we propose an OOD generalized disentangled graph contrastive model (OOD-GCL), which is capable of learning disentangled graph-level representations with self-supervision that can handle distribution shifts between training and testing graph data. Specifically, we first introduce a disentangled graph encoder to map each input graph into the corresponding factorized graph representation. Each channel is specifically designed to capture features from a specific disentangled latent factor, thereby accurately representing the information of that disentangled latent factor. Then we propose a tailored disentangled invariant self-supervised learning module. It can achieve sufficient predictive ability of the representations and make sure the representations other than from one specific channel are invariant to the environments partitioned by this latent factor for excluding the information corresponding to this latent factor. Thus the representations are encouraged to be disentangled so as to best characterize the aspect pertinent to a latent factor of the graph and achieve OOD generalized predictions when finetuning in downstream tasks. We further provide comprehensive theoretical analyses to show that our proposed model can learn disentangled graph representations and achieve OOD generalization with a strong guarantee. Extensive empirical evaluations are performed on several well-established graph benchmarks. The results demonstrate that the representations generated by our model lead to significant enhancements in generalization performance for downstream graph classification tasks under distribution shifts, outperforming the state-of-the-art baselines.
+
+The contributions of this paper are summarized as follows.
+
+- We focus on a novel self-supervised graph out-of-distribution (OOD) generalization problem and propose a tailored model to learn disentangled graph-level
+
+representations with self-supervision that can handle distribution shifts between training and testing graph data. To the best of our knowledge, we are the first to study self-supervised graph OOD generalization with theoretical guarantees.
+
+- We present a disentangled graph encoder with a tailored training strategy to capture the multiple aspects of the input graph. We further propose an invariance regularized contrastive learning module so that the spurious correlations between the latent factors can be eliminated in the graph representations and achieve representation disentanglement for OOD generalized predictions in downstream tasks.
+
+- We theoretically show that our model can provably learn disentangled graph representations, and make OOD generalization based on the disentangled graph representations.
+
+- Extensive empirical results demonstrate the effectiveness of our proposed OOD-GCL against a range of leading-edge baselines on various benchmark datasets under distribution shifts.
+
+The rest of the paper is organized as follows. We introduce some preliminaries in Section 2. Subsequently, in Section 3, we describe the technical details of our proposed OOD-GCL. The experimental settings and results are present in Section 4. We review some related works in Section 5. Finally, we conclude this work in Section 6.
+
+# 2. Preliminaries
+
+Let  $\mathbf{G} = \{G_n\}_{n=1}^N$  denote the input graph dataset, where  $G_{n}$  is the  $n$ -th graph. The multi-channel graph encoder maps the input graph into its disentangled representation  $\mathbf{Z}_n = \Phi(G_n)$ . Assuming that there are  $K$  latent factors behind the input graph,  $\mathbf{Z}_n = [\mathbf{Z}_n^1, \mathbf{Z}_n^2, \dots, \mathbf{Z}_n^K]$  consists of  $K$  disentangled components. Let  $\mathbf{Z}^k$  denote the representation corresponding to the  $k$ -th latent factor for all input graphs. Each  $\mathbf{Z}_n^k = \Phi^{(k)}(G_n) \in \mathbb{R}^{\Delta d}$ , where  $k \in \{1, \dots, K\}$ ,  $\Delta d = d / K$ , and  $d$  is the representation dimensionality. For simplification, we use  $\mathbf{Z}_n^{-k} \in \mathbb{R}^{(K-1) \times \Delta d}$  to denote the concatenated representation except for the representation  $\mathbf{Z}_n^k$  corresponding to the  $k$ -th latent factor, i.e.,  $\mathbf{Z}_n^{-k} = [\mathbf{Z}_n^1, \mathbf{Z}_n^2, \dots, \mathbf{Z}_n^{k-1}, \mathbf{Z}_n^{k+1}, \dots, \mathbf{Z}_n^K]$ .
+
+Following the invariant learning literature (Li et al., 2022d;b), we make the assumption:
+
+Assumption 1. There exists a portion of information inside input graph  $G$  such that it not only has sufficient abilities in predicting the graph labels, but has invariant relations with the labels under distribution shifts, i.e. satisfying: (a) invariance assumption:  $P^{e}(Y|\Psi (G)) = P^{e^{\prime}}(Y|\Psi (G))$
+
+for any environment (domain)  $e$ ,  $e'$  that graph comes from, and  $\Psi(\cdot)$  is to capture the invariant information; and (b) sufficiency assumption:  $Y = \omega(\Psi(G)) + \epsilon$ , where  $\omega(\cdot)$  is one function and  $\epsilon$  is random noise.
+
+Intuitively, if we accurately capture the latent factors that satisfy the invariance assumption and sufficiency assumption, and only make predictions based on these "invariant & sufficient latent factors", our predictions can be generalized across different environments under distribution shifts, and the representations corresponding to these latent factors have sufficient abilities in predicting the graph labels. Therefore, the problem can be solved by accurately identifying which latent factor belongs to the "invariant & sufficient latent factors" and making OOD generalized predictions only based on the disentangled representations corresponding to these latent factors.
+
+# 3. Method
+
+In this section, we present the proposed OOD-GCL in detail, whose framework is shown in Figure 1. We first present the disentangled graph encoder to identify the complex latent factors and output the factorized representations capturing the multiple aspects of graphs. Then, we propose the disentangled invariant self-supervised learning strategy on tailored contrastive tasks and invariance regularization. Next, we describe how to obtain the predictions based on the learned disentangled representations for specific downstream tasks. Lastly, we provide some theoretical analyses of our proposed method.
+
+# 3.1. Disentangled Graph Encoder
+
+In this subsection, we present the disentangled graph encoder for producing the factorized graph representation  $\mathbf{Z}_n = [\mathbf{Z}_n^1,\mathbf{Z}_n^2,\dots ,\mathbf{Z}_n^K ]$  for the input graph  $G_{n}$ . After that, we can infer which latent factor belongs to the "invariant & sufficient latent factors" and finally make OOD generalized predictions only based on the representations corresponding to these factors.
+
+Typically, graph neural networks (GNNs) leverage the topology of the graph and attributes of nodes to infer the representation vector  $\mathbf{h}_v$  for every node  $v$ . This is achieved through a message-passing framework, where the representation of a node is recursively refined by aggregating the representations of the adjacent nodes. The message-passing of the  $l^{\mathrm{th}}$  message-passing layer is formulated as (Li et al., 2021a):
+
+$$
+\mathbf {h} _ {v} ^ {l} = \operatorname {C O M} ^ {l} \left(\mathbf {h} _ {v} ^ {l - 1}, \operatorname {A G G} ^ {l} \left(\left\{\mathbf {h} _ {u} ^ {l - 1}: u \in \mathcal {N} (v) \right\}\right)\right), \tag {1}
+$$
+
+where  $\mathbf{h}_v^l$  is the representation of node  $v$  at the  $l^{\mathrm{th}}$  layer and  $\mathbf{h}_v^0$  is the input node features. COM and AGG denote the combination and aggregation function, respectively.
+
+$\mathcal{N}(v)$  denotes the adjacent nodes of node  $v$ . For simplicity, we denote GNN as the message-passing layer in Eq. (1). Define  $\mathbf{H}^l$  as the set of node embeddings  $\{\mathbf{h}_v^l | v \in V\}$  subsequent to the  $l^{\text{th}}$  GNN layer, with the set of nodes  $V$  in the graph. After the  $L$  conventional message-passing layers, our model further adopts a layer for graph disentanglement to output factorized graph representations by extracting features attributed to distinct latent factors through the individual channel. Following (Li et al., 2021a), the complex multiple latent factors are extracted by  $K$  different message-passing channels, thereby encapsulating various aspects of the input graph. Specifically, each channel independently employs a  $\mathrm{GNN}_k$ , using its unique parameters to propagate information:  $\mathbf{H}_k^{L+1} = \mathrm{GNN}_k(\mathbf{H}^L, A)$ , with  $A$  representing the adjacency matrix of the graph. The embeddings  $\mathbf{H}_k^{L+1}$  are exclusively associated with the  $k$ -th latent factor for disentanglement. Each channel also employs its own READOUT function, essentially a pooling function, to summarize the node embeddings into a graph-level representation:  $\mathbf{h}_{G_n,k} = \mathrm{READOUT}_k(\{\mathbf{H}_k^{L+1}\})$ . Finally, each channel produces a factor-specific graph representation through an individual MLP:  $\mathbf{Z}_n^k = \mathrm{MLP}_k(\mathbf{h}_{G_n,k})$ . Following (Li et al., 2021a), the disentangled graph encoder comprises  $K$  message-passing channels, which are more specialized in capturing different aspects of the graph, compared with the traditional graph encoders (Kipf & Welling, 2017; Velicković et al., 2018; Xu et al., 2019), that are fundamentally holistic in nature. This design facilitates the identification of intrinsic latent factors and enables the encapsulation of diverse aspects of graphs, which provides a basis for further identifying invariant and sufficient latent factors for OOD generalization.
+
+# 3.2. Disentangled Invariant Self-supervised Learning
+
+Different from traditional contrastive learning approaches, our proposed OOD-GCL introduces an innovative task centered around sufficiently discriminative instance identification of invariant factors. Our model not only draws similar instances closer together while pushing dissimilar ones apart in the representation space to learn sufficiently predictive representations, but it also facilitates the integration of factor-specific information into these representations. This integration enhances the disentanglement of these latent factors, thereby improving the identification of invariant and sufficient latent factors for out-of-distribution (OOD) generalization.
+
+In detail, the formation of real-world graphs is usually driven by multiple latent factors consisting of invariant and sufficient latent factors that can be OOD generalized as well as the other variant latent factors under distribution shifts. Based on the disentangled graph encoder, we can obtain the disentangled graph representations  $\mathbf{Z} \in \mathbb{R}^{N \times K \times \Delta d}$  for the whole graph dataset, where  $\mathbf{Z}_n^k \in \mathbb{R}^{\Delta d}$  be the  $k$ -th chan
+
+![](images/2a76adc9c9a1b186c91b261a4e390494a2023036689121dc978329079f58c008.jpg)  
+Figure 1. The framework of OOD-GCL. It consists of the following stages: (1) The input graph  $G_{n}$  is subjected to graph augmentations, resulting in an augmented view  $G_{n}^{\prime}$ . Both  $G_{n}$  and  $G_{n}^{\prime}$  are subsequently processed by a shared multi-channel disentangled graph encoder to produce the factorized graph representations. (2) The disentangled invariant self-supervision learning module is to maximize the predictive ability of the disentangled graph representations and make sure the disentangled graph representations other than that from the  $k$ -th channel are invariant under the environment partition inferred by the  $k$ -th latent factor, to exclude the information of the  $k$ -th latent factor for enhancing representation disentanglement. (3) At the downstream finetuning stage, our model can achieve theoretically guaranteed OOD generalization ability with a linear predictor based on the learned disentangled graph representations.
+
+nel's disentangled representation for  $G_{n}$ . We first infer the environment partition  $\mathcal{E}^k$  by clustering the representation corresponding to the  $k$ -th latent factor for all graphs  $\mathbf{Z}^k$  as follows:
+
+$$
+\mathcal {E} ^ {k} = \operatorname {C l u s t e r} \left(\mathbf {Z} ^ {k}\right). \tag {2}
+$$
+
+Here the motivation is that since the disentangled graph representation of each channel only captures the specific information in terms of each latent factor, the representations should naturally form multiple clusters (i.e., partitions) according to their intrinsic semantic information of the latent factor.
+
+Specifically, we adopt the k-means (MacQueen et al., 1967) clustering algorithm by setting the cluster number  $C$  in an adaptive manner. The Eq. (2) is implemented by:
+
+$$
+C ^ {*} = \operatorname {a r g m a x} _ {C} \operatorname {S i l} \left(\mathcal {E} ^ {k}\right), \tag {3}
+$$
+
+where  $\mathrm{Sil}(\cdot)$  is the Silhouette Score (Rousseeuw, 1987) that is a commonly used metric for evaluating the quality of clustering. It measures how similar each sample is to its own cluster compared to other clusters. The score ranges from  $-1$  to  $1$ , where a higher score indicates better-defined clusters. By calculating this Silhouette Score for different numbers of clusters, we can adaptively determine the optimal number of clusters by maximizing the score, thus
+
+ensuring the most appropriate cluster configuration. Finally, we obtain the environment partition with the optimal number of clustering, i.e.,
+
+$$
+\mathcal {E} ^ {k} = \mathrm {k} - \operatorname {m e a n s} \left(\mathbf {Z} ^ {k}; C ^ {*}\right). \tag {4}
+$$
+
+We expect the inferred environment partition can reflect the intrinsic clusters in semantic space of the latent factor that could be the label information in the downstream tasks (Wang et al., 2021a).
+
+After obtaining the inferred environment partition, we utilize the disentangled representation  $\mathbf{Z}_n^{-k}$  that is from all channels except for the  $k$ -th channel to calculate the contrastive loss which can be regarded as discriminating oneself from several negative samples. We define the loss function  $\ell_e^k$  in the  $e$ -th environment (i.e., cluster) inferred by the disentangled representation of the  $k$ -th latent factor as follows:
+
+$$
+\ell_ {e} ^ {k} = - \sum_ {n = 1} ^ {N} \mathbb {1} _ {[ \psi_ {n} = e ]} \log \frac {\exp \phi \left(\mathbf {Z} _ {n} ^ {- k} , \mathbf {Z} _ {n} ^ {\prime - k}\right)}{\sum_ {n ^ {\prime} \neq n} \mathbb {1} _ {[ \psi_ {n ^ {\prime}} = e ]} \exp \phi \left(\mathbf {Z} _ {n} ^ {- k} , \mathbf {Z} _ {n ^ {\prime}} ^ {\prime - k}\right)}, \tag {5}
+$$
+
+where  $e \in \{1, \ldots, |\mathcal{E}^k|\}$ ,  $|\mathcal{E}^k|$  denotes the number of clusters, and  $n' \in \{1, \ldots, N\}$ .  $\mathbb{1}_{[\psi_n = e]} \in \{0, 1\}$  is an indicator function evaluating to 1 iff.  $\psi_n = e$  and  $\psi_n$  is the environment of the  $n$ -th graph considering the current en
+
+vironment partition.  $\phi$  is the cosine similarity with temperature  $\tau$ , i.e.,  $\phi (\mathbf{a},\mathbf{b}) = \cos (\mathbf{a},\mathbf{b}) / \tau$  and  $\cos (\mathbf{a},\mathbf{b}) = \mathbf{a}^{\top}\mathbf{b} / (\| \mathbf{a}\|_{2}\| \mathbf{b}\|_{2})$ .  $\mathbf{Z}_n^{\prime -k}$  is the corresponding representation from the augmented view of the  $n$ -th graph except for the  $k$ -th channel. We adopt the common augmentation strategies following (You et al., 2020) to obtain  $\mathbf{Z}_n^{\prime -k}$ . Overall, the training objective is formulated as:
+
+$$
+\ell = \sum_ {k = 1} ^ {K} \ell^ {k}, \tag {6}
+$$
+
+where
+
+$$
+\ell^ {k} = \sum_ {e \in \mathcal {E} ^ {k}} \ell_ {e} ^ {k} + \lambda \operatorname {I n v} \left(\ell_ {e} ^ {k}; \mathcal {E} ^ {k}\right). \tag {7}
+$$
+
+$\mathrm{Inv}(\cdot)$  denotes the invariance regularization. In our method, we adopt V-REx (Krueger et al., 2021) as our invariance regularization and  $\lambda$  is the invariance coefficient. Note that our method is also compatible with the other invariance regularization, e.g., IRM (Arjovsky et al., 2019), IGA (Koyama & Yamaguchi, 2020), etc.
+
+Intuitively, the first contrastive loss term in Eq. (7) is to encourage sufficient predictive ability of the disentangled graph representations, while the second invariance regularization term in Eq. (7) is to make sure the disentangled graph representations other than that from the  $k$ -th channel are invariant under the environment partition inferred by the  $k$ -th latent factor, so as to exclude the information of the  $k$ -th latent factor for representation disentanglement.
+
+Finally, we iteratively update  $\mathcal{E}^k$  by Eq. (4) and optimize the disentangled graph encoder by Eq. (6) until convergence.
+
+# 3.3. Downstream Finetuning
+
+Based on the disentangled invariant self-supervised learning, the graph encoder can be optimized to make the representation disentangled in a self-supervised manner. After that, when deployed in the specific downstream tasks, the disentangled graph representations are further fed into a linear predictor  $\omega$  to calculate the graph labels. We finetune the disentangled graph representations and linear predictor to make predictions to the graph labels as:
+
+$$
+\hat {Y} _ {n} = \omega (\mathbf {Z} _ {n}). \tag {8}
+$$
+
+Note that although more powerful predictors (e.g., MLP, etc.) can also be utilized to map the disentangled graph representations into the labels, we show that such a simple linear predictor is good enough to achieve OOD generalization in the next subsection.
+
+# 3.4. Theoretical Analyses
+
+Here we provide some theoretical analyses from two aspects, which show: (1) our model can provably learn disentangled
+
+graph representations, and (2) our model can make OOD generalization based on the output disentangled graph representations.
+
+First, we analyze why our model can learn disentangled graph representations. Note that in this work, we follow (Wang et al., 2021a) to adopt the established definition of disentanglement (Higgins et al., 2018).
+
+Definition 1. (Disentangled Representation (Higgins et al., 2018; Wang et al., 2021a)) Let  $\mathcal{U}$  denote the semantic space. The semantic information potentially corresponds to the label of the downstream task. Let  $\mathcal{Z}$  denote the representation space. Let  $\mathcal{G}$  denote the group that acts on  $\mathcal{U}$ . Assume that there is a direct product decomposition where  $\mathcal{G}$  is decomposed into  $\mathcal{G}^1 \times \ldots \times \mathcal{G}^K$ , and  $\mathcal{U}$  is decomposed into  $\mathcal{U}^1 \times \ldots \times \mathcal{U}^K$ , with  $\mathcal{G}^k$  acting on  $\mathcal{U}^k$  respectively. The representation is disentangled if it satisfies the following equivariant property and decomposable property in the meantime.
+
+1. Equivariant property:  $\forall g\in \mathcal{G},\forall u\in \mathcal{U},f(g\cdot u) = g\cdot f(u)$ , where  $f:\mathcal{U}\to \mathcal{Z}$  is the representation function.  
+2. Decomposable property: there is a decomposition  $\mathcal{Z} = \mathcal{Z}^1 \times \ldots \times \mathcal{Z}^K$ , where each  $\mathcal{Z}^k$  is affected only by  $\mathcal{G}^k$  and unaffected by all  $\mathcal{G}^{k'}$ ,  $k \neq k'$ , and  $k, k' \in \{1, \ldots, K\}$ .
+
+Intuitively, the equivariant property means that the action of  $\mathcal{G}$  on semantic space  $\mathcal{U}$  is equivariant to the action on  $\mathcal{Z}$ . The decomposable property means that the representation corresponding to one latent factor is not affected by changing the semantic information of the other latent factors.
+
+Let  $g$  denote a group element in  $\mathcal{G}$  and  $g$  can be decomposed by  $(g^1, g^2, \ldots, g^K)$ , where  $g^k \in \mathcal{G}^k$ . The goal of our method is to learn the disentangled graph representations w.r.t.  $\prod_{k=1}^{K} \mathcal{G}^k$ . The group action  $g \in \mathcal{G}$  on the semantic space  $\mathcal{U}$  is equivariant to its action on the representation  $\mathcal{Z}$ . Furthermore, the representation space  $\mathcal{Z}$  is decomposed into  $K$  parts:  $\mathcal{Z}^1 \times \ldots \times \mathcal{Z}^K$ , where  $\mathcal{Z}^k$  is affected only by  $g^k \in \mathcal{G}^k$  and unaffected by all  $g^{k'} \in \mathcal{G}^{k'}(k' \neq k)$ . Inspired by the literature (Wang et al., 2021a), in the following theorem, we prove that optimizing the objective function of our method can lead to disentangled graph representations, formulated by this definition.
+
+Theorem 1. The graph representation is disentangled w.r.t.  $\prod_{k=1}^{K} \mathcal{G}^k$  if the objective function Eq. (7) reaches the minimum for any  $k$ -th latent factor,  $k \in \{1, \ldots, K\}$ .
+
+The proof is shown in Appendix due to the page limit.
+
+Next, we prove that the disentangled graph representations can lead to OOD generalized predictions under mild assumptions. Following the literature (Xu et al., 2022), we want to estimate the coefficient of the linear predictor to calculate the label given the disentangled graph representations.
+
+Theorem 2. Given the disentangled graph representations, (1) if the  $k$ -th latent factor does not belong to the invariant and sufficient latent factors to predict the label  $Y$ , then the coefficient of the linear predictor  $w^k$  equals zero; and (2) if the  $k$ -th latent factor belongs to the invariant and sufficient latent factors to predict the label  $Y$ , then the coefficient of the linear predictor  $w^k$  does not equal zero.
+
+The proof is also shown in Appendix. Intuitively, for achieving OOD generalization, we aim to identify the invariant and sufficient latent factors under distribution shifts, so that we can learn a graph model that is generalized to unknown testing distribution. As relationships between entangled graph representations usually change under distribution shifts, the model will fail to achieve OOD generalization if we use traditional entangled representations. Therefore, it is critical to find the invariant and sufficient latent factors and adopt their disentangled representations for making predictions, so that it can relieve the negative impact from the other latent factors that are not in the invariant and sufficient latent factors under distribution shifts.
+
+# 4. Experiments
+
+In this section, we conduct experiments on real-world datasets to show the effectiveness of the proposed OOD-GCL model for handling distribution shifts on graphs.
+
+# 4.1. Experimental Setup
+
+Datasets. We adopt real-world benchmark datasets for the graph classification task, including the datasets from graph OOD generalization benchmark GOOD (Gui et al., 2022) and the datasets from Open Graph benchmark (Hu et al., 2020). Following (Sui et al., 2023), the datasets in the experiment consist of:
+
+- Motif. Each graph in this dataset consists of a motif and a base graph, where the ground-truth label only depends on the motif. The distribution shift is induced by different spurious correlations between the label and the base graph, as well as different sizes.  
+- CMNIST. Each graph is converted from an image in MNIST (LeCun et al., 1998) which is to be classified into the corresponding handwritten digit. The distribution shift exists on node features by colorizing the digits differently.  
+- Molbbbp and Molhiv. The distribution shift exists on graph structural properties (scaffolds) or sizes. The provided split separates structurally different molecules with different scaffolds or sizes into different subsets.
+
+Note that all of the datasets consist of two data split strategies to create different distribution shifts except for CM-
+
+NIST, following the well-established settings (Gui et al., 2022; Sui et al., 2023).
+
+Baselines. We compare our OOD-GCL with several representative state-of-the-art methods, which can be divided into three groups:
+
+- General invariant learning methods, including standard ERM, IRM (Arjovsky et al., 2019), GroupDRO (Sagawa et al., 2019), V-REx (Krueger et al., 2021).  
+- Graph OOD generalization methods, including DIR (Wu et al., 2022b), CAL (Sui et al., 2022), GSAT (Miao et al., 2022), OOD-GNN (Li et al., 2022a), StableGNN (Fan et al., 2021), CIGA (Chen et al., 2022), DisC (Fan et al., 2022), DropEdge (Rong et al., 2019), GREA (Liu et al., 2022), FLAG (Kong et al., 2022), M-Mixup (Wang et al., 2021b),  $\mathcal{G}$ -Mixup (Han et al., 2022), AIA (Sui et al., 2023).  
+- Graph self-supervised learning methods, including InfoGraph (Sun et al., 2019), GraphCL (You et al., 2020), GMI (Peng et al., 2020), CNC (Zhang et al., 2022a).
+
+Implementation Details. The number of epochs for pretraining and finetuning is chosen from  $\{50, 100, 200\}$ . The Adam optimizer (Kingma & Ba, 2014) is adopted for gradient descent. The evaluation metric is accuracy for Motif and CMNIST datasets and ROC-AUC for Molbbp and Moltiv datasets. The dimensionality of the representations  $d$  is chosen from [128, 256, 512]. The invariance regularizer coefficient  $\lambda$  is chosen from  $\{10^{-4}, 10^{-2}, 10^{0}\}$ . The number of disentangled channels  $K$  is chosen from  $\{2, 3, 4, 5\}$ . We report mean results and standard deviations of ten runs.
+
+# 4.2. Main Results
+
+The comparisons of different methods on the real-world graph datasets under distribution shifts are shown in Table 1. We have the following observations.
+
+- Graph OOD generalization methods achieve better performance than general invariant learning methods, e.g., IRM, V-REx, etc. Such performance gains demonstrate the importance of considering and addressing the complex structural distribution shifts on graphs in the model design.  
+- Graph self-supervised learning methods, despite lacking tailored designs for handling graph distribution shifts, exhibit comparable performance to the graph OOD generalization methods in some comparisons. This suggests that the inherent capacity of graph self-supervised learning to capture truly predictive structural information in graph data contributes to its effectiveness in tackling distribution shifts.
+
+Table 1. Comparisons of different methods on the real-world graph datasets under distribution shifts. Numbers after the  $\pm$  signs represent the standard deviations. The best results are in bold. The results show that our method outperforms the baseline methods consistently, including general invariant learning methods, graph OOD generalization methods, and graph self-supervised learning methods.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">Motif</td><td>CMNIST</td><td colspan="2">Molbbbp</td><td colspan="2">Molhiv</td></tr><tr><td>base</td><td>size</td><td>color</td><td>scaffold</td><td>size</td><td>scaffold</td><td>size</td></tr><tr><td>ERM</td><td>68.66±4.25</td><td>51.74±2.88</td><td>28.60±1.87</td><td>68.10±1.68</td><td>78.29±3.76</td><td>69.58±2.51</td><td>59.94±2.37</td></tr><tr><td>IRM</td><td>70.65±4.17</td><td>51.41±3.78</td><td>27.83±2.13</td><td>67.22±1.15</td><td>77.56±2.48</td><td>67.97±1.84</td><td>59.00±2.92</td></tr><tr><td>GroupDRO</td><td>68.24±8.92</td><td>51.95±5.86</td><td>29.07±3.14</td><td>66.47±2.39</td><td>79.27±2.43</td><td>70.64±2.57</td><td>58.98±2.16</td></tr><tr><td>V-REx</td><td>71.47±6.69</td><td>52.67±5.54</td><td>28.48±2.87</td><td>68.74±1.03</td><td>78.76±2.37</td><td>70.77±2.84</td><td>58.53±2.88</td></tr><tr><td>DIR</td><td>62.07±8.75</td><td>52.27±4.56</td><td>33.20±6.17</td><td>66.86±2.25</td><td>76.40±4.43</td><td>68.07±2.29</td><td>58.08±2.31</td></tr><tr><td>CAL</td><td>65.63±4.29</td><td>51.18±5.60</td><td>27.99±3.24</td><td>68.06±2.60</td><td>79.50±4.81</td><td>67.37±3.61</td><td>57.95±2.24</td></tr><tr><td>GSAT</td><td>62.80±11.41</td><td>53.20±8.35</td><td>28.17±1.26</td><td>66.78±1.45</td><td>75.63±3.83</td><td>68.66±1.35</td><td>58.06±1.98</td></tr><tr><td>OOD-GNN</td><td>61.10±7.87</td><td>52.61±4.67</td><td>26.49±2.94</td><td>66.72±1.23</td><td>79.48±4.19</td><td>70.46±1.97</td><td>60.60±3.77</td></tr><tr><td>StableGNN</td><td>57.07±14.10</td><td>46.93±8.85</td><td>28.38±3.49</td><td>66.74±1.30</td><td>77.47±4.69</td><td>68.44±1.33</td><td>56.71±2.79</td></tr><tr><td>CIGA</td><td>66.43±11.31</td><td>49.14±8.34</td><td>32.22±2.67</td><td>64.92±2.09</td><td>65.98±3.31</td><td>69.40±2.39</td><td>59.55±2.56</td></tr><tr><td>DisC</td><td>51.08±3.08</td><td>50.39±1.15</td><td>24.99±1.78</td><td>67.12±2.11</td><td>56.59±10.09</td><td>68.07±1.75</td><td>58.76±0.91</td></tr><tr><td>DropEdge</td><td>45.08±4.46</td><td>45.63±4.61</td><td>22.65±2.90</td><td>66.49±1.55</td><td>78.32±3.44</td><td>70.78±1.38</td><td>58.53±1.26</td></tr><tr><td>GREA</td><td>56.74±9.23</td><td>54.13±10.02</td><td>29.02±3.26</td><td>69.72±1.66</td><td>77.34±3.52</td><td>67.79±2.56</td><td>60.71±2.20</td></tr><tr><td>FLAG</td><td>61.12±5.39</td><td>51.66±4.14</td><td>32.30±2.69</td><td>67.69±2.36</td><td>79.26±2.26</td><td>68.45±2.30</td><td>60.59±2.95</td></tr><tr><td>M-Mixup</td><td>70.08±3.82</td><td>51.48±4.91</td><td>26.47±3.45</td><td>68.75±0.34</td><td>78.92±2.43</td><td>68.88±2.63</td><td>59.03±3.11</td></tr><tr><td>G-Mixup</td><td>59.66±7.03</td><td>52.81±6.73</td><td>31.85±5.82</td><td>67.44±1.62</td><td>78.55±4.16</td><td>70.01±2.52</td><td>59.34±2.43</td></tr><tr><td>AIA</td><td>73.64±5.15</td><td>55.85±7.98</td><td>36.37±4.44</td><td>70.79±1.53</td><td>81.03±5.15</td><td>71.15±1.81</td><td>61.64±3.37</td></tr><tr><td>InfoGraph</td><td>69.73±2.74</td><td>50.17±3.72</td><td>33.84±1.52</td><td>70.39±1.34</td><td>80.82±0.49</td><td>67.51±3.65</td><td>60.16±2.12</td></tr><tr><td>GraphCL</td><td>71.34±3.76</td><td>52.77±5.89</td><td>32.81±1.71</td><td>69.36±1.32</td><td>80.64±0.78</td><td>70.12±2.19</td><td>57.19±3.18</td></tr><tr><td>GMI</td><td>65.26±1.98</td><td>51.85±3.49</td><td>30.24±5.98</td><td>69.38±1.02</td><td>77.67±0.30</td><td>68.95±2.92</td><td>59.86±2.78</td></tr><tr><td>CNC</td><td>70.34±2.73</td><td>50.64±4.86</td><td>32.41±1.28</td><td>68.16±1.25</td><td>76.19±3.52</td><td>69.13±3.71</td><td>58.67±3.41</td></tr><tr><td>OOD-GCL</td><td>76.31±2.37</td><td>60.12±2.35</td><td>41.89±2.03</td><td>72.86±1.97</td><td>83.12±3.21</td><td>74.13±2.01</td><td>63.41±2.59</td></tr></table>
+
+- Our method outperforms the baseline methods consistently, including general invariant learning methods, graph OOD generalization methods, and graph self-supervised learning methods. This superiority can be attributed to the ability to effectively identify underlying invariant and sufficient latent factors that are crucial for preserving truly predictive graph properties and learning disentangled representations. Unlike the baseline methods, our approach employs disentangled invariant self-supervised learning during the pretraining stage, leading that the graph representations can be disentangled and OOD generalized. This unique design enables our method to more effectively generalize to unknown distribution shifts, ultimately resulting in superior performance across a range of graph prediction tasks under distribution shifts. Overall, the results highlight the significance of learning disentangled representations in addressing challenges posed by graph distribution shifts.
+
+# 4.3. Time Complexity Analysis
+
+Now we analyze the time complexity of our method to show its efficiency. The time complexity of the proposed
+
+ODG-CCL is  $O(|E|d + |V|d^2)$ , where  $|V|$  represents the number of nodes,  $|E|$  denotes the number of edges, and  $d$  is the dimensionality of the representations. In particular, we employ the message-passing GNN to instantiate the GNN components in the disentangled graph encoder, which also exhibits a complexity of  $O(|E|d + |V|d^2)$ . As for the disentangled invariant self-supervised learning module, the time complexity is  $O(|\mathcal{B}|^2d)$ , where  $|\mathcal{B}|$  is the batch size. Since  $|\mathcal{B}|$  is small constant, the overall time complexity of OOD-GCL is  $O(|E|d + |V|d^2)$ . In comparison, the time complexity of the other GNN-based baselines is also  $O(|E|d + |V|d^2)$ . Therefore, the time complexity of our OOD-GCL is comparable with the existing baselines. Although our method exhibits effectiveness, it does not induce a higher time cost.
+
+# 4.4. Ablation Study
+
+In this subsection, we conduct ablation studies to verify the effectiveness of the key modules in our proposed OOD-GCL. We consider the following ablated variants:
+
+- Variant "w/o Inv." means that we set  $\lambda = 0$  in Eq. (7),
+
+![](images/da2222674b8f203160c4dc33d5752ec2b7616bbf8b9706d67b9113cfd910341b.jpg)  
+(a) Invariance coefficient  $\lambda$
+
+![](images/a907d64622898a694a76580ada1852dd9de9613be1b79190fa65ce5541f0aed3.jpg)  
+(b) Number of factors  $K$
+
+![](images/d235db1234d973a2c0fe2a7a8b75c7074b3d2fd70eab4f778ce4e0d5158d6844.jpg)  
+(c) Dimensionality  $d$
+
+![](images/db2f7f3a1be3d0b5c3a0f8c6562d8e556f1960c3127b69112caba5d884b88359.jpg)  
+(d) Number of layers  $L$
+
+![](images/139b1c90a6410137cdd99e018b5beec65ae2c3c4e4bd48d189c52eda56a99033.jpg)  
+Figure 2. Hyperparameter sensitivity analysis on CMNIST dataset. The blue lines denote the results of our method and the red dashed lines are the results of the best baseline.  
+Figure 3. Ablation studies. The ablated version "w/o Inv." removes the invariance regularization by setting  $\lambda = 0$  in Eq. (7), and "w/o Disen." removes the disentangled module and degenerates to the entangled version.
+
+namely removing the invariance regularization.
+
+- Variant "w/o Disen." means that we remove the disentanglement design and the model is degenerated to the entangled version.
+
+The results of OOD-GCL and its variants are shown in Figure 3. Notably, there is a noticeable performance drop for the variant "w/o Inv.", indicating the effectiveness of inferring and identifying the latent factor behind graphs for each channel accurately. The invariance regularization term ensures that the  $k$ -th part of disentangled graph representations only includes the information of the  $k$ -th latent factor. So it will affect the disentanglement if this invariance regularization is removed. In the case of "w/o Disen.", the absence of disentanglement leads to entangled latent factors within the graph representations. This entanglement poses challenges in characterizing distinct aspects of the graphs and hinders the ability to learn informative graph representations and achieve OOD generalization.
+
+# 4.5. Hyperparameter Sensitivity
+
+In this subsection, we analyze the sensitivity of some important hyperparameters. First, the hyperparameter  $\lambda$  in Eq. (7) has a moderate influence on the performance, since  $\lambda$  serves a role in encouraging the model to capture the invariance among different environments. To explore the sensitivity of our method on this hyperparameter, we adjust  $\lambda \in \{10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}\}$ , while maintaining the default value of the other hyperparameters unchanged. The results on the CMNIST dataset are shown in Figure 2, while the results on the other datasets show similar patterns. The results show that a small  $\lambda$  may struggle to capture disentangled latent factors to output representations accurately, leading to challenges in achieving OOD generalized predictions. Conversely, a large  $\lambda$  can lead that the learned representations are not informative enough to capture the graph properties. Besides, we also conduct sensitivity analysis on the other hyper-parameters, including the number of factors  $K \in \{2, 3, 4, 5\}$ , the dimensionality  $d \in \{128, 256, 512, 1024\}$ , and the number of GNN layers  $L \in \{2, 3, 4, 5, 6\}$ . The results show that our method outperforms the best baseline in a wide range of hyperparameters, showing our method is not sensitive to hyperparameter choices.
+
+# 5. Related Work
+
+OOD Generalization on Graphs. A basic assumption for most machine learning methods is that there exists an identical distribution between training and testing data. However, this assumption is frequently violated in real-world scenarios due to inevitable distribution shifts, thereby introducing substantial challenges to the model's generalization performance, particularly in out-of-distribution (OOD) scenarios (Shen et al., 2021). The model performance decline becomes noticeable when it lacks enough OOD generalization capability. To tackle this problem, several general invariant learning methods have been developed (Arjovsky et al., 2019; Krueger et al., 2021; Koyama & Yamaguchi, 2020; Chang et al., 2020; Creager et al., 2021; Liu et al.,
+
+2021a). Recently, OOD generalization on graphs (Li et al., 2022b) has also attracted much attention, considering the complex distribution shifts existing on the graph structure and node feature. Several famous graph OOD generalization work (Wu et al., 2022c;a; Li et al., 2022d; Fan et al., 2022; Sui et al., 2022; Liu et al., 2022; Sui et al., 2023; Zhang et al., 2022b; 2023b;c;d; Li et al., 2023; Mao et al., 2024) have been proposed, in general focusing on learning invariant graph patterns among different environments for tackling the distribution shifts. Most of the approaches typically adopt a supervised manner to get rid of the spurious correlations between the variant patterns and the labels and facilitate the invariant patterns for predictions. However, they heavily rely on sufficient task-dependent annotated labels to learn OOD generalized graph representations, which could be unavailable in practice. To this end, in this paper, we focus on the self-supervised graph OOD generalization problem, namely learning GNNs capable of achieving relatively stable performances under distribution shifts without graph labels. This problem is not explored in the literature.
+
+Contrastive Learning on Graphs. Self-supervised learning is one machine learning technique to learn informative representations by training the model using the data itself rather than the external annotated labels (Liu et al., 2021b). Recently, self-supervised learning has been revolutionized by the development of contrastive learning, which primarily focuses on instance discrimination as a pretext task (Jaiswal et al., 2020). The methodologies on this technique are widely developed (Oord et al., 2018; Wu et al., 2018; He et al., 2020; Chen et al., 2020; Tsai et al., 2021). In the meantime, the application of contrastive learning on graph data has also been explored in various research works (Sun et al., 2019; Qiu et al., 2020; Hassani & Khasahmadi, 2020; You et al., 2020; Zhang et al., 2021; Li et al., 2021a; Ren & Liu, 2020; Peng et al., 2020; Li et al., 2022e;c). The fundamental principle of these approaches is to enhance the similarity or agreement between different views of the input graph. Nevertheless, current methods in graph contrastive learning primarily focus on general settings and ignore the diverse latent factors within the complex graph data. Consequently, such approaches may not adequately preserve intrinsic and truly predictive graph characteristics, leading to the potential fragility to distribution shifts.
+
+Disentangled Representation Learning. Disentangled representation learning is expected to capture the latent factors behind the observed data, where these latent factors are represented as factorized vectors (Bengio et al., 2013; Higgins et al., 2018; Wang et al., 2022a; Pfau et al., 2020; Winter et al., 2022; Shu et al., 2018). It has found widespread applications in various domains, including computer vision (Hsieh et al., 2018; Wang et al., 2021a; 2023b; Chen et al., 2023), recommendation systems (Wang et al.,
+
+2022b; 2023a; Zhang et al., 2023a; Wang et al., 2024), etc. Noteworthy advancements in this field also involve the application of disentangled representation learning for graph data through the development of disentangled graph neural networks (Ma et al., 2019; Li et al., 2021b; 2022c; Yang et al., 2020; Zhang et al., 2023e; 2024b;a). For instance, (Ma et al., 2019) introduce a neighborhood routing mechanism within graph convolution to identify latent factors influencing node-to-neighbor connections. (Liu et al., 2020) strive to enhance the independence of graph latent factors by eliminating their mutual dependence, leveraging a kernel-based metric. Some works (Li et al., 2021b; 2022c; Wang et al., 2021a) extend the disentangled representation learning in a self-supervised manner, but they fail to tackle the distribution shifts for OOD generalization with a theoretical guarantee. We draw inspirations from the concept of disentangled representation learning (Higgins et al., 2018; Wang et al., 2021a; 2022a) to address the graph out-of-distribution generalization problems and bridge the disentanglement and OOD generalization on graphs from a theoretical view.
+
+# 6. Conclusion
+
+In this paper, we study the self-supervised graph OOD generalization problem. The key challenge lies in disentangling invariant and variant latent factors behind the graphs without graph labels for tackling distribution shifts. To address the challenge, we propose a novel OOD generalized disentangled graph contrastive learning model (OOD-GCL), which is able to learn disentangled graph representations with self-supervision that can generalize under distribution shifts. We conduct comprehensive theoretical analyses to prove that our method can learn disentangled graph representations and achieve OOD generalization. The extensive experiments on several real-world benchmarks demonstrate the superiority of the proposed method over state-of-the-art baselines.
+
+# Acknowledgements
+
+This work is supported by the National Key Research and Development Program of China No.2023YFF1205001, National Natural Science Foundation of China (No. 62222209, 62250008, 62102222, 62206149), Beijing National Research Center for Information Science and Technology under Grant No. BNR2023RC01003, BNR2023TD03006, and Beijing Key Lab of Networked Multimedia.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Proof
+
+# A.1. Proof of Theorem 1
+
+Proof. We prove that our learned graph representation is disentangled by showing it satisfies the equivariant property and decomposable property in Definition 1. The reasoning process is inspired by Lemma 1 in (Wang et al., 2021a), but we extend it into a more general scenario that the representation is disentangled w.r.t.  $\prod_{k=1}^{K} \mathcal{G}^k$  instead of disentangled w.r.t. some  $\mathcal{G}^k$  and the others in (Wang et al., 2021a), thanks to our tailored model design.
+
+First, we prove that the graph representation learned from our method satisfies the equivariant property. Consider that the objective function Eq. (7) reaches the minimum for any  $k$ -th latent factor,  $k \in \{1, \dots, K\}$ . Assume that there are representations  $\mathbf{Z}_n$  and  $\mathbf{Z}_{n'}$  of two different graphs satisfying  $\mathbf{Z}_n = \mathbf{Z}_{n'}$ . The similarity between the two graphs can be further reduced by outputting different representations for the  $n$ -th graph and the  $n'$ -th graph for reaching a lower loss in Eq. (7). It contradicts that the objective function reaches the minimum. Therefore, the sample-equivariant property (Wang et al., 2021a) can be achieved, i.e., different graphs have different representations. We assume that the transition from the  $n$ -th graph to the  $n'$ -th graph in semantic space is from the group action and denote this action as  $g \in \mathcal{G}$  that is transitive on representation space, i.e.  $g \cdot \mathcal{Z}_n = \mathcal{Z}_{n'}$ . Therefore, the sample-equivariant graph representations can be an approximation of group-equivariant graph representations (Wang et al., 2021a).
+
+Then, we prove that the graph representation learned from our method satisfies the decomposability property. Consider that the objective function Eq. (7) reaches the minimum for any  $k$ -th latent factor,  $k \in \{1, \dots, K\}$ . Assume that  $\exists \mathbf{Z}^k$  is affected by not only the action  $g^k$  but also one different action  $g^{k'}$  and consider obtaining the environment partition by clustering the representation  $\mathbf{Z}^{k'}$ . The similarity between two graphs measured by the part  $\mathbf{Z}^k$  can be further reduced by excluding the action  $g^{k'}$  for reaching a lower loss in Eq. (7). Furthermore, since  $\mathbf{Z}^k$  is also affected by  $g^{k'}$  in addition to  $g^k$ , the invariant regularization term in Eq. (7) can also be further reduced by excluding the information of the  $k'$ -th latent factor. It contradicts that the objective function reaches the minimum. Finally, we conclude that each  $\mathbf{Z}^k$  is affected only by  $g^k$  corresponding to the  $k$ -th latent factor and unaffected by the actions corresponding to the other latent factors.
+
+Overall, the learned graph representation of our method can be disentangled.
+
+# A.2. Proof of Theorem 2
+
+Proof. The OOD generalization means that the model can make predictions only based on the invariant and sufficient disentangled graph latent factors. Therefore, given the disentangled graph representations  $\mathbf{Z} = [\mathbf{Z}^1,\dots ,\mathbf{Z}^K]$ , the coefficient  $\mathbf{w} = [w^{1},\ldots ,w^{K}]$  of the linear predictor for predicting label  $Y$  can be regarded to identify the invariant and sufficient latent factors and the predictions can be OOD generalized with the corresponding disentangled graph representations. Following the reasoning line of Theorem 5.1 and Theorem 5.2 in (Xu et al., 2022), we assume that the ordinary least square method is adopted, the mean of each disentangled representation is zero, and the independence among the disentangled representations. We derive the solutions of the coefficient  $\mathbf{w} = [w^{1},\dots ,w^{K}]$ .
+
+Specifically, we have
+
+$$
+w ^ {k} = \frac {\operatorname {C o v} \left(\mathbf {Z} ^ {k} , Y\right)}{\operatorname {V a r} \left(\mathbf {Z} ^ {k}\right)} = \frac {\mathbb {E} \left[ \mathbf {Z} ^ {k} Y \right]}{\operatorname {V a r} \left(\mathbf {Z} ^ {k}\right)} = \frac {\mathbb {E} \left[ \mathbf {Z} ^ {k} \mathbb {E} \left[ Y \mid \mathbf {Z} \right] \right]}{\operatorname {V a r} \left(\mathbf {Z} ^ {k}\right)}, \tag {9}
+$$
+
+where Cov and Var denote the covariance and variance respectively, and  $\mathbb{E}$  denotes the expectation.
+
+On the one hand, if  $k$ -th latent factor does not belong to the invariant and sufficient latent factors to predict the label  $Y$ , then  $\mathbb{E}[Y|\mathbf{Z}]$  can be represented as the function  $\Omega(\mathbf{Z}^{-k})$ , where  $\mathbf{Z}^{-k} = [\mathbf{Z}^{1}, \ldots, \mathbf{Z}^{k-1}, \mathbf{Z}^{k+1}, \ldots, \mathbf{Z}^{K}]$ . Therefore,
+
+$$
+w ^ {k} = \frac {\mathbb {E} \left[ \mathbf {Z} ^ {k} \Omega (\mathbf {Z} ^ {- k}) \right]}{\operatorname {V a r} (\mathbf {Z} ^ {k})} = 0. \tag {10}
+$$
+
+The coefficient  $w^{k}$  is zero, meaning that the prediction does not rely on the representation corresponding to the  $k$ -th latent factor. The spurious correlation between this latent factor and the label can be removed, leading to OOD generalized predictions.
+
+On the other hand, if  $k$ -th latent factor belongs to the invariant and sufficient latent factors to predict the label  $Y$ , then
+
+$\mathbb{E}\left[Y|\mathbf{Z}\right]$  can be represented as the function  $\Theta (\mathbf{Z}^k)$ . Therefore,
+
+$$
+w ^ {k} = \frac {\mathbb {E} \left[ \mathbf {Z} ^ {k} \Theta (\mathbf {Z} ^ {k}) \right]}{\operatorname {V a r} \left(\mathbf {Z} ^ {k}\right)} \neq 0. \tag {11}
+$$
+
+The coefficient  $w^{k}$  is not zero, meaning that the prediction relies on the representation corresponding to the  $k$ -th latent factor. The invariant and sufficient correlation between this latent factor and the label can be identified accurately, leading to OOD generalized predictions.
+
+![](images/3371ee6e9fa484e905c08c9f955de935e731d9daab06756158e5f9a1154c83bd.jpg)

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2024/Disguised Copyright Infringement of Latent Diffusion Models/full.md

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+# Disguised Copyright Infringement of Latent Diffusion Models
+
+Yiwei Lu  $^{1,2}$  Matthew Y.R. Yang  $^{1}$  Zuoqiu Liu  $^{1}$  Gautam Kamath  $^{1,2}$  Yaoliang Yu  $^{1,2}$
+
+# Abstract
+
+Copyright infringement may occur when a generative model produces samples substantially similar to some copyrighted data that it had access to during the training phase. The notion of access usually refers to including copyrighted samples directly in the training dataset, which one may inspect to identify an infringement. We argue that such visual auditing largely overlooks a concealed copyright infringement, where one constructs a disguise that looks drastically different from the copyrighted sample yet still induces the effect of training Latent Diffusion Models on it. Such disguises only require indirect access to the copyrighted material and cannot be visually distinguished, thus easily circumventing the current auditing tools. In this paper, we provide a better understanding of such disguised copyright infringement by uncovering the disguises generation algorithm, the revelation of the disguises, and importantly, how to detect them to augment the existing toolbox. Additionally, we introduce a broader notion of acknowledgment for comprehending such indirect access. Our code is available at https://github.com/watml/disguised_copyright_infringement.
+
+# 1. Introduction
+
+Generative models, especially the recent advanced Latent Diffusion Models (LDM) (Rombach et al. 2022), have shown tremendous ability to generate new images, even of creative or artistic form according to text prompts. Such models are trained on a large corpus of data, which may consist of copyrighted material. Additionally, prior works have established that such generative models are prone to regurgitating content from their training data (Ippolito et al. 2023; Zhang et al. 2021; Carlini et al. 2022; Vyas et al.
+
+*Equal contribution  ${}^{1}$  School of Computer Science,University of Waterloo,Waterloo,Canada  ${}^{2}$  Vector Institute. Correspondence to: Yiwei Lu <yiwei.lu@uwaterloo.ca>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+2023; Somepalli et al. 2023a; Somepalli et al. 2023b), which may also be copyrighted.
+
+In this paper, we will focus on copyright law within the jurisdiction of the United States. To establish a copyright violation, two factors must be present. First, the accused must have had access to the copyrighted material. Second, the accused must produce content that bears "substantial similarity" to the copyrighted material (reproducing). Note that the definition of substantial similarity can be ambiguous. Within the context of images, its definition appears to be relatively broad (Steinberg v. Columbia Pictures Industries, Inc. 1987), and in particular encompasses near-exact copies.
+
+Turning our attention to the former "access" criterion: the natural way to establish that a model had access to a particular piece of copyrighted material is to inspect its training data. For example, in the case of Andersen v. Stability AI Ltd. (Dist. Court 2023), the case was allowed to proceed based on the fact that copyrighted images were found in LAION-5B (Schuhmann et al. 2022) (the training data used for Stable Diffusion) using haveibeentrained.com.
+
+We challenge the perspective that such visual auditing is sufficient to establish access to copyrighted material. Our results show that it is possible to conceal copyrighted images within the training dataset for LDMs. Specifically, LDMs are equipped with a fixed encoder for dimension reduction such that the diffusion learning process occurs in the latent space. This structure can be maliciously exploited to generate disguised copyrighted samples: given a copyrighted image, we show how to generate a disguise such that it is visually different from the copyrighted sample but shares similar latent information. The closeness of the two samples in the latent space can be quantitatively measured by a distance function, or qualitatively revealed by a concept extraction tool called textual inversion (Gal et al. 2022), both of which we will demonstrate in our empirical study.
+
+Our study reveals the possibility of creating a new training dataset that does not appear to directly or blatantly contain any copyrighted data. Nonetheless, if a model is trained on this derivative training dataset, it would behave similarly as if the copyrighted data were present. Such disguises may still exhibit copyright infringement, although only accessing proprietary data indirectly. In Figure 1, we display a comparison between the previous copyright infringement
+
+![](images/7cdc648c42a9509190d70a87391f96cba4e2e811eb76c0963213429a53e9ec1c.jpg)  
+Figure 1: An overview of conventional (with direct access to copyrighted material) and disguised (with indirect access) copyright infringement for latent diffusion models. For direct access, training an LDM-based model on copyrighted material  $x_{c}$  and reproducing  $x_{c}$  is subjected to copyright infringement. For indirect access, one trains the same model on disguised samples  $x_{d}$ , which are drastically different from  $x_{c}$ , but is still able to reproduce  $x_{c}$  during inference.
+
+phenomenon (direct access) with the disguised copyright infringement (indirect access). Clearly, there was still access to the copyrighted material in the latter training pipeline, which raises the following question:
+
+What constitutes access? How to quantify it?
+
+We answer the first question by introducing a notion of acknowledgment, which refers to a criterion that any sample that contains similar latent information as that of a copyrighted sample should be considered acknowledging it, despite possible visual dissimilarity. To quantify acknowledgment in practice, a deeper inspection than visually auditing the training set is required. Thus we further propose a two-step detection method: (1) a feature similarity search for screening suspects; (2) an encoder-decoder examination to confirm disguises, which augments the existing criterion. In summary, we make the following contributions:
+
+- We challenge the current "access" criterion and point out its insufficiency in more delicate cases of copyright infringement;  
+- We propose an algorithm that demonstrably crafts disguised copyrighted data to conceal the content (or concepts) of copyrighted images in the training set;  
+- We show disguised data contain copyrighted information in the latent space, such that by finetuning them on textual inversion or DreamBooth, or training on LDM, the model reproduces copyrighted data during inference;
+
+- We propose methods to detect such disguises, which further encourage the expansion and quantification of "access" in the context of copyright infringement.
+
+# 2. Background
+
+# 2.1. Diffusion Models
+
+We focus on latent diffusion models (Rombach et al. 2022) as they are ideal for text-to-image generation. We first recall the objective of regular diffusion models (Sohl-Dickstein et al. 2015):
+
+$$
+\mathcal {L} _ {\mathrm {D M}} = \mathbb {E} _ {x, \varepsilon , t} \left[ \| \varepsilon - \varepsilon_ {\theta} (x _ {t}, t) \| _ {2} ^ {2} \right],
+$$
+
+where  $\varepsilon \sim \mathcal{N}(0,\mathbb{I})$ ,  $t$  is the timestep uniformly sampled from  $\{1,\dots,T\}$ , and  $x_{t}$  is a noisy version of the input sample  $x$  at timestep  $t$ . Briefly, diffusion models are probabilistic models designed to learn a data distribution  $p(x)$  by gradually denoising a normally distributed variable, which corresponds to learning the reverse process of a fixed Markov chain of length  $T$ .
+
+Note that the intermediate  $x_{t}$  are all in pixel space, which makes the training and inference of diffusion models expensive. To address this problem, Rombach et al. (2022) propose to perform the diffusion process in the latent space, namely latent diffusion models (LDM). Specifically, LDMs utilize a pre-trained (and fixed) autoencoder architecture which consists of an encoder  $\mathcal{E}$  and a decoder  $\mathcal{D}$ , where  $\mathcal{E}$  is only used during training and  $\mathcal{D}$  is only used during inference. The objective of LDM can be expressed as:
+
+$$
+\mathcal {L} _ {\mathrm {L D M}} = \mathbb {E} _ {x, \varepsilon , t} \left[ \| \varepsilon - \varepsilon_ {\theta} (\mathcal {E} (x _ {t}), t) \| _ {2} ^ {2} \right].
+$$
+
+When we perform text-to-image tasks, a text condition (or prompt)  $y$  needs to be considered. Such a text condition is compressed into the latent space with a pre-trained text-embedding model  $c_{\theta}(\cdot)$  (typically a BERT (Devlin et al. 2018) text encoder) and fed into LDM training as a conditioning vector:
+
+$$
+\mathcal {L} _ {\mathrm {L D M - C}} = \mathbb {E} _ {x, y, \varepsilon , t} \left[ \| \varepsilon - \varepsilon_ {\theta} (\mathcal {E} (x _ {t}), t, c _ {\theta} (y)) \| _ {2} ^ {2} \right]. \tag {1}
+$$
+
+Intuitively, LDMs perform the diffusion training process on a compressed version of the training data, while the compression procedure is deterministic. As a result, it is possible that two training samples share similar latent representations, but differ significantly visually. Based on this intuition, we propose a method to generate disguised data, demonstrating that the use of pre-trained autoencoders may conceal the inclusion of copyrighted data in the training set.
+
+# 2.2. Data Poisoning attacks
+
+Data poisoning attacks refer to the threat of contaminating part of the training data such that a machine learning model trained on the mixture of clean and poisoned data is influenced toward certain behaviors. Existing data poisoning attacks mainly focus on poisoning supervised models (classifiers). Based on the objective, there exist (1) indiscriminate attacks (e.g., Biggio et al. 2012; Koh and Liang 2017; Koh et al. 2022; Muñoz-González et al. 2017; Lu et al. 2022; Lu et al. 2023; Lu et al. 2024), which seek to decrease the overall test accuracy; (2) targeted attacks (e.g., Shafahi et al. 2018; Aghakhani et al. 2021; Guo and Liu 2020; Zhu et al. 2019; Geiping et al. 2021) that only alter the prediction on specific test examples; (3) backdoor attacks (e.g., Gu et al. 2017; Tran et al. 2018; Chen et al. 2017; Saha et al. 2020) that trigger if a particular pattern appears in an example; (4) unlearnable examples (e.g., Liu and Chawla 2010; Huang et al. 2021; Yu et al. 2022; Fowl et al. 2021b; Fowl et al. 2021a; Sandoval-Segura et al. 2022; Fu et al. 2021) that aim to reduce the utility of the data towards training a model. Moreover, the recent advanced Nightshade (Shan et al. 2024) shows some success in poisoning LDMs on prompt-specific text-to-image tasks; Wan et al. (2023) and Jiang et al. (2023) also explore data poisoning attacks against large language models.
+
+Copyright infringement can be regarded as a special case of data poisoning, where one includes some "poisoned data" as a subset of a training set  $\chi$ , such that after training a LDM on  $\chi$ , the model reproduces a copyrighted sample  $x_{c}$ . For direct access, "poisoned data" simply refers to direct copies of  $x_{c}$ . In this paper, we also examine indirect access, where one can construct the "poisoned data" so they are visually different from  $x_{c}$  using an adaptation of the data poisoning attack in Shafahi et al. (2018).
+
+# 2.3. Textual Inversion
+
+Art generation and related copyrighted material reproduction frequently happen in LDM-based generative tools. For example, textual inversion (Gal et al. 2022) is a popular tool for modifying personalized images with language-guided LDMs. Textual inversion uses a small set of images (typically three to five), which depicts the target concept  $S_{*}$  (which is a placeholder in a text prompt, e.g., "A photo of  $S_{*}$ ) across multiple settings such as varied backgrounds or poses. The concept  $S_{*}$  is tokenized, converted to an embedding  $\nu_{*}$ , and then encoded as part of the text embedding  $c_{\theta}(y)$ . The embedding  $\nu_{*}$  is acquired by optimizing the LDM loss in Equation 1 over the training images:
+
+$$
+\nu_ {*} = \underset {\nu} {\operatorname {a r g m i n}} \mathcal {L} _ {\mathrm {L D M - C}} \tag {2}
+$$
+
+Optimization is performed using the same training scheme as the original LDM while keeping  $\varepsilon_{\theta}$ ,  $c_{\theta}$  and  $\mathcal{E}$  fixed. After obtaining  $\nu_{*}$ , it can be combined with various text prompts (typically derived from the CLIP ImageNet templates (Radford et al. 2021)) for generation. Overall, textual inversion is a great tool for capturing particular concepts and may also be used to reproduce copyrighted content.
+
+# 3. Generating Disguises
+
+We describe how to conceal copyrighted images within the training dataset. Specifically, we demonstrate that one can construct a disguised sample  $x_{d}$  that is visually distinct from a copyrighted sample  $x_{c}$ , but contains essentially similar information in the latent space (measured via the distance between feature representations). Specifically, we first recall the feature matching attack (Shafahi et al. 2018) designed for targeted data poisoning attacks, and then discuss how to adapt the attack to generate disguises.
+
+# 3.1. Feature Matching Attack
+
+We recall that targeted attacks aim to change the prediction (e.g., causing misclassification) of a model (typically a classifier) on a targeted test sample  $x_{t}$  (with label  $y_{t}$ ) by training on a mixture of the existing clean data and poisoned data. Specifically, Shafahi et al. (2018) propose a feature matching attack, where a poisoned sample  $x_{p}$  is acquired by making imperceptible changes to a base sample  $x_{b}$  with label  $y_{b}$  (different than  $y_{t}$ ) such that the feature representation of the poisoned sample  $f(x_{p})$  matches that of the target sample  $f(x_{t})$ , where the model  $f$  is a (usually fixed) pretrained feature extractor. By training the model (e.g., with a softmax layer on top of  $f$ ) on  $x_{p}$ , it is likely to misclassify the target sample  $x_{t}$  as the wrong (base) label  $y_{b}$ .
+
+Note that the generality of the feature matching attack is restricted in the context of classification tasks because it is only applicable when the feature extractor  $f$  is fixed, which
+
+![](images/4b6803aa58051d36ed091a190fa86ade85d519a39fe58be5d313950e15c3f242.jpg)  
+Figure 2: An illustration on the algorithm to generate disguises. We aim to optimize the loss  $\mathcal{L}$  consisting of an input space constraint that measures the distance between the base image  $x_{b}$  and the disguise  $x_{d}$  in the input space, and a feature space constraint that measures the distance between the copyrighted  $x_{c}$  and  $x_{d}$  in the feature space.
+
+may only be applicable for fine-tuning or transfer learning. However, we will show how it can be used to craft disguised copyrighted images in the next paragraph.
+
+Adapting to copyright infringement: Training LDMs on a dataset  $\chi$  amounts to performing a regular diffusion learning process on a latent dataset  $\zeta$ , where  $\zeta = \mathcal{E}(\chi)$ . Suppose there exists a set of  $N$  copyrighted data  $\{x_c^i\}_{i=1}^N \subseteq \chi$ , which we aim to substitute with the same amount of disguised data  $\{x_d^i\}_{i=1}^N$  such that the latent dataset  $\zeta$  stays intact. Compared with data poisoning, the target sample  $x_t$  is set to  $x_c$ , the feature extractor  $f$  is set to  $\mathcal{E}$ , the base sample can be any uncopyrighted image different from  $x_c$ , the sample we aim to optimize is  $x_d$ , and the feature matching algorithm can be adapted. Conveniently, the encoder model  $\mathcal{E}$  is pre-trained and its weights are fixed during the training of diffusion, thus making the above attack realistic.
+
+# 3.2. Disguised Copyrighted Image Generation
+
+Now we introduce our disguise generation algorithm for individual images. Formally, given a target copyrighted image  $x_{c}$ , a base image  $x_{b}$  which is chosen to be visually different from  $x_{c}$ , distance measures  $D_{1}(\cdot)$  and  $D_{2}(\cdot)$  for input space and latent space respectively, and a fixed pretrained encoder  $\mathcal{E}$ , we aim to construct a disguised image  $x_{d}$  such that it satisfies:
+
+$$
+D _ {1} = D _ {1} \left(x _ {b}, x _ {d}\right) \leq \gamma_ {1}, D _ {2} = D _ {2} \left(\mathcal {E} \left(x _ {c}\right), \mathcal {E} \left(x _ {d}\right)\right) \leq \gamma_ {2},
+$$
+
+# Algorithm 1 Disguise Generation
+
+Input: copyrighted image  $x_{c}$ , base image  $x_{b}$ , pre-trained encoder  $\mathcal{E}$ , input threshold  $\gamma_{1}$ , feature threshold  $\gamma_{2}$ , distance measure on input space  $D_{1}(\cdot)$ , distance measure on feature space  $D_{2}(\cdot)$ , hyperparameter on input space constraint  $\alpha$ , learning rate  $\eta$ .
+
+1 Initialize disguise  $x_{d}$  with base image  $x_{b}$  
+2 repeat  
+3  $D_{1}\gets D_{1}(x_{b},x_{d})$  // image distance   
+4  $D_{2}\leftarrow D_{2}(\mathcal{E}(x_{c}),\mathcal{E}(x_{d}))$  // feature   
+distance   
+5  $\mathcal{L}\gets \alpha D_1 + D_2$  // calculate loss   
+6  $x_{d}\gets x_{d} - \eta \frac{\partial\mathcal{L}}{\partial x_{d}}$  // update disguise   
+7  $x_{d}\gets \mathrm{Proj}_{\Gamma}(x_{d})$  // project to admissible   
+set  
+8 until  $D_{1}\leq \gamma_{1}$  and  $D_{2}\leq \gamma_{2}$  
+9 return disguise  $x_{d}$
+
+where  $\gamma_{1}$  is the input threshold that measures the visual distance of  $x_{d}$  compared with  $x_{b}$ ,  $\gamma_{2}$  is the threshold that measures the featur distance between  $x_{c}$  and  $x_{d}$ , which can be both determined empirically. We can then express the objective function as follows:
+
+$$
+\operatorname * {a r g m i n} _ {x _ {d}} \alpha D _ {1} + D _ {2},
+$$
+
+where  $\alpha$  is a tunable hyperparameter for controlling the tradeoff. To illustrate more, the extreme case  $\alpha = 0$  refers to the scenario where there is no input space constraint on  $x_{d}$  while  $\alpha = \infty$  refers to no feature space constraint. Our algorithm is summarized in Algorithm 1 and Figure 2.
+
+Connection to Nightshade: Shafahi et al. (2018) introduced the idea of attacking a model by manipulating training datapoints so that their feature representation resembles that of a different point. They applied this principle to targeted data poisoning attacks against classifiers, whereas Shan et al. (2024) (in their work "Nightshade") and we apply it to LDMs. To highlight some of the differences between our work and that of Shan et al. (2024), there are crucial technical differences in how we instantiate this algorithm. Specifically, Nightshade intends to stop LDM from producing the image containing the correlated concept. In contrast, we aim to urge LDM to reproduce the target image (copyrighted image) through indirect access. Furthermore, the two works employ this algorithm towards very different ends. Nightshade aims to sabotage the connection between an image and its corresponding text prompt, thus requiring a pair of (image, text) as input. In contrast, our method only requires the form of images as input.
+
+![](images/b1b3f75c82b0776e005050f3603cb6fce539fa751a02c9593159cfd569c9d970.jpg)
+
+![](images/433eb71d51503146fa4a5edea1e9fa6b8ac40663617ec81d4d5fe1855dd2cb19.jpg)
+
+![](images/7360f7015374be81eff0b37eaddb1191fa8659b02fae38ff62fad2c519dc746c.jpg)
+
+![](images/41e298e298b420c96a8167229bb732f3cef6a38285cefee7ddac7e29b5ea842e.jpg)
+
+![](images/310895355a80d18bfd9f47c1b81db8e424a07692d7a2a8ae8b1cdce24d8e1b3c.jpg)
+
+![](images/8c6347c6756e6d6725d77eee4b7ac037b355459a69806c8ae308eb0a46530264.jpg)  
+$x_{c}$  
+Disguises  $x_{d}$  
+Inference on textual inversion by learning on  $x_{d}$  only
+
+# 4. Revealing Disguises
+
+Recall that the set of disguised data  $\{x_d^i\}_{i = 1}^N$  is constructed to substitute their copyrighted counterpart  $\{x_c^i\}_{i = 1}^N$  such that the latent dataset  $\zeta$  is not changed much. To evaluate whether this goal is accomplished, we extract the latent information contained in the acquired samples  $x_{d}$  using textual inversion to qualitatively reveal the disguises. We also report the feature threshold  $\gamma_{2}$  for quantitative analysis.
+
+Moreover, examination of textual inversion also demonstrates how disguised copyright infringement may occur in the wild, i.e., reproducing copyrighted materials using products that utilize pre-trained LDMs to generate arts, which bring major concerns for human artists. Such LDM-based methods usually require only a few images as the training set  $\chi$ , and can be easily substituted by disguises entirely. In Appendix F, we will further extend our evaluation to DreamBooth (Ruiz et al. 2023) and LDM training.
+
+# 4.1. Experimental settings
+
+Note that the "copyrighted" images  $x_{c}$  used in our experiments may not be actually copyrighted, but used as a substitute to demonstrate disguised copyright infringement.
+
+LDM: We adopt the official PyTorch implementation<sup>1</sup> of conditional LDM (Rombach et al. 2022) and acquire the pre-trained weights<sup>2</sup>(including that of the encoder  $\mathcal{E}$ , the denoising U-Net  $\varepsilon_{\theta}$  and the text embedding  $c_{\theta}(\cdot)$ ) of a 1.45B parameter KL-regularized LDM-8 (8 denotes downsampling factor) model conditioned on language prompts on LAION
+
+![](images/2b2d741b8f0839772c1d0e8c76f286b86e63ef8649590b1d7c0ebb979bb3c264.jpg)
+
+![](images/629209b35728890bb78769537cfb2f144fbf2ec2510b2e20a481a58e972e688e.jpg)
+
+![](images/1b38e9c1ef6bf2bae1397b662d3ca38d9bd50df762c5df8d6d26b8fe0523e241.jpg)
+
+![](images/1ac14373fc9f11f7e28c3c1a581134e35a87168dabf436e143ab61d85cdf78d9.jpg)
+
+![](images/4fd339994008a6394150abb164c312d0109ed9864378047edf5c16994ebb235e.jpg)
+
+![](images/d8a32244d4b4dfe91abcbbc05a3ee96bb2924b67041eb36e060a11a03b001102.jpg)
+
+![](images/4179857d64c9ed7ed5402b3701340717dd1353c915660b032a8384b86d5533d3.jpg)  
+Figure 3: The disguised symbol on textual inversion. The first row from left to right: Column (1) the designated copyrighted symbol; Columns (2)-(5) four disguises  $x_{d}$  generated with different  $x_{b}$ . The second row: images generated by textual inversion after training on the above  $x_{d}$ .  
+Figure 4: We show the disguised copyrighted content on textual inversion. The first row: the designated copyrighted image  $x_{c}$  (The Sunflowers by Vincent Van Gogh); the second row: three disguises  $x_{d}$  generated with different  $x_{b}$ ; the third row: images generated by textual inversion after training on the above  $x_{d}$ .
+
+![](images/a2a48d960bf9d60023dc43ebbc15e0a2d26a231acf2029e58388fd5f103145ad.jpg)
+
+![](images/1b3d396864bb7d2c2c598a9220316331d72e05e61154715bf1e925c2baff2551.jpg)
+
+400M (Schuhmann et al. 2021). We apply the pre-trained encoder  $\mathcal{E}$  in Algorithm 1 to generate disguises, and the entire pre-trained LDM for textual inversion. Note that the LDM is not re-trained in our experiments.
+
+Generating Disguises: Throughout our experiments, we apply the pre-trained  $KL$ -regularized encoder  $\mathcal{E}$ , and set the input distance measure  $D_{1}(\cdot)$  as a sum of the multi-scale structural similarity index (MS-SSIM) loss (Wang et al. 2003) and  $L_{1}$  loss following the analysis of (Khare et al. 2021), and the feature distance measure to be the  $L_{2}$  loss:  $D_{2}(\mathcal{E}(x_{c}),\mathcal{E}(x_{d})) = \| \mathcal{E}(x_{c}) - \mathcal{E}(x_{d})\|_{2}$ . The choice of the copyrighted image  $x_{c}$ , base image  $x_{b}$ , the input threshold  $\gamma_{1}$ , the feature threshold  $\gamma_{2}$  and the hyperparameter on the input constraint  $\alpha$  are task-dependent (for different copyrighted material), and we specify them in their corresponding paragraphs below. We set the admissible set to be in the range of  $[0,1]$  as the legitimate (normalized) image pixel value and run the algorithm for 100000 epochs (early stop if the stopping criteria are reached) for all experiments.
+
+Textual inversion: After acquiring the disguises  $x_{d}$ , we feed them (we generate  $N = \{3,4\}$  disguises for extracting the same concept, following the normal recipe) into textual inversion (Gal et al. 2022) and optimize the embedding  $\nu_{*}$  with Equation 2. We adopt the official PyTorch implementation
+
+![](images/c2b2d24202f815a02181869c51f6eecb9f5ab51c16057dc33eb430687322f33d.jpg)  
+Figure 5: We show the disguised copyrighted style on textual inversion. The first row: the designated copyrighted style  $x_{c}$  (in the style of The Starry Night by Vincent Van Gogh); the second row: disguises  $x_{d}$  generated with different  $x_{b}$ ; the third row: images generated by textual inversion after training on the above  $x_{d}$ .
+
+tion $^3$ , which utilizes the pre-trained  $KL$ -regularized LDM-8 with the same encoder  $\mathcal{E}$ . Note that training textual inversion requires an initial word as input, which is an initialization for the concept  $S_{*}$ . Throughout our experiments, we choose the initial word to match the visual appearance of the disguises (thus the concept of the base image) and we observe that changing the initial word does not lead to a significant difference for the algorithm. For generating new images, we select from the pool of prompts included in the existing implementation of textual inversion $^4$  which we specify for each task below.
+
+# 4.2. Concepts learned on disguises
+
+We demonstrate the concepts learned by textual inversion on the disguises with respect to different copyrighted patterns. (1) a copyrighted symbol; (2) copyrighted content; and (3) style. We note that although the style is generally not copyrightable, there are still ongoing debates about the ethics of using copyrighted artwork to train a model that can imitate an artist's style<sup>5</sup>. For this reason, we also examine
+
+3https://github.com/rinongal/textual_inversion  
+4https://github.com/rinongal/textual_inversion/blob/main/ldm/data/personalized.py  
+<sup>5</sup>https://news.bloombergglaw.com/ip-law/a i-limitating-artist-style-drives-call-to-r ethink-copyright-law
+
+the case of hiding an artist's style in disguised images.
+
+Disguised symbol: We first show the easiest scenario to generate disguises, namely copyrighted symbol. In Figure 3, we pick the symbol "A" as  $x_{c}$  (note that here we abstract  $x_{c}$  as the symbol "A", in our experiment, it is combined with each base image to create the corresponding copyrighted image  $x_{c}$ , see Figure 8 in Appendix A) and choose four images of a water bottle as base images  $x_{b}$ . The base images do not need to be carefully constructed and are photos taken by ourselves. Each disguise  $x_{d}$  is created by taking a pair of  $x_{c}$  and  $x_{b}$  as input to Algorithm 1. We set  $\gamma_{1} = 0.05$ ,  $\gamma_{2} = 0.35$  (normalized to the range of [0, 1]) as the threshold of input and feature distance, respectively, and  $\alpha = 8000$ . By feeding  $x_{d}$  into textual inversion with the text prompt "a photo of a *", we reproduce the target symbol "A" without being exposed to the semantic information of the copyrighted content.
+
+Disguised content We then show a more challenging task to generate disguised content. In Figure 4, we pick three drawings of The Sunflowers<sup>6</sup> by Vincent Van Gogh as  $x_{c}$  (first row). Due to the difficulty of the task, we cannot choose any image as  $x_{b}$ . Thus we first blur  $x_{c}$  such that they lose their semantic information and retain the color pattern, then we add simple sketches of houses<sup>7</sup> as base images  $x_{b}$  (second row). In Figure 14 in Appendix A, we show the background is essential for our purpose, where the disguises are ineffective with a white background. We set  $\gamma_{1} = 0.08$ ,  $\gamma_{2} = 0.26$ ,  $\alpha = 4000$ . By feeding  $x_{d}$  into textual inversion with the text prompt "a photo of a *", we recover the content of the sunflowers (third row) without containing the semantic information of  $x_{d}$ .
+
+Disguised style Finally, we show the results for style scraping. In Figure 5, we pick three drawings<sup>8</sup> in the style of The Starry Night by Vincent Van Gogh as  $x_{c}$  (first row). The base images  $x_{b}$  (second row) are the target images with another style (watercolor), generated with AdaIN-based (Huang and Belongie 2017) style transfer<sup>9</sup>. We set  $\gamma_{1} = 0.03, \gamma_{2} = 0.33, \alpha = 2000$ . By feeding  $x_{d}$  into textual inversion with the text prompt "a painting in the style of *", we reproduce the style of  $x_{c}$  (The Starry Night) while textual inversion only learns from images that visually resemble watercolor style.
+
+![](images/245a20d77c2dba1549ab32e6f5733a68586010587702702b7fcc92b0c2457140.jpg)  
+Figure 6: The effect of tuning the hyperparameter  $\alpha$  on disguises (Rows 1-2: symbol; Rows 3-4: style). Rows 1 and 3 show disguises generated with different  $\alpha$ ; Rows 2 and 4 show the images generated by textual inversion by training on the disguises above. Note that the criterion on the input and feature threshold may not be satisfied in this series of experiments and we only train the disguise generation algorithm until convergence.
+
+In summary, for disguised symbols, generating the disguises  $x_{d}$  with any base image  $x_{b}$  is easy. However, for generating disguised content and style, one needs to choose the base image carefully for successful optimization.
+
+# 4.3. Visual appearance of disguises
+
+Next, we control the visual appearance of  $x_{d}$  by tuning  $\alpha$ , which is the weighting parameter of the input space constraint. In Figure 6, we show the tradeoff of tuning the hyperparameter  $\alpha$ : (1) a smaller  $\alpha$  indicates weaker input space constraint and it could lead to an ineffective disguise, where  $x_{d}$  still visually contains the copyrighted material; (2) in contrast, a bigger  $\alpha$  shifts the optimization focus away from feature matching to generate latent embeddings distinct from the copyrighted content's. In the latter case, the tradeoff is that despite having a strong disguise that visually hides the copyrighted content, the textual inversion process may not successfully learn the copyrighted material, thus rendering our attack pointless. Note that there is no
+
+input constraint for the extreme case  $\alpha = 0$ , and the visual appearance of disguises largely depends on the initialization (we initialize with  $x_{b}$  above) and we show the results for different initialization in Figure 15 in Appendix A.
+
+# 4.4. Detection
+
+Next, we introduce a two-step detection method specifically for disguised samples that go beyond browsing through the training set, e.g., haveibeentrained.com.
+
+(1) Feature similarity search: Quantitatively, a disguised sample has a similar feature representation with that of a copyrighted sample, i.e.,  $D_{2}(\mathcal{E}(x_{c}),\mathcal{E}(x_{d})) \leq \gamma_{2}$ . As we have provided the feature threshold  $\gamma_{2}$  which is sufficient for replicating the copyrighted content for different tasks, one can use this as a reference threshold to detect possible disguises. Specifically, given an encoder  $\mathcal{E}$  (which is usually easily accessible), a copyrighted image  $x_{c}$  which needs examination for infringement, one simply goes through the training set and computes  $\mathcal{E}(x)$  for every single sample and
+
+![](images/fd19b722d60166a18cb48ede13c6f9fd95f47db8f60195977bda7824d4fbc1e2.jpg)  
+Figure 7: Results for the encoder-decoder examination. The first row shows the disguises  $x_{d}$  and the second row includes the output of the autoencoder:  $\mathcal{D}(\mathcal{E}(x_d))$ . Columns 1-3, 4-6, and 7-9 show results for symbol, content, and style, respectively. The autoencoder reveals the copyrighted information contained in the disguises.
+
+compare with that of  $x_{c}$ . However, such a search alone only extracts suspects, which may not be true disguises. To rule out the sheer chance of collisions in the feature space, we perform another examination below.
+
+(2) Encoder-decoder examination: The encoder architecture in LDMs is part of an autoencoder architecture (e.g., the KL-based VAE), where the encoder and decoder are used separately for encoding and inference. Consequently, the decoder  $\mathcal{D}$  can be naturally used to detect disguises qualitatively. Specifically, for a well-trained autoencoder,  $\mathcal{D}(\mathcal{E}(x_c)) \approx x_c$ , while for disguises  $\mathcal{E}(x_d) \approx \mathcal{E}(x_c)$ , thus we have  $\mathcal{D}(\mathcal{E}(x_d)) \approx x_c$ . In Figure 7, we show that the encoder-decoder architecture is a great detection tool for disguises, where the output of the autoencoder reveals the copyrighted content hidden in  $x_d$ .
+
+# 4.5. A broader definition of access
+
+Our experiments have demonstrated that by generating disguises, one can indirectly acquire access to copyrighted material as part of the training data to fuel LDM-based models, which could reproduce the copyrighted samples during inference or deployment. We expand the definition of direct access to acknowledgment to cover such scenarios as possible copyright infringement: any training data  $x$  that contains similar latent information as that of copyrighted image  $x_{c}$ , measured by their similarity in the latent feature representation, even visually different from  $x_{c}$ , shall be considered to have an acknowledgment of  $x_{c}$ . The notion of acknowledgment may also augment existing regulatory frameworks (e.g., the White House Executive Order on AI (Biden 2023)) on AI governance in terms of data quality evaluation. Additionally, our quantification of acknowledgment (the detection method) provide a timely tool for auditing beyond black-box access, which was pointed out to be insufficient in (Casper et al. 2024).
+
+# 5. Conclusion
+
+In this paper, we review the current access criterion of containing copyrighted material in the training set (direct access) in copyright infringement of generative models and point out its insufficiency by introducing disguised copyright infringement (indirect access). Specifically, such an infringement is realized by injecting disguised samples into the training set, which urges LDMs to produce copyrighted content. Such disguises are generated with a simple algorithm and demonstrated to share the same concept with their target copyrighted images using the textual inversion tool. To alleviate the concern on the disguises, we expand the current visual auditing (browsing the training set) with additional tools, i.e., feature similarity search and encoder-decoder examination to better identify these disguises. Furthermore, we propose a broader definition of acknowledgment to cover this new type of copyright violation.
+
+Limitations and future work: One interesting future work is to quantify the number of disguises needed for reproducing in large-scale training, which can be further linked to the quantification of memorization of such models (Carlini et al. 2022; Somepalli et al. 2023a; Somepalli et al. 2023b; Carlini et al. 2023; Ippolito et al. 2023; Zhang et al. 2021)). Additionally, although our algorithms can generate descent disguises, we believe there is still room for improvement for optimization. Finally, one extension we didn't touch is the possibility of "chopping" copyrighted data and hiding it in several images. It is intriguing to explore whether it is possible to generate such a smuggler's dataset and detection towards it.
+
+Learning from noisy data A simultaneous and independent work (Daras et al. 2024) considers learning LDMs from noisy data. Although the techniques are very different, our works share a similar implication: the training dataset may not immediately resemble the generations produced, thus al
+
+lowing copyright issues to be disguised from an auditor who manually inspects the dataset. In the work of Daras et al. (2024), the training data is the original data with Gaussian noise applied to it. In our case, the training data has the original data hidden in the latent space. In Section 4.5, we argue that if training data contains the same latent information as some copyrighted data, then it should be acknowledged. Daras et al. (2024) show that even when the latent space is corrupted the copyrighted information could still be contained, which could raise a stronger disguised copyright infringement attack.
+
+Acknowledgment and fair use Finally we discuss the anticipated impact of our definition of acknowledgment on judicial decisions. Specifically, we believe acknowledgment will not directly affect the current fair use doctrine or change its decision. To further illustrate the above claim, we first propose two principles: (1) copyright infringement is not fair use: although generative AI law is not rigorously defined, we generally assume that "reproducing the copyrighted content" is not fair use when there's direct access; (2) disguised infringement does not (appear to) use the copyrighted data: there is no decision of fair use doctrine here as there is no apparent "usage" of the copyrighted data, not to mention "fair use". Thus the decision is on the "originality" of the output (reproduction of the copyrighted content) but not fair use. As a result, the notion of acknowledgment aims to identify disguised copyright infringement in the second principle and challenge the decision of "originality". Based on the acquired outcome (reproduction), there would be no scenarios that acknowledgment can change the decision of fair use according to the first principle.
+
+# Impact Statement
+
+In this paper, we adopt a simple algorithm to disguise images, which we realize may be applied as a concealed copyright infringement tool. We are surprised at how easy it is for Generative AI to circumvent the copyright law, which was originally designed for regulating human scrapers. This seemingly undetectable infringing threat may bring disadvantages for human artists, who are already losing the battle with GenAI. Furthermore, disguised images may not only be a concern to copyright owners but also a threat to Generative AI companies: if these disguises are accidentally collected as part of the training data, copyright infringement could be triggered unconsciously and unwillingly. As computer scientists, we are obligated to reveal the existence of such possible unlawful acts and provide technical tools to identify such behaviors. We expect our paper to be a prudent tool such that disguised copyright infringement and its corresponding detection methods are recognized by law experts.
+
+# Acknowledgements
+
+We thank the reviewers for the critical comments that have largely improved the presentation and precision of this paper. We gratefully acknowledge funding support from NSERC and the Canada CIFAR AI Chairs program. YY is also supported by the Ontario early researcher program. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute.
+
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+conference on computer vision and pattern recognition, pp. 10684-10695.  
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+
+# A. Additional Experiments on Revealing Disguises
+
+We provide additional experimental results on revealing disguises to further support our observations.
+
+Textual inversion on  $x_{c}$  and  $x_{b}$ : Recall that Figure 3, Figure 4 and Figure 5 show the disguised images  $x_{d}$  and the new images generated by textual inversion. To further show the effect of our algorithm on generating disguises, we directly perform a textual inversion (with the same setting in our main experiments) on the copyrighted images  $x_{c}$  and the base images  $x_{b}$  below. Firstly, for copyrighted symbols:
+
+Learning on the copyrighted images  $x_{c}$ :
+
+![](images/f5db6fcc7fb3eb44c07a3282853d00ee90e002a7b8d2e708b601ce0d54940e73.jpg)  
+Figure 8: Textual inversion on images  $x_{c}$  with the copyrighted symbol "A". The first row shows the copyrighted images  $x_{c}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{c}$ .
+
+Learning on the base images  $x_{b}$ :
+
+![](images/5c810a6fecf411639231365d9706a3f79a05a14373ed3231649d3a13dd33f097.jpg)  
+Figure 9: Textual inversion on base images  $x_{b}$  (different images of a bottle). The first row shows the base images  $x_{b}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{b}$ .
+
+Comparing with Figure 3, we observe that although the disguises  $x_{d}$  look visually similar to their corresponding base images  $x_{b}$ , they contain drastically different latent information.
+
+Next, we show the results for copyrighted content in the same format:
+
+Learning on the copyrighted images  $x_{c}$ :
+
+![](images/d541fe36a4195296e4d2d828f38268fc732ae214ce46794b09b83cb10df807ae.jpg)
+
+![](images/ad46ef1c98bd2e2c414554c579a87f1c00773f9f6af23f46166ad8149242fc56.jpg)
+
+![](images/c6d5bbfcab7c0961ade50a22b1052657775caf2f2f50d16824407941258ee743.jpg)
+
+![](images/c0bd02d42eee2bd4861caebc4b43ae5f919b82b3f0c6a61dd62b7fe3695626d6.jpg)  
+Figure 10: Textual inversion on copyrighted images  $x_{c}$  (The Sunflowers by Vincent Van Gogh). The first row shows the copyrighted images  $x_{c}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{c}$ .
+
+![](images/1bf891d6fc18d6b92d95820b770ac20c864b1de1657b19c17482e2b7b41b9592.jpg)
+
+![](images/cacc4e75a334a855c80c521a97f1e719df37ea15ebfe9b6db6c73c19578f7602.jpg)
+
+![](images/e56753732b3535d0d337697f0d53eec0e34c1ad40372a261dee368b2af227810.jpg)
+
+Learning on the base images  $x_{b}$ :
+
+![](images/c6c941730738cbe96428058ef32bebc5b8a97557a5849f1197837589d55ee591.jpg)
+
+![](images/69ee14721c171b589a090ddefd1a95e84e5f69d6f8023a7f642c8b3470241b3d.jpg)
+
+![](images/d2df1c982589da16e7e4cb8124f69723208cc63bb040ac9880fb608568df3620.jpg)
+
+![](images/7673fb262ef0eddacf965cd8e81a1372d79ff773f28ccfcb231cbcecdb47b41d.jpg)  
+Figure 11: Textual inversion on base images  $x_{b}$  (images of sketched houses with the blurry version of the corresponding  $x_{c}$  as the background). The first row shows the base images  $x_{b}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{b}$ .
+
+![](images/72bc442e972b39735fd0b4811d3f3cbf2916bf5e0e1b4007d6de3d5884f397dc.jpg)
+
+![](images/8ed5a9895d4720c13f196df318fda51e7d7f04273517d699b80db21b1ca0e072.jpg)
+
+![](images/ac66dfd9546d4fb0887849f80fc06c4c4b0f4c9505e605bffd89082d2d257f68.jpg)
+
+Compared to Figure 4, we observe that the background (blurry version of the  $x_{c}$ ) does not provide any information regarding the copyrighted images  $x_{c}$ , and the success of the disguises is accomplished by our generating algorithm.
+
+Finally, we show the results for style scraping:
+
+Learning on the copyrighted images  $x_{c}$ :
+
+![](images/aa876914eca891bb044f87e36a82bb788abfcba299c6d9380394c1f674c45229.jpg)
+
+![](images/ce644d058e725b4d433a0c1b9347ea769686be2f078d94cf589a88a7b39284e0.jpg)
+
+![](images/cb0d5e085d8371c6240ebc71d06b6928284e94852770e0bced9f38eead6e6c49.jpg)
+
+![](images/2eeb4bfd5b5aeaff71de033771e42b05856829ac201ebd97955de2c1fa583b86.jpg)  
+Figure 12: Textual inversion on copyrighted images  $x_{c}$  (images in the style of The Starry Night by Vincent Van Gogh). The first row shows the copyrighted images  $x_{c}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{c}$ .
+
+![](images/19812b1c97080c7a1492b2bd34caecb557330dff96f0580f3db4c12d47fc9488.jpg)
+
+![](images/fd41cd1530f227e01598954c1f81e11e097c588e1ecbfda15e539e87b7647430.jpg)
+
+![](images/8b065df7a6a21645cb820538858a6c3e716527b3ed4ba68eccf7521e853e6fa7.jpg)
+
+Learning on the base images  $x_{b}$ :
+
+![](images/8e327b49a200aa0cab5f819de6687c9f4e1b46ddb80a6b2fb3635efa46c8ced2.jpg)
+
+![](images/8903f3ee63100cd35ef780a8675dd5b302f22d488230210ac495ab2f6a8b95ff.jpg)
+
+![](images/8f03694d11da597a97e62f8b78aab0cb3f45e9bd2a5cc4e2c03468099fdef167.jpg)
+
+![](images/d453027e2aa47353d75843637972544cf77ff306f66e029c15d200ea5fc51069.jpg)  
+Figure 13: Textual inversion on base images  $x_{b}$  (images in watercolor style). The first row shows the base images  $x_{b}$ , and the second row shows new images generated by textual inversion after learning on the above  $x_{b}$ .
+
+![](images/ba67999d5e1b2e73732e3c8299cc35be8871a08f531a4e4558e5f616af7babd6.jpg)
+
+![](images/7467c76f337fd4d1c93cd6e4634a90e63d7d47481695fb9150a0adc23e39f22f.jpg)
+
+![](images/e4538ed4696ee449b52d1a2e6337475c9b936ce095b639c864233230e4493cde.jpg)
+
+Compared to Figure 5, we again confirm that the disguises  $x_{d}$  generated for style scraping look visually similar to their corresponding base images  $x_{b}$  but contain latent information similar to  $x_{c}$ .
+
+Choice of  $x_{b}$  for generating disguised content: Recall that for generating disguises in Figure 4, we choose a blurry version of  $x_{c}$  as background to retain the color pattern. Here we demonstrate the necessity of the background by substituting it for a white background as base images  $x_{b}$ . In Figure 14, we observe that the disguises fail in reproducing the copyrighted content on textual inversion.
+
+![](images/dd35c7c29eef0c1031a15a65a935f69d837536d5d517d6fb7fe9769b63923ed2.jpg)  
+Figure 14: Failure to learn the copyrighted content (The Sunflowers) from a disguised sketch of the house with a white background. The first row: disguises; the second row: images generated by textual inversion.
+
+Initialization for unbounded disguises: In Figure 6 (first column), we show the disguises generated without any input constraints. Note that the disguise (for the copyrighted symbol) still contains the base image as background as the initialization of the disguise is  $x_{b}$ . To confirm, we show two different initializations (zero tensor and Gaussian noise) in Figure 15 and observe that the background diminishes while the disguises still contain the copyrighted symbol.
+
+![](images/2d9f199d2d63e411a9e479b385b871ada753ad49727a7bf8afe20a4e0ed6c621.jpg)
+
+![](images/0780506f36715d446017e655acc13568e62b5fdf3878ffff4a585bd5fb09fabf.jpg)  
+Disguises  $x_{d}$
+
+![](images/026f610031bf4d0024ce9abf6e6df23806fef22c8105f86ab8453514d565e059.jpg)
+
+![](images/dd96e44b13ac451d1e8ac77ef4c64e908f106f02122a2c6da21fa1549ae196ab.jpg)  
+Inference on textual inversion by learning on  $x_{d}$  only  
+Figure 15: Textual inversion on the disguised symbol without any input constraints. We deploy different initializations for generating disguises: the first row: zero tensor; the second row: Gaussian noise.
+
+# B. Data augmentation
+
+In textual inversion, images are horizontally flipped with  $50\%$  probability. In this section, we explore how this affects the success of our disguises. The top two rows of Figure 16 show the results of training textual inversion on our disguise as well as its horizontally flipped augmentation. For the simpler concept of watermarking, textual inversion learns a concept similar to the intended one even when the input image is flipped. On the other hand, when the concept becomes more difficult,
+
+textual inversion struggles to learn the concept under a horizontal flip augmentation (top right of Figure 16).
+
+![](images/90f4b117de61629c7f813256d0b5f3e4c7a949979ffbd693b4beff9c2b1d6305.jpg)  
+Disguise  $x_{d}$
+
+![](images/92e98cfbc29764dd40d1898919c4b93d2f0337d9f6b095db768301366346417d.jpg)  
+Learn on  $x_{d}$  
+Learn on  $hflip(x_d)$
+
+![](images/dccbcfa13722e7bf5afe722b011c87e887633c6665c63d2a68e513150582926f.jpg)  
+Disguise  $x_{d}$
+
+![](images/5d28d61357bc33e93a77489a52d85ca966e314d283d0e99729b774a8ed76d2ff.jpg)  
+Learn on  $x_{d}$  
+Learn on  $hflip(x_d)$
+
+![](images/87b95d390ff91013e3bf2f02ae0a0a7a2fd1fbe7a0b18ffe88601171eb862a2e.jpg)  
+Disguise  $x_{d}$
+
+![](images/07e6621ec344e18e5f87c321cd86704c823de02de7c68cc323bb9f549b049a49.jpg)  
+Learn on  $x_{d}$  
+Learn on  $hflip(x_d)$
+
+![](images/c5646fb71838276404aa4f5329d6491a27e94032fdbc38a981d2c948084b8113.jpg)  
+Disguise  $x_{d}$
+
+![](images/f5bf91a40e3d64ac1e0a700c101e78467998e1ed5f83ae4bb4e3284410a22433.jpg)  
+Learn on  $x_{d}$  
+Learn on  $hflip(x_d)$  
+Figure 16: Textual inversion on different disguises. Top left: textual inversion on a single watermarking disguise. Top right: textual inversion on a single content disguise. Bottom row: same as top row, but with robust disguises.
+
+To ensure that a horizontal flip augmentation would not damage the quality of our poison, we construct a new robust poison that additionally penalizes the distance between the features of the horizontally flipped poison and the horizontally flipped copyrighted image  $x_{c}$ . This can be done with a simple change to the objective from
+
+$$
+\underset {x _ {d}} {\operatorname {a r g m i n}} \alpha D _ {1} (x _ {b}, x _ {d}) + D _ {2} (\mathcal {E} (x _ {c}), \mathcal {E} (x _ {d}))
+$$
+
+to
+
+$$
+\operatorname * {a r g m i n} _ {x _ {d}} \alpha D _ {1} (x _ {b}, x _ {d}) + D _ {2} (\mathcal {E} (x _ {c}), \mathcal {E} (x _ {d})) + D _ {2} (\mathcal {E} (h f l i p (x _ {c})), \mathcal {E} (h f l i p (x _ {d}))).
+$$
+
+The bottom two rows of Figure 16 demonstrate the effectiveness of the robust poison and its qualitative improvement under horizontal flipping.
+
+Table 1: The encoder-decoder examination's threshold  $\zeta$ , mean construction losses, false positive rate, and binary AUROC across symbol, content, and style disguises. The second group of rows is for the setting of mixture with base images, and the third group of rows is for the setting of mixture with 100 randomly selected Imagenette images.
+
+<table><tr><td></td><td>Symbol</td><td>Content</td><td>Style</td></tr><tr><td>Exd(Lreconstruct xd))</td><td>0.0856</td><td>0.1632</td><td>0.2667</td></tr><tr><td>Threshold ζ</td><td>0.0703</td><td>0.1412</td><td>0.2506</td></tr><tr><td>Exb(Lreconstruct xb))</td><td>0.0502</td><td>0.0256</td><td>0.0551</td></tr><tr><td>FPR (xb misclassified)</td><td>1/4</td><td>0/3</td><td>0/3</td></tr><tr><td>AUC</td><td>0.8750</td><td>1</td><td>1</td></tr><tr><td>Exclean(Lreconstruct(xclean))</td><td>0.0655</td><td>-</td><td>-</td></tr><tr><td>FPR (xclean misclassified)</td><td>42/100</td><td>2/100</td><td>0/100</td></tr><tr><td>AUC</td><td>0.7550</td><td>0.9933</td><td>1</td></tr></table>
+
+# C. Quantitative detection
+
+In Section 4.4, we show how to qualitatively distinguish a disguise with encoder-decoder examination. In this section, we further examine two quantitative detection scenarios with/without the presence of the copyrighted image  $x_{c}$ :
+
+Without the presence of  $x_{c}$ : Specifically, we observe that in Figure 7, for disguises  $x_{d}$ , the input  $x_{d}$  and output of the autoencoder  $\mathcal{D}(\mathcal{E}(x_d))$  are significantly different, i.e., the reconstruction loss  $D_{1}(\mathcal{D}(\mathcal{E}(x_{d})),x_{d})$  is "big" (recall that  $D_{1}$  is the distance measure in the input space, a sum of MS-SSIM loss and  $L_{1}$  loss). This property differs from normal images, which are expected to have low reconstruction loss. Leveraging this intuition, we develop a detection criterion for disguised images:
+
+$$
+x \text {i s a d i s g u i s e i f} \mathcal {L} _ {\text {r e c o n s t r u c t}} (x) := D _ {1} (\mathcal {D} (\mathcal {E} (x)), x) \geq \zeta
+$$
+
+for some threshold  $\zeta$ . The choice of  $\zeta$  differs on different tasks (depending on the choice of the task, which we specify below). Note that this detection mechanism does not require any knowledge of the copyrighted image and is simply a data sanitization method that can be applied to any dataset.
+
+In Table 1, we show the effectiveness of our proposed method across the three tasks we performed in Section 4 (copyrighted symbol, content, and style). Row 2 shows the mean of the reconstruction loss for the disguises we generated for each task; Row 3 shows the choice of threshold  $\zeta$  as the lowest reconstruction loss out of all the disguised images such that there are no false negatives. Rows 4-6 examine the detection performance in terms of FPR (False Positive Rate) and AUC (Are Under the Curve) within the pool of the mix of the disguises  $x_{d}$  and their corresponding base images  $x_{b}$ ; Rows 7-9 performs the same examination on a larger pool of the mix of the disguises  $x_{d}$  and a subset of 100 images randomly chosen from ImageNet.
+
+We observe that for copyrighted symbols, as the copyrighted images  $x_{c}$  (see Figure 8) and the base images  $x_{b}$  (see Figure 9) share the same background and only differ on the symbol, the reconstruction loss of  $x_{d}$  is low (mean of 0.0856) and falls into the range of that of clean images, thus having a higher FPR and a lower AUC than the other two tasks.
+
+With the presence of  $x_{c}$ : In practical scenarios (e.g., examination in court), one may already have acquired the copyrighted images  $x_{c}$  and want to find its (possible) corresponding disguise from a dataset. In such scenarios, one may directly use the detection method introduced in Section 4.4. We repeat the above experiment using our feature similarity search as a first step, which we recall, screen disguises using the criterion  $D_{2}(\mathcal{E}(x_{c}),\mathcal{E}(x_{d}))\leq \gamma_{2}$ . We acquire perfect AUC score and obtain no false positives across all tasks, which indicate that the second step (encoder-decoder examination) is not required for this specific task.
+
+# D. Circumventing the detection?
+
+A stronger disguise may be generated to circumvent our detection method (e.g., the encoder-decoder examination). This can be possibly accomplished by modifying the disguise generation objection. Recall that our previous objective is:
+
+$$
+\operatorname * {a r g m i n} _ {x _ {d}} \alpha D _ {1} (x _ {b}, x _ {d}) + D _ {2} (\mathcal {E} (x _ {c}), \mathcal {E} (x _ {d})),
+$$
+
+Here we wish to also minimize the input distance between  $x_{d}$  and its autoencoder output  $D_{1}(\mathcal{D}(\mathcal{E}(x_{d})),x_{d}))$ , thus we can modify our objective to:
+
+$$
+\operatorname * {a r g m i n} _ {x _ {d}} \alpha (D _ {1} (x _ {b}, x _ {d}) + D _ {1} (\mathcal {D} (\mathcal {E} (x _ {d})), x _ {d})) + D _ {2} (\mathcal {E} (x _ {c}), \mathcal {E} (x _ {d}))
+$$
+
+We perform experiments on the copyrighted symbols and observe in Figure 17 that, the additional term adds a constraint such that  $x_{d}$  and  $\mathcal{D}(\mathcal{E}(x_d))$  are visually similar (row 2). However, performing textual inversion on these images also fails to reveal the copyrighted symbol. We conclude that this method might not be ideal, and additional study might be required.
+
+# E. Additional experiments on disguised style
+
+In this section, we exhibit additional experimental results on style transfer by extending the choices of the (designated) copyrighted style  $x_{c}$  and base images  $x_{b}$ .
+
+Disguises in the style of Claude Monet: Recall that in Section 4, we choose the base images  $x_{b}$  as the ones with the watercolor style. In the following experiments, we aim to change  $x_{b}$  to images with the style of Claude Monet and still generate disguises  $x_{d}$  that implicitly contain the style of Vincent Van Vogh. In Figure 18, we first create the  $x_{c}$  using style transfer (Huang and Belongie 2017), which is San Giorgio Maggiore at Dusk (1908-1912) by Claude Monet transferring to the style of The Starry Night by Vincent Van Gogh. In Figure 19, we create the disguise  $x_{d}$  (row 1, column 2) with the
+
+![](images/4da44df739903f5dcbb0890425f5f08046ae5d713629fb678db610c425ce338c.jpg)  
+Figure 17: An attempt to evade the encoder-decoder detection on symbols. The first row shows the disguises  $x_{d}$ , and the second row includes the output of the autoencoder  $\mathcal{D}(\mathcal{E}(x_d))$  that is unable to reveal the copyrighted information contained in the disguises. The third row shows the (unsuccessful) outcome of applying textual inversion to learn the hidden symbol.
+
+visual appearance of the base  $x_{b}$  and show that the images generated by textual inversion after training on  $x_{d}$  still contain the style of The Starry Night.
+
+![](images/8baf803ae31da941d1bc15ca667bd5be599d5717d92c4335ba52d374b92fbd07.jpg)  
+(1) San Giorgio Maggiore at Dusk
+
+![](images/f5aed6693cf83a11e0fbdcb0e7993bdb191bcc4211e198c55f5c0eded4038885.jpg)  
+(2) The Starry Night  
+Figure 18: Creating the copyrighted image  $x_{c}$  (third column) using AdaIN-based style transfer (Huang and Belongie 2017).  $x_{c}$  is San Giorgio Maggiore at Dusk by Claude Monet transferring to the style of The Starry Night by Vincent Van Gogh. Scraping the style of other artists: In the previous experiments, we choose the style of The Starry Night as the designated copyrighted style. Next, we show the results for disguised style scraping of another artist Cecily Brown<sup>10</sup>. In Figure 20, we
+
+![](images/07639f07334e1f92e37bda2d4e711ac396a5e44b2401f266f8d48e7c42962ec5.jpg)  
+$x_{c}$  : (1) with the style of (2)
+
+![](images/972e20e7342eecb23ab1e97ef93603181cebfc6910f8bae3eb8e1eefe3bb266e.jpg)  
+Base  $x_{b}$
+
+![](images/a92b6ac1dbcc28f4fe57d93e52ad76e49b87f45d203514b0e838b39d88dc4736.jpg)
+
+![](images/a5a4ea11e42a0f232f6c1532abc0813853879bff385457c52824adaefe67d53f.jpg)  
+$x_{c}$
+
+![](images/1f6ffe3656e0ed4dd3cbcfcbdad976e3523d491911d0b4008c8871f2f2256997.jpg)  
+Images generated by textual inversion after training on the  $x_{d}$
+
+![](images/e2c197b213a86936a0867d7c876a1dbbbba8631f1ecb3a6263040f70a6bf2bd4.jpg)  
+Disguise  $x_{d}$
+
+![](images/e9f6d9652edc859214136ada1146ba5625f8df05c704a01a37f3967b56efbc82.jpg)  
+Figure 19: Additional results on the disguised copyrighted style with textual inversion, where we generate disguise  $x_{d}$  that visually resembles the style of Claude Monet but containing the style of Vincent Van Gogh in the latent space.
+
+![](images/b369d354a09ce9a91b758b01985a770a824e34d6819268d399ae0032bbfd0a64.jpg)
+
+first create the  $x_{c}$  using style transfer (Huang and Belongie 2017), which is Taj Mahal (adamkaz/Getty Images) transferring to the style of the painting with serial number 56542 by Cecily Brown<sup>11</sup>. In Figure 21, we create the disguise  $x_{d}$  (row 1, column 2) with the visual appearance of the base  $x_{b}$  (row 1, column 1) and show that the images generated by textual inversion after training on  $x_{d}$  contain the style of the style of Cecily Brown.
+
+Finally, in Figure 22, we compare two different disguises  $x_{d}$  and observe images that visually look similar can contain drastically different styles in the feature space, which again demonstrates the threat of disguised copyright infringement.
+
+![](images/6d3fb5d3ba04f9e8d6ab57fe8d591187610059e8e575ab82f3b9e6f86b42df8b.jpg)  
+(1) Taj Mahal (watercolor)
+
+![](images/0799b31dfa615ea1bf204b55c427a7072a95b3cbf647c396b253e1ff0a77e900.jpg)  
+(2) Cecily Brown-56542
+
+![](images/7a128a220cbfc08cd63cfbaf066cacd77b83f34806395d3f80dce6b8d9b70bed.jpg)  
+$x_{c}$  : (1) with the style of (2)  
+Figure 20: Creating the copyrighted image  $x_{c}$  (third column) using AdaIN-based style transfer (Huang and Belongie 2017).  $x_{c}$  is Taj Mahal transferring to the style of Cecily Brown.
+
+Choice of  $x_{b}$ : In previous experiments, we construct the disguises  $x_{d}$  by choosing the base image  $x_{b}$  to have a different style from  $x_{c}$  but containing the same content. In Figure 23, we also explore the possibility of choosing a random base image from ImageNet (row 1, column 1). Our results show that although the style is partially learned by textual inversion from the disguise  $x_{d}$ , it also loses the structural information of  $x_{c}$ .
+
+![](images/80b37d66ddaec7f76acbe28e2e7255399586e60d0ebf32629507d3158fd90bb5.jpg)
+
+![](images/2d0280707c58d1918e100488d1797bf919782fbf2e75b6f14a9618f3edda3ad1.jpg)  
+Figure 21: Additional results on the disguised copyrighted style with textual inversion, where we generate disguise  $x_{d}$  that visually resembles the watercolor style but containing the style of Cecily Brown in the latent space.
+
+![](images/497df848a8631492f17a656bb14811898453908859cd2aa330524e0525a80282.jpg)  
+Taj Mahal in watercolor style
+
+![](images/6db57c1557b670673aef9619e183ef662b268cd7ca3896d893a9188d3c795e31.jpg)  
+Taj Mahal in The Starry Night style
+
+![](images/ad62acbdd55f4dfa553d5b61d036ad1bf080f7bece644abed2aaf30418db6732.jpg)  
+Taj Mahal in Cecily Brown style
+
+![](images/3d78459b95aab8ecf8a4851fb531e33328afa21b4da4595b06891c5281c32195.jpg)  
+Taj Mahal in watercolor style
+
+![](images/8fcf0f8782903214cf8e6590c5052f7a3217a66ac8c227de17bddaf218adbb93.jpg)  
+$x_{d}$  (The Starry Night style)
+
+![](images/60f554a698aa19ab2fb6cafac14f5cb0defb20708a27bda9f6342812b6a7f579.jpg)  
+$x_{d}$  (Cecily Brown style)  
+Figure 22: Comparison between different disguises  $x_{d}$  we generated that contain drastically different styles in the feature space, but are visually similar in the input space.
+
+![](images/97847b47ccaddc8544b9155ab5271cadbf625217712010675d2ebf149a1ecae8.jpg)
+
+![](images/59425ffaa904c9d33f9dc53cffc66aea51a1ea73315d00c12618c65d0f68ab25.jpg)  
+Base  $x_{b}$  
+Disguise  $x_{d}$  
+$x_{c}$  
+$\mathcal{D}(\mathcal{E}(x_d))$  
+Images generated by textual inversion after training on the  $x_{d}$  
+Figure 23: Additional results on the disguised copyrighted style with textual inversion, where we generate disguise  $x_{d}$  that visually resembles a random base image but intend to contain the style of The Starry Night in the latent space.
+
+# F. Disguises in the Wild
+
+We have previously revealed the concept hidden in the disguised data using textual inversion, which is also an LDM-based application that can possibly generate copyright materials. In this section, we further extend our evaluation to other LDM-based pipelines that change the parameters of the U-Net. Specifically, we consider DreamBooth (Ruiz et al. 2023), which fine-tunes a pre-trained LDM model with only a few disguised images (similar to textual inversion); and mixed-training that contains both clean data and a small portion of disguises.
+
+# F.1. Evaluation on DreamBooth
+
+Main results: In Section 4, we reveal the latent (copyrighted) information contained in the acquired samples  $x_{d}$  using textual inversion, where the diffusion model (i.e., the U-Net in LDM) is not retrained. To further demonstrate the effectiveness of the disguises  $x_{d}$  and examine disguised copyright infringement in other scenarios, we further perform evaluation on DreamBooth (Ruiz et al. 2023), an LDM-based fine-tuning method on a small set of images designed for subject-driven generation. Specifically, we choose five images from the DreamBooth dataset<sup>12</sup> as the (designated) copyrighted image  $x_{c}$ , and generate their corresponding disguises  $x_{d}$  using Algorithm 1 by choosing the noisy version of  $x_{c}$  as the base images  $x_{b}$ , training 10000 epochs and setting the weight parameter  $\alpha = 1000$ . After generating the disguises  $x_{d}$ , we apply DreamBooth for fine-tuning (on  $x_{d}$  only) and inference following the general recipe<sup>13</sup>. In Figure 24, we show the copyrighted images  $x_{c}$  in the first row, their corresponding disguises  $x_{d}$  in the second row, and images generated by DreamBooth by fine-tuning with the above  $x_{d}$  in the third row. Our results qualitatively confirm that copyrighted information in  $x_{c}$  can be reproduced by fine-tuning on  $x_{d}$  with DreamBooth.
+
+Evaluation on model utility: To further examine the practicality of the disguises, we want to demonstrate that the model utility is not depreciated after training (or fine-tuning) on  $x_{d}$ . To accomplish this task, we perform the same text-to-image generation task with the prompt "a photo of an astronaut riding a horse on Mars" before/after training on  $x_{d}$  with LDM. Specifically, the "before" model is Stable Diffusion v1-4, and the "after" model is the acquired model by fine-tuning the "before" model on the dog disguise (second row, first column in Figure 24). We show our results in Figure 25 for 6
+
+![](images/40e90f696680b4cea345b327c94f17481c95742ecffbd7321e70c87e851bc2d5.jpg)  
+Images generated by DreamBooth by fine-tuning with the above disguises
+
+![](images/e5696870ecd94dbd6e75a6d5e871e32aa24f50c210f11e14f2b582b670ddc09e.jpg)  
+Figure 24: We show the disguised copyright infringement with DreamBooth. The first row: the designated copyrighted image  $x_{c}$  from the DreamBooth dataset; the second row: the corresponding disguises  $x_{d}$  generated with our Algorithm 1; the third row: images generated by DreamBooth after training on the above disguises  $x_{d}$ .
+
+![](images/45224c302aa38740cc1656050be14d4c31c655b125f54acbbba4e484840b4b7d.jpg)  
+Images generated by LDM before fine-tuning with our disguises using DreamBooth  
+Images generated by LDM after fine-tuning with our disguises using DreamBooth  
+Figure 25: Utility validation: images generated by LDM using the prompt "a photo of an astronaut riding a horse on Mars" before/after fine-tuning with our disguises in Figure 24 (row 2, column 1) with DreamBooth.
+
+inferences and observe that the generation ability of the model is not compromised. We will show in the next section that this observation extends to LDM training.
+
+# F.2. Mixed-training on CelebA-HQ
+
+In the previous experiments, we have performed textual inversion and fine-tuning with DreamBooth where the disguises  $x_{d}$  are the entire training set. In this section, we extend our experiments to more challenging settings where the disguises  $x_{d}$
+
+![](images/606bf0a8ea1c2e1b930ff1eb7c32656ddde278a2d84379a511dd8babf8d030e4.jpg)  
+Target images  $x_{c}$  with the copyrighted symbol
+
+![](images/24ddd1c35b251b6628ee8f26fc08b3e62ecf680c3ef711f7742870ea500acbc7.jpg)
+
+![](images/77d810ca437f313202b7f723a78bdfc0c27102d44c2ab06801531a833dac360c.jpg)  
+Disguises  $x_{d}$
+
+![](images/e1f065c9e3421ccb1ce720d82fe3453d99ce8c80667d52d3bf705446cccc6c0c.jpg)
+
+![](images/64f70322db141877f5ef3d701353b24a0ab300ee77e4373764fd355331fd5855.jpg)  
+Images generated by LDM by training on clean images (1000) and disguises (100)  
+Figure 26: Disguised copyright infringement during mixed-training for unconditional generation on CelebA-HQ. The first row: the designated copyrighted image  $x_{c}$  with a red symbol "A"; the second row: the corresponding disguises  $x_{d}$ ; the third to fifth rows: images generated by LDM by training on 1000 clean images and 100 disguises (each disguise above contributes 20 copies). Images highlighted with a red box indicate reproductions of the copyrighted symbol.
+
+only constitute a small portion of the training set and are overwhelmed by the clean samples. Specifically, we randomly take a subset of 1000 images from CelebA-HQ/256 $^{15}$  as clean training samples, and another subset of 5 samples (outside of the clean set) and add a (designated) copyrighted symbol "A" to all of them to construct  $x_{c}$  (first row in Figure 26). Next, we construct the corresponding disguises  $x_{d}$  (second row in Figure 26) to visually remove the symbol using Algorithm 1 and duplicate each disguise 20 times to construct a disguise set containing 100 images. Overall, we acquire a training set of 1100 images where the disguise fraction is  $\frac{1}{11}$ . We initialize the model with the CompVis official model $^{16}$  to preserve reasonable generation performance and train the model on the mixed dataset for 200 DDIM steps. In Figure 26 (rows 3-5) we show the images generated by the model after training (inference for 20 times) and observe 3 out of 20 images contain the copyrighted symbol, which demonstrated the effectiveness of the disguises for mixed-training scenarios.

2024/Disguised Copyright Infringement of Latent Diffusion Models/images.zip

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2024/Disparate Impact on Group Accuracy of Linearization for Private Inference/65264f8f-bd85-42cb-bcb3-d9992ec3bcc1_origin.pdf

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2024/Disparate Impact on Group Accuracy of Linearization for Private Inference/full.md

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+# Disparate Impact on Group Accuracy of Linearization for Private Inference
+
+Saswat Das\*1 Marco Romanelli\*2 Ferdinand Fioretto
+
+# Abstract
+
+Ensuring privacy-preserving inference on cryptographically secure data is a well-known computational challenge. To alleviate the bottleneck of costly cryptographic computations in non-linear activations, recent methods have suggested linearizing a targeted portion of these activations in neural networks. This technique results in significantly reduced runtimes with often negligible impacts on accuracy. In this paper, we demonstrate that such computational benefits may lead to increased fairness costs. Specifically, we find that reducing the number of ReLU activations disproportionately decreases the accuracy for minority groups compared to majority groups. To explain these observations, we provide a mathematical interpretation under restricted assumptions about the nature of the decision boundary, while also showing the prevalence of this problem across widely used datasets and architectures. Finally, we show how a simple procedure altering the fine-tuning step for linearized models can serve as an effective mitigation strategy.
+
+# 1. Introduction
+
+Private Inference is the process of performing inference tasks on encrypted or private data, ensuring that the data remains confidential throughout the process. It has found application in ML settings where sensitive data is required for inference but should not be revealed to the model, for instance, when the model is owned by a cloud service provider and is not necessarily trusted. Although promising, cryptographic computations are notoriously computationally expensive when applied to nonlinear functions (Mishra et al., 2020), and, in the context of ML inference, Rectifier Linear Function (ReLU) activations are often identified as the pri
+
+![](images/7e4e04ea062d2a580addc7108bb44630dfc424d6bdeb3b7ea3b26ad4d411e50f.jpg)  
+Figure 1. ResNet18 accuracy on UTKFaces across various ReLU linearization budgets using SNL (Cho et al., 2022).  $100\%$  ReLUs corresponds to the original model. Subgroup sample sizes are shown above the corresponding bars. The accuracy for the majority group remains almost unchanged while other groups, in particular the minority one, are diversely impacted.
+
+mary cause of this computational burden (Cho et al., 2022; Jha et al., 2021; Kundu et al., 2023).
+
+To address this challenge, various approaches have proposed to approximate these non-linear activations using linear surrogates. The objective is to develop a new model that minimizes the number of ReLU activations while maintaining the highest possible accuracy. Over time, the proposed frameworks have evolved in sophistication—it has been demonstrated that not only the quantity of ReLU activations matters (Jha et al., 2021), but their placement within the network is also crucial (Cho et al., 2022). Recent methods have advanced to the point where they can manage both the ReLU budget and the distribution of linearized neurons across the network, resulting in faster inference times with minimal sacrifice in accuracy (Kundu et al., 2023).
+
+This paper builds on this body of work and observes that while these methods are indeed effective at trading run-time reduction for marginal global accuracy losses, these accuracy decreases are unevenly distributed among different subgroups. We find that the accuracy loss is more pronounced for underrepresented subgroups and that this impact intensifies as more ReLUs are linearized. This effect is depicted in Figure 1, which shows the accuracy impact across age groups on a facial recognition task. Remarkably, while the majority group (left-most) remains relatively un
+
+harmed, minorities exhibit considerable reduction. These results are far from "expected": they challenge the assumption that linearization uniformly affects the learning model and underscore the need for different approaches in model optimization, particularly in diverse datasets.
+
+Contributions. This paper makes the following contributions: (1) It observes, for the first time, a Matthew effect on the reduction of precision caused by the targeted linearization process adopted to reduce the number of ReLU functions in private inference methods. Further, it shows that such disparity is exacerbated as the number of approximated ReLU functions grows. (2) Next, it presents a mathematical interpretation of this phenomenon, which relates these disparity effects with the approximation capabilities of ReLU functions when assumptions on the space of the decision boundaries can be made. (3) It further shows that the effect of ReLU reduction on fairness is vastly present for commonly deployed algorithms and models when trained on unbalanced datasets. (4) Finally, the paper proposes a simple yet effective mitigation strategy that can be applied to any framework for ReLU reduction during the fine-tuning phase, showing favorable results in terms of fairness and overall accuracy reduction.
+
+# 2. Related Work
+
+Frameworks for ML Private Inference. The relationship between presence of ReLU activations and the inference time of a private inference model was first investigated by Mishra et al. (2020). The seminal solution proposed in this work is based on automatic generation of neural network architecture configurations, that navigates the performance-accuracy trade-offs over a given cryptographic protocol. Building on this intuition, Ghodsi et al. (2020); Lou et al. (2021) propose to selectively replace ReLU functions with, computationally more efficient, polynomial operations. DeepReDuce (DR) (Jha et al., 2021) takes this approach a step further by proposing a multi-step optimization, where portions of the network are replaced with linear functions, and then finetuned to recover the accuracy of the original model. The main drawback of these methods is the limited control over the "distribution" of the ReLU throughout the net. To tackle this issue, Selective Network Linearization (SNL) (Cho et al., 2022) proposes a custom activation node, where the linear and rectified behavior are parametrized and learned during training through a parametric mask. In line with this paradigm, the approach in Kundu et al. (2023) introduces a novel measure of nonlinearity sensitivity that helps reduce the need for manual efforts in identifying the best ReLU budget and placement.
+
+Piece-wise Approximation with ReLU Functions. In addition to observing disparate impacts from methods that reduce the number of ReLU functions in neural networks,
+
+our work seeks to explain these effects by linking them to the expressiveness and approximation capabilities of a Deep Neural Network (DNN). These studies trace back to seminal works in Cybenko (1989); Hornik (1991). The widespread use of ReLU activations to introduce non-linearity in the learning of decision boundaries has motivated early studies about their approximation power (Montúfar et al., 2014; Pan & Srikumar, 2016). Several papers have observed the effect of ReLU activations in the learning process (Hanin & Rolnick, 2019; Lin & Jegelka, 2018; Lu et al., 2017; Shen et al., 2022; Yarotsky, 2017). Among others, Arora et al. (2018); Liu & Liang (2021) distinguish themselves for contributions to the field, analyzing the connection between the number of ReLU activations and that of the approximated piecewise functions, along with a ReLU-dependent bound on the approximation error.
+
+Privacy and Fairness Tradeoff in Machine Learning. Among other works on the tradeoff between privacy and fairness in machine learning, Bagdasaryan et al. (2019) provides mostly empirical evidence of the disparate effect of differentially private training algorithms on fairness, while Farrand et al. (2020) further investigates this phenomenon by considering different levels of differential privacy and unbalance in the datasets. These works provide empirical evidence to show the noise injection and the gradient clipping involved in differentially private Stochastic Gradient Descent impact the overall accuracy, and especially that of underrepresented classes. The privacy framework we consider is extremely different, as the diverse impact is not a direct consequence of the privacy preservation framework design, but of the architectural adjustments made to speed up cryptographic secure inference. Such adjustments impact the approximation capabilities of the model resulting in disparate accuracy across different groups.
+
+Finally, model compression has also been shown to induce fairness issues. Empirical observations reported that quantization, network compression, and knowledge distillation could amplify the unfairness in different learning tasks (Lukasik et al., 2022; Hooker et al., 2020a;b; Joseph et al., 2020; Blakeney et al., 2021; Ahn et al., 2022). Inspired by these works, we show how ReLU linearization techniques may adversely affect the fairness of the resulting predictors.
+
+# 3. Settings
+
+The paper considers a dataset  $S_{T}$  consisting of  $N$  individual data points  $(x_{i},a_{i},y_{i})$ , with  $i\in [N]$  drawn i.i.d. from an unknown distribution  $\Pi$ . Therein,  $x_{i}\in \mathcal{X}$  is a feature vector,  $a_{i}\in \mathcal{A} = [M]$ , for some finite  $M$ , is a demographic group attribute, and  $y_{i}\in \mathcal{Y}$  is a C-class label  $(\mathcal{Y} = [C])$ . The goal is to learn a classifier  $f_{\theta}:\mathcal{X}\to \mathcal{Y}$ , where  $\pmb{\theta}$  is a vector of real-valued parameters. The classifier  $f$  is often defined through a soft classifier  $h_\theta :\mathcal{X}\rightarrow \mathbb{R}^C$ , that, for a
+
+given input  $\pmb{x}$ , produces scores for each label in  $\mathcal{V}$ , defining  $f_{\theta}(\pmb{x}) = \operatorname{argmax}_{j} h_{\theta}(\pmb{x}; j)$  where  $h_{\theta}(\pmb{x}; j)$  denotes the  $j$ -th component of  $h_{\theta}(\pmb{x})$ . The model quality is measured in terms of a loss function  $\ell: \mathcal{V} \times \mathcal{V} \to \mathbb{R}_+$  and the training minimizes the empirical risk function  $J_{(\theta; S_T)}$ :
+
+$$
+\boldsymbol {\theta} ^ {*} = \underset {\boldsymbol {\theta}} {\operatorname {a r g m i n}} J (\boldsymbol {\theta}; S _ {T}) = \frac {1}{| S _ {T} |} \sum_ {(\boldsymbol {x}, a, y) \in S _ {T}} \ell \left(h _ {\boldsymbol {\theta}} (\boldsymbol {x}), y\right).
+$$
+
+The paper studies the disparate impacts of classifiers  $f_{\theta}$  when it is subject to a ReLU partial reduction. The fairness notion employed is that of accuracy parity (Barocas et al., 2023), which holds when the classifier's misclassification rate is conditionally independent of the protected group. That is, for any  $\bar{a} \in \mathcal{A}$
+
+$$
+\Pr \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \neq y | a = \bar {a}\right) = \Pr \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \neq y\right).
+$$
+
+In other words, this property advocates for equal errors of the classifier on different subgroups of inputs. Empirically, it is measured by comparing the accuracy rates over an evaluation set  $S_{E}$  comprised of data points drawn from the same distribution  $\Pi$  as the training set.
+
+# 4. Why Network Linearization May Increase Unfairness?
+
+This section introduces theoretical insights to elucidate the causes behind the observed disparity effects resulting from the linearization of ReLU reduction. We start by recalling some fundamental results that will help us develop our analysis in the rest of the paper. The first results (Liu & Liang, 2021) defines a connection between the number of piecewise functions and the approximation error of convex functions. Throughout the section,  $\mathcal{F}$  denotes the set of strict convex univariate functions.
+
+Proposition 4.1. Consider a function  $f \in \mathcal{F}$  and let  $f_{n}^{\star}$  be its optimal approximation through a piecewise linear function with  $n$  segments. Then, the approximation error  $\Delta(f_{n}^{\star})$  is bounded by the number of its segments, and it decreases at a rate of  $O\left(\frac{1}{n^{2}}\right)$ .
+
+The second result (Arora et al., 2018) upper-bounds the number of piecewise linear functions that can be represented by a ReLU network.
+
+Proposition 4.2. Given a  $\text{ReLU} \mathbb{R} \to \mathbb{R}$ $DNN$  with  $k$  hidden layers, and layer widths  $\omega_1, \ldots, \omega_k$ , the number of attainable linear pieces is at most  $2^{k-1} \cdot (\omega_1 + 1) \cdot \omega_2 \cdot \dots \cdot \omega_k$ .
+
+This bound links the network's capability to approximate complex decision boundaries with the count of ReLU functions it contains (see Appendix B.1 for an illustrative example). Together with the previous result on the approximation error linked to a specific number of pieces (Proposition 4.1),
+
+![](images/932c8d47b1cc69db25dacd39ac89ea7840b3aac77b9e8d491decf615c4baf723.jpg)  
+Figure 2. Decision boundary estimated by a ReLU model (left) and a linearized model (right). The ground truth decision boundary (black dotted line) separates majority (blue) from minority (red) samples. The linearized model shows an inferior approximation of the decision boundary, and therefore a lower prediction accuracy. Moreover, its decision boundary is deeper into to the minority class resulting in accuracy disparities.
+
+it sets the stage for our subsequent analysis. We will illustrate that linearizing a number of ReLU functions not only leads to a potential decrease in accuracy when approximating nonlinear boundaries but it may also result in varying degrees of accuracy impact across distinct groups.
+
+We start with an illustrative experiment, detailed in Figure 2. It analyzes a dataset requiring a non-linear decision boundary for class separation (black dotted line), representing a protected group in fairness terms, with majority (blue) and minority (red) classes. We compare two DNN models with identical structures but different activation functions: a ReLU model and a modified model where half of the ReLU activations are replaced with linear activations, as shown in Figure 2a and Figure 2b, respectively. Notice that, after training both models from scratch, the ReLU model achieves better boundary approximation and accuracy (99.3% globally, 100.0% for the majority class, and 93.0% for the minority class), while the modified model shows comparable global accuracy (99.0%) but much lower fairness, indicated by reduced accuracy for the minority class (88.0%) compared to the original ReLU model.
+
+# 4.1. Decision Boundary Approximation and Fairness
+
+These insights indicate that there exists classification instances where the quantity of ReLU functions within a DNN enhances the approximation of decision boundaries, thereby improving prediction accuracy. Although empirical results often apply to much more complex settings than those which can be analyzed formally, they align with the forthcoming theoretical findings. In the following, we will use  $\theta$  to denote the parameters of the original ReLU network, which is assumed to have  $R$  ReLU functions, and  $\tilde{\theta}^r$  those associated to the linearized model, which retains  $r$  ReLU functions. We omit the superscript when it is sufficient to denote a network retaining  $1 \leq r < R$  ReLU functions.
+
+![](images/4026f1eda41cccfb08e0e42b36a2211fcb0a872b7e844373e9b6e01bb6868a83.jpg)  
+Figure 3. Grad Norms for classification on UTKFace with race labels using ResNet18 and SNL-based (Cho et al., 2022) ReLU linearization across various ReLU budgets. Subgroup sizes are reported on top of each bar group.
+
+Proposition 4.3. Let  $f \in \mathcal{F}$  be the optimal decision boundary for a classification task, and assume that  $\theta$  implements  $f_{n}^{\star}$ , i.e., the minimal error approximation. For a sample set  $S^{a} = \{(x,\bar{a},y) \sim \Pi |\bar{a} = a\}$  with data points drawn from  $\Pi$  containing exclusively members of group  $a \in \mathcal{A}$ , it holds
+
+$$
+R (a) = J (\tilde {\boldsymbol {\theta}}; S ^ {a}) - J (\boldsymbol {\theta}; S ^ {a}) \geq 0,
+$$
+
+for any  $a \in \mathcal{A}$ , where all the models are assumed to be trained to achieve their best possible performance.
+
+The above follows directly from the optimality of  $\theta$ . In other words, for this class of boundaries, any degree of ReLU linearization results in models with higher loss, within the class of decision boundaries considered.
+
+The next result is adapted from Tran et al. (2021) which exploits a similar approximation in the context of differentially private stochastic gradient descent. We upper bound the drop in loss for a given group  $a \in \mathcal{A}$  due to ReLU linearization by two key interpretable components: the gradient norm of the samples in group  $a$ , and the maximum eigenvalues of the Hessian of the loss function associated with such a group.
+
+Theorem 4.4. For a sample set  $S^a = \{(x, \bar{a}, y) \sim \Pi | \bar{a} = a\}$  with data points drawn from  $\Pi$  containing exclusively members of group  $a \in \mathcal{A}$ , the difference between the risk functions of some protected group  $a$  of a model  $\theta$  trained on a ReLU network and one  $\tilde{\theta}$  trained on a linearized ReLU network, is bounded by:
+
+$$
+\begin{array}{l} R (a) \leq \| \boldsymbol {g} _ {a} \| \times \| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \| + \frac {1}{2} \lambda (\boldsymbol {H} _ {a}) \times \| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \| ^ {2} \\ + O \left(\| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \| ^ {3}\right), \tag {1} \\ \end{array}
+$$
+
+where  $\pmb{g}_a = \nabla J(\pmb{\theta}; S_T^a)$  describes the gradient norm of samples in group  $a$ ,  $\pmb{H}_a^\ell = \nabla^2 J(\pmb{\theta}; S_T^a)$  is the Hessian of the loss function associated with group  $a$ , and  $\lambda(\Sigma)$  denotes the maximum eigenvalue of matrix  $\Sigma$ .
+
+![](images/c536f0fafd2438564117d52747f0a632f140ca7149a5b6cdc010a9274cc1b1d8.jpg)  
+Figure 4. ResNet-18: Highest eigenvalues of Hessians for UTK-Face with race labels obtained from the base model.
+
+The proof is based on the Taylor expansion of the loss function around the parameters  $\theta$  and is relegated to Appendix A.2. In the above, and throughout the paper, the gradients are understood to be over the model's parameters.
+
+There are two major takeaways from Proposition 4.2. For given parameter settings  $\theta$  and  $\tilde{\theta}$ , the first component of the bound is influenced by the gradient norm of the loss function for each group  $a \in \mathcal{A}$ . Informally, the gradient provide information on how close the model is to local optima; larger gradients suggest sub-optimality, while smaller ones indicate proximity to these optima. Typically, in converged models, underrepresented groups tend to have larger gradient norms, implying a greater distance from local optima (cf. Appendix A.1). This aspect is highlighted in Figure 3, where each shade represents a linearized model variant with varying counts of retained ReLU functions. Here, it's evident that gradient norms are higher for underrepresented groups, decreasing as more ReLU functions are retained. Next, the bound's second term is governed by the Hessian of the loss function for each group  $a \in \mathcal{A}$ . The Hessian reflects the local curvature, or sharpness, of the loss function. Generally, smaller Hessian eigenvalues suggest (with certain caveats) a flatter curve for the group loss, potentially indicating better generalizability to unseen samples.
+
+Proposition 4.5. Consider a binary classifier  $f_{\theta}$  trained using binary cross entropy loss. For any group  $a \in \mathcal{A}$  the maximum eigenvalue of the group Hessian  $\lambda(H_a)$  can be upper bounded by
+
+$$
+\lambda \left(\boldsymbol {H} _ {a} ^ {\ell}\right) \leq Z _ {1} + Z _ {2}, \tag {2}
+$$
+
+where,
+
+$$
+Z _ {1} = \frac {1}{\left| S ^ {a} \right|} \sum_ {\left(\boldsymbol {x}, a, y\right) \in S ^ {a}} \underbrace {h _ {\boldsymbol {\theta}} (\boldsymbol {x}) \left(1 - h _ {\boldsymbol {\theta}} (\boldsymbol {x})\right)} _ {\text {P r o x i m i t y t o d e c i s i o n b o u n d a r y}} \times \| \nabla h _ {\boldsymbol {\theta}} (\boldsymbol {x}) \| ^ {2},
+$$
+
+$$
+Z _ {2} = \underbrace {| f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y |} _ {E r r o r} \times \lambda \left(\nabla^ {2} h _ {\boldsymbol {\theta}} (\boldsymbol {x})\right).
+$$
+
+The proof (see Appendix A.2) relies on derivations of the Hessian associated with the model loss function and Weyl inequality with the notion of proximity to decision boundary is derived by (Cohen et al., 2019). As a consequence, groups with small Hessians eigenvalues (those generally distant from the decision boundary and highly accurate) tend to be less sensitive to the effects of the ReLU reduction rate. Conversely, groups with large Hessians eigenvalues tend to be affected by the ReLU reduction rate to a greater extent, typically resulting in larger excessive losses.
+
+Figure 4 illustrates the theoretical insights presented in Proposition 4.5: crucially, the value of  $\lambda\left(H_{a}^{\ell}\right)$  corresponding to the underrepresented groups are consistently higher than those corresponding to highly represented groups.
+
+We note that the bound in Theorem 4.4 depends also on the distance  $\|\tilde{\pmb{\theta}} - \pmb{\theta}\|$  between the ReLU reduced model and the original model. Generally, it is not trivial to establish a formal relationship between such distance and the rate of ReLU linearization; indeed such a formalization depends inherently on the model and the ReLU distribution induced by the linearization scheme adopted. However, our empirical analysis provides a strong sign for such trend to occur, as it will be further elaborated in Section 5.2, Figure 8.
+
+# 5. Empirical Results on the Fairness Analysis
+
+We next present the key findings from our empirical analysis on how ReLU linearization affects the fairness of linearized models. These findings build upon and extend the theoretical insights from earlier sections to scenarios involving high-dimensional and non-convex decision boundaries. In summary, we show that (1) model linearization produces disparate degradation of performance across groups and that such disparity is enhanced with the larger amounts of linearization; (2) the relative magnitude of gradient norms for a group serves as a reliable indicator of the group's accuracy decline resulting from ReLU linearization; (3) groups located farther from (or closer to) the decision boundary are less (more) likely to experience accuracy reduction at varying levels of ReLU linearization<sup>1</sup>.
+
+Datasets and Models. We adopt three datasets:
+
+- UTKFace (Zhang et al., 2017). Containing over 20,000 face images with annotations for age, gender, and race is widely used for computer vision tasks. Our experiments use age and race as protected groups.  
+- SVHN (Digits) (Netzer et al., 2011). Containing 60,000  $32 \times 32$  RGB images of digits of house number plates.  
+- CIFAR-10 (Krizhevsky & Hinton, 2009). Containing 60,000  $32 \times 32$  RGB images of 10 classes (airplanes, cars,
+
+![](images/66559d2a2953ccbe2f346dc6e16a9829008c3a1bcf5920b0d85571e7a9bf690c.jpg)  
+(a) SNL linearization
+
+![](images/512c7ea40738224175a2fbc4871f5305cb33255f8ae3e43b505f271743645d65.jpg)  
+(b) DR linearization  
+Figure 5. Global test accuracy on UTKFace with age labels using ResNet18 for SNL and DR. The red horizontal line represents the original (no ReLU linearization) global test accuracy.
+
+birds, cats, deer, dogs, frogs, horses, ships, and trucks).
+
+To assess the effect of network linearization on groups with various representations, we evaluate several models and architectures on unbalanced datasets. While UTKFace and SVHN are naturally unbalanced, we unbalanced CIFAR-10 by selecting  $90\%$  of the samples for each of 5 classes, and retaining  $10\%$  of the samples for each of the remaining classes. All reported metrics are average over 10 random seeds. The UTKFace dataset includes images with a large variation in age, ranging from 0 to 116 years old. As customary in demography, the ages are categorized into four groups: 2-12 (young children), 13-17 (adolescents), 18-64 (working-age individuals), and over 64 (senior citizens) for the experiments in this paper and these groups contain, respectively,  $10.14\%$ ,  $3.63\%$ ,  $77.56\%$ , and  $8.67\%$  of the total samples in the dataset. Data points corresponding to ages 0 and 1 are considered outliers and thus omitted for this task. Race classification on UTKFace utilizes the dataset's provided labels: white, black, Asian, native Indian, and others which correspond to  $42.51\%$ ,  $19.09\%$ ,  $14.49\%$ ,  $16.77\%$ , and  $7.14\%$  of the total samples in the dataset, respectively. Finally, digit recognition studies are conducted on the naturally imbalanced SVHN dataset. This dataset contains 10 digits from 0 to 9 which have  $6.75\%$ ,  $18.92\%$ ,  $14.45\%$ ,  $11.60\%$ ,  $10.18\%$ ,  $9.39\%$ ,  $7.82\%$ ,  $7.64\%$ ,  $6.89\%$ , and  $6.36\%$  of the total samples in the dataset, respectively
+
+In this analysis two popular architectures are considered: ResNet18 and ResNet34 (He et al., 2015). These architectures are the main ones used in the linearization frameworks of (Jha et al., 2021; Cho et al., 2022).
+
+Linearization Methods. The paper considers two ReLU linearization techniques, representing the current state of the art: Selective Network Linearization (SNL) (Cho et al., 2022) and DeepReDuce (DR) (Jha et al., 2021). Both methods aim to linearize ReLUs in neural networks, but their approaches differ: SNL selectively linearizes specific ReLUs, whereas DR provides for a much coarser linearization strategy. In particular, DR does not allow to control the
+
+![](images/985dac5cacbc609d392a00a6843efb81363a5a9e3b7c1be6ab68fa7eda1660e8.jpg)
+
+![](images/15aa4e2f8284eeabf0b9ca2dd3dfb0c3501d4a1253a5324cc6d547f433cd7aed.jpg)
+
+![](images/bc82c6992fe623ce98cefd48ab89b8b80116a085907bf104a5809614399748ea.jpg)
+
+![](images/d7a629c20e6c09cf452835fc547c3b7d47b6551eae23ef4cd85f1d597dcf7c1c.jpg)
+
+![](images/22f60e6f445a12a2bdcc95e84934c901bfc8516d78b851e48a30b9e17c272bdd.jpg)  
+Figure 6. SNL: Relative test accuracy drop for different datasets, models, and ReLU linearization methods. 1st col: ResNet18 trained on SVHN (top) and CIFAR-10 (bottom) with SNL; 2nd col: ResNet18 trained on UTKFace age (top) and race (bottom) labels with SNL; 3rd col: ResNet34 trained on UTKFace age (top) and race (bottom) labels with SNL; 4th col: ResNet18 trained on UTKFace age (top) and race (bottom) labels with DR; The results show that performance on minority groups is affected disparately by ReLU linearization across choices of datasets, model architectures, and linearization methods.
+
+![](images/9f667456ae32d366393b4910b6061bf159c26baccc6deeb0d64dde8bf9f10579.jpg)
+
+![](images/7a1cfd2a6347121d7be60f4062a8db3ac356f2b6655724e96e6631c2ec851660.jpg)
+
+![](images/3ceeb39db42543d3b3a18a3ff6dfcb3702109bbf29bebc1e5927cde85922eba3.jpg)
+
+"distribution" of the ReLU functions across the architecture, but only to linearize blocks of contiguous nodes. This fundamental difference is expected to result in distinct patterns of disparity observed with each method. As a matter of fact, DR is not inherently data-driven, and its performance is not only affected by the number of residual ReLU functions, but also by the block that the practitioner decides to linearize. In turn, this results in different linearized model configurations as reported in Figure  $5\mathrm{b}^2$
+
+In the following, the term "ReLU budget" is used to refer to the number of ReLUs retained in a ReLU-linearized model across both methods. However, note that specifying a desired ReLU budget is feasible only with SNL. In contrast, with DR, the number of retained ReLUs depends on the specific layers linearized. Additionally, while ReLU linearization may lead to a loss in utility, notice that the models and ReLU budgets used for the analysis yield a minor decrease in global test accuracies (as shown in Figure 5) indicating that these configurations represent practical, deployable ReLU-linearized models.
+
+# 5.1. Impact on Accuracy
+
+In the primary empirical findings, the study highlights the relative decreases in test accuracy for various groups under
+
+different ReLU budgets, as shown in Figure 6. For datasets with more classes like CIFAR-10 and SVHN, the analysis focuses on the two most and two least represented classes in the test sets to maintain clarity and prevent plot overcrowding. The relative drop in test accuracy serves as a measure of accuracy loss due to ReLU linearization. The larger this value, the greater the performance decline. Let  $\bar{J} (\pmb {\theta};S) = 1 / |S|\sum_{(\pmb {x},y)\in S}\mathbb{1}[f_{\pmb{\theta}}(\pmb {x}) = y]$ , measure the classifier  $\pmb{\theta}$  accuracy over set  $S$ , then the relative drop in test accuracy for group  $a$  is measured as:
+
+$$
+\frac {\bar {J} (\boldsymbol {\theta} ; S ^ {a}) - \bar {J} (\tilde {\boldsymbol {\theta}} _ {r} ; S ^ {a})}{\bar {J} (\boldsymbol {\theta} ; S ^ {a})} \times 1 0 0.
+$$
+
+The results reveal a consistent trend: majority groups generally show resilience to ReLU linearization while minority groups experience increasingly significant accuracy losses with more linearization. This pattern holds true across various architectures and datasets. For example, using SNL on ResNet-34 trained on UTKFace with race labels, white individuals experience almost no impact in accuracy drop across different ReLU linearization budgets, while the "other" race group suffers an almost 5-fold accuracy drop. Similarly, for DR on ResNet-18 trained on the same dataset, the accuracy for white individuals remains almost unaffected while the "other" race group suffers an almost 10-fold accuracy drop, with a near  $100\%$  accuracy drop for the lowest ReLU budget. Indeed, a similar observation holds for DR for UTKFace with age labels.
+
+Recall that Theorem 4.4 highlighted the connection between a drop in loss (as a differentiable proxy for accuracy) experienced by a given group, under a certain ReLU linearization
+
+level, and two key characteristics: the gradient norms associated with such groups at convergence and the distance to the decision boundary. While these results hold for a restricted classes of functions, the next two sections provide further evidence on such relationships on highly non linear models for a variety of linearization techniques and architectures. Further experiments on different architectures and wider variants of the ResNet models are reported in Figure 7.
+
+# 5.2. Gradient Norms and Fairness
+
+Firstly, we notice Theorem 4.4 highlights how both the group gradient and the group Hessian terms are multiplicatively influenced by the distance  $\|\tilde{\theta} - \theta\|$  between the model with reduced ReLU functions and the original model. Figure 8 (left) illustrates that this distance increases with the increase in linearized ReLU functions, suggesting that the reduction in ReLU activations amplifies the effect on the gradient norm. Next, Figure 8 (middle) presents a scatterplot that contrasts accuracy against gradient norms at the highest level of linearization examined. It reveals an inverse relationship between gradient norms and accuracy: data points for the majority group align with lower gradient norms and higher accuracy, whereas points for minority groups are linked with higher gradient norms and lower accuracy. Taken together, the increasing distance between reduced and original model parameters, alongside the varied impact on gradient norms across different groups, highlight the dynamics of gradient norms as a factor explaining the exacerbation of unfairness post-ReLU linearization.
+
+# 5.3. Distance to the Decision Boundary and Fairness
+
+Next, we look into the influence of the group Hessian on the accuracy reduction resulting from ReLU linearization. Recall that Proposition 4.5 establishes a link between the group Hessian and the distance to the decision boundary: notably, a smaller distance (or more proximity) to the decision boundary correlates with higher bounds for the associated group Hessian. This relationship renders the distance to the decision boundary an insightful, interpretable metric. Figure 8 (right) elucidates this concept. It illustrates the relationship between each test sample's distance to the decision boundary and its accuracy. The data points are color-coded according to group membership, revealing a strong correlation between the distance to the boundary and accuracy levels. Specifically, groups identified as majorities tend to have a larger distance to the boundary (and high accuracy), whereas minority groups exhibit a smaller distance.
+
+This observation, in conjunction with what observed in Figure 8 (left), underscores the key relation between distance to the decision boundary plays and fairness issues following ReLU linearization. Understanding of this interplay is crucial to design the proposed mitigation technique, reviewed
+
+in the next section.
+
+# 6. Mitigation and Impact on Fairness
+
+So far, we have presented an in-depth analysis of the disparate effects that linearization techniques produce on the fairness of unbalanced datasets, showing how reduced ReLU budgets negatively affect the accuracy over underrepresented groups. We will now present a solution to extend ReLU linearization to contexts where fairness matters along with Private Inference. Recall from Proposition 4.5 the relationship between a group's Hessian eigenvalues, their proximity to the decision boundary, and the corresponding errors made by the model for that group. Since, as shown in the previous section, the distance to the decision boundary of a sample directly relates to their accuracy drop across various ReLU linearization rates, our strategy leverages this intuition and introduces "fairness regularizers" to equalize the distances to the decision boundary of various groups. These regularizers, capture the differences in losses between groups and population, (and by proxy the distance to the boundary). The method leverages Lagrangian duality principles (Fioretto et al., 2020) for implementation and it works off-the-shelf with the Knowledge-Distillation (KD) based finetuning step, offering a seamless integration into existing algorithms.
+
+Algorithm 1 reports the proposed fairness-aware finetuning method, replacing the standard finetuning step of SNL and DR. Let us introduce the main notation. Therein,  $\mathcal{L}^{\mathrm{KD}}: \mathbb{R}^C \times \mathbb{R}^C \times \mathbb{R}^d \to \mathbb{R}$  is the loss function associated with the KD finetuning, which, in addition to a batch of data  $B$ , takes as input two classifiers  $h_{\tilde{\theta}}^r$  and  $h_\theta$ , and returns a scalar;  $\lambda_e$  is the vector of Lagrangian multipliers, each associated to a fairness constraint violation, expressed as the distance between the loss computed over a group and that computed over the whole dataset. These multipliers are updated at each epoch  $e$ ;  $\mu$  is the multiplier associated with the Lagrangian update step; and  $\alpha$  is the learning rate. Furthermore, we call  $\mathcal{L}_a^{\mathrm{KD}}$  the loss function, when evaluated over the samples of a specific group  $a \in \mathcal{A}$ . Crucially, the proposed algorithm allows the student model, i.e., the linearized one, to be trained with knowledge flowing from the original model, while at the same time being constrained against excessive degradation of performance over underrepresented groups.
+
+Let  $\bar{J} (\tilde{\pmb{\theta}}^r;S) = 1 / |S|\sum_{(\pmb {x},\pmb {y})\in S}\mathbb{1}\left[f_{\tilde{\pmb{\theta}}^r}(\pmb {x}) = \pmb {y}\right]$ , measure the classifier  $\theta$  accuracy over set  $S$ , then the relative drop in test accuracy for group  $a$  is measured as:
+
+$$
+\frac {\bar {J} \left(\tilde {\boldsymbol {\theta}} ^ {r} ; S ^ {a}\right) - \bar {J} \left(\tilde {\boldsymbol {\theta}} _ {m i t} ^ {r} ; S ^ {a}\right)}{\bar {J} \left(\tilde {\boldsymbol {\theta}} ^ {r} ; S ^ {a}\right)} \times 1 0 0,
+$$
+
+where, with a slight abuse of notation,  $\tilde{\pmb{\theta}}^r$  represents the model parameter before the mitigation, and,  $\tilde{\pmb{\theta}}_{mit}^{r}$  the parameter of the corresponding linearized model finetuned with our mitigation algorithm.
+
+![](images/bfd97d16ed815e1b73f7a8d55f3b805a09f09214d26e1d17e315fcbfbac26471.jpg)  
+(a) SNL on UTKFace (race) for (b) SNL on UTKFace (age) for (c) SNL on UTKFace (race) for (d) SNL on UTKFace (age) for VGG-19. VGG-19. WideResNet. WideResNet.
+
+![](images/31b8f61629e158d9f41282e6cd607df722b0e369dbefb17ff7ebc4d1cbc87bc8.jpg)
+
+![](images/be3c94e9f60cc97df8a9da4177bc8d1d9bf6e74bfff52cdee44a464ca628434c.jpg)
+
+![](images/1c93c14bfc4c094b8f3e906033aa1943514916593310d7cbc7a3635ab987482e.jpg)
+
+![](images/8369e4398574891a75270948e8c824b0bec733a7f97b6392e5d151dd560cf741.jpg)  
+Figure 7. Results for VGG-19 and WideResNet models on UTKFace with SNL. We observe that the performance is consistent with the results obtained for the ResNet18 and ResNet34 models: more linearization produces higher accuracy loss for underrepresented groups, while the accuracy remains almost unchanged in the majority group. We observe that the phenomenon of disparity in the accuracy for underrepresented groups is evident on WideResNet-22-8, a wider variant of ResNet.
+
+![](images/b1f181cee4f2d5f9e70b35adde88d6139053912c74e431abe9bb8a2a06dfff33.jpg)  
+Figure 8. SNL on UTKFace with race labels and ResNet-18: left:  $\|\tilde{\pmb{\theta}}^r - \pmb{\theta}\|$  vs. ReLU budget; middle: group accuracy vs. gradient norm for  $2\%$  ReLUs retained; right: group accuracy vs. distance to decision boundary for  $2\%$  ReLUs retained.
+
+![](images/2288875ed7aa4776ad79f6a0d09d19ee2f2e03f040eb5b1407288b9e5c9f3207.jpg)
+
+# Algorithm 1 Fairness Mitigation for ReLU Linearization
+
+Require:  $h_{\tilde{\theta}^r}, h_\theta$ , Multiplier  $\mu > 0$ , step size  $\eta$ , loss function  $\mathcal{L}^{\mathrm{KD}}$ , group-wise loss function  $\mathcal{L}_a^{\mathrm{KD}}$ , dataset  $S_T$ , set of groups  $\mathcal{A}$
+
+$\lambda_{1}\gets (0)_{a\in \mathcal{A}}$  {array of zeros with size  $|\mathcal{A}|\})$
+
+for epoch  $e = 1,2,\dots$  do
+
+for all mini-batch  $B \subseteq S_T$  do
+
+$$
+\rho_ {\tilde {\boldsymbol {\theta}} ^ {r}, \boldsymbol {\theta}, B} \leftarrow | \mathcal {L} ^ {\mathrm {K D}} (f _ {\tilde {\boldsymbol {\theta}} ^ {r}}, f _ {\boldsymbol {\theta}}, B) - \mathcal {L} _ {a} ^ {\mathrm {K D}} (f _ {\tilde {\boldsymbol {\theta}} ^ {r}}, f _ {\boldsymbol {\theta}}, B) |
+$$
+
+$$
+\tilde {\boldsymbol {\theta}} ^ {r} \leftarrow \tilde {\boldsymbol {\theta}} ^ {r} - \alpha \nabla_ {\tilde {\boldsymbol {\theta}} ^ {r}} \left(\mathcal {L} \left(f _ {\tilde {\boldsymbol {\theta}} ^ {r}}, f _ {\boldsymbol {\theta}}, B\right) + \boldsymbol {\lambda} _ {e} ^ {\top} \rho_ {\tilde {\boldsymbol {\theta}} ^ {r}, \boldsymbol {\theta}, B}\right)
+$$
+
+end for
+
+$$
+\boldsymbol {\lambda} _ {e + 1} \leftarrow \boldsymbol {\lambda} _ {e} + \mu | \mathcal {L} ^ {\mathrm {K D}} (f _ {\tilde {\theta} ^ {r}}, f _ {\boldsymbol {\theta}}, S _ {T}) - \mathcal {L} _ {a} ^ {\mathrm {K D}} (f _ {\tilde {\theta} ^ {r}}, f _ {\boldsymbol {\theta}}, S _ {T}) |
+$$
+
+end for
+
+Figure 9 illustrates the effect of the mitigation strategy on the UTKFace dataset for both SNL (left) and DR (right) methods, with results segmented by age (top) and race (bottom) labels. Negative values indicate accuracy improvements for a group compared to the baseline accuracy of the original model at the same level of ReLU reduction. There are two key aspects: First, the mitigation consistently enhances accuracy across all groups and linearization levels tested, with minimal negative effects on majority classes. Second, notice how the minority groups improve their relative performance, a trend that is especially evident for DR at higher levels of
+
+ReLU linearization. Furthermore, there is an observable reduction in gradient norms and an augmentation in the distance to the decision boundary for minority groups under the mitigation approach (see Appendix D.1). These results illustrate the potential for effective mitigations to significantly improve the performance for minority groups without detriment to majority groups, thus substantially reducing the unfairness produced by ReLU linearization.
+
+Finally, an ablation study on the impact of the parameter  $\mu$  in the mitigation algorithm is presented in Appendix D.2.
+
+# 7. Discussion
+
+Figure 9 demonstrates the effectiveness of the mitigation method, highlighting the importance of choosing the right approach for practitioners aiming to enforce fairness predictably. A data-driven ReLU linearization method like SNL, which carefully considers the impact of linearization on model utility and fairness, stands out as preferable over simpler methods such as DR. In DR, ReLUs may be linearized arbitrarily across any layer without regard for potential impacts on the model performance.
+
+This is evident in Figure 6, where the smallest groups under DR exhibit significantly poorer performance compared to
+
+![](images/f1ecada93b08312942971a561af806f456be4250ca00785769b18b60c16d02b0.jpg)
+
+![](images/6524efddd02b4ffb0aa145441a67b83ea4513990ac66faa702225b828c533520.jpg)
+
+![](images/372552f04c255093be5771c83b138d32c231bae2bb13a9e40c46547a060611e2.jpg)  
+Figure 9. Mitigation: Relative test accuracy drop for different linearization methods. left: ResNet18 trained on UTKFace age (top) and race (bottom) labels with SNL; right: ResNet18 trained on UTKFace age (top) and race (bottom) labels with DR.
+
+![](images/e43add487de44800c3d029e5fbd190bf97bb72b4a01e4553ed1f97e0d08e21ea.jpg)
+
+others and when SNL is applied. The unpredictable utility and fairness outcomes with DR make it a less suitable option for achieving fair models predictably. In contrast, SNL offers consistent and predictable outcomes, enabling practitioners to more easily apply fair ReLU linearization and select suitable parameters for effective mitigation.
+
+Limitations. Our theoretical framework proposes a bound for the residual loss of a DNN model and its linearized counterpart, elucidating their relationship with group norms, group Hessians, and the norm of the weight distance between the two models' parameters. While our analysis introduces proxies to articulate the connection between group size, linearized ReLU proportion, and accuracy drop, examining the actual relationship between these quantities remains an open question. Addressing this gap presents an avenue for future research, with an opportunity to formalize fundamental limits of disparate impact mitigation strategies through non-trivial lower bounds on loss differences. We note that the proposed mitigation strategy necessitates access to protected group information. Although this information is frequently available, the requirement may restrict the applicability of the proposed mitigation in contexts where such information may be missing. To address this limitation, future research could explore the integration of our mitigation approach with tailored loss functions to promote fairness without relying on demographic information (cf. Lahoti et al. (2020)) or with a context of distributionally robust objectives (cf. Hashimoto et al. (2018)).
+
+# 8. Conclusion
+
+This study observed that while ReLU linearization strategies effectively reduce computational costs and inference times, they inadvertently exacerbate accuracy disparities across different subgroups, particularly affecting underrepresented ones. This phenomenon was found consistent across various datasets and architectures. The empirical analysis was grounded in theoretical insights, which highlighted two key factors responsible for the observed unfairness. We showed that as ReLU functions are approximated, the gradient for underrepresented groups tend to increase, suggesting that the model becomes more sensitive to input variations for these groups, leading to a higher misclassification rates. This effect is exacerbated for underrepresented groups due to their typically smaller distance from the decision boundary in the space that these models operate within. This distance to the boundary, when combined with altered gradient norms due to ReLU linearization, results in a disproportionate impact on the fairness of the model's outcomes.
+
+Motivated by such observations, the paper proposed a mitigation solution which acts on regularizing the finetuning step of ReLU linearization strategies through a Lagrangian dual approach. This simple solution was found effective in balancing the computational benefit of ReLU linearization with the imperative of fairness.
+
+# Acknowledgements
+
+This research is partially supported by NSF grants 2133169, 2232054 and NSF CAREER Award 2143706. Fioretto is also supported by an Amazon Research Award and a Google Research Scholar Award. Its views and conclusions are those of the authors only.
+
+# Impact Statement
+
+Our contributions shed light on the Matthew effect within accuracy reduction caused by ReLU linearization, offering a mathematical intuition of the observed disparities, demonstrating the widespread presence of these effects across algorithms and models, and proposing a viable mitigation strategy. The proposed strategy not only addresses fairness concerns but also manages to preserve overall accuracy to a commendable extent.
+
+The evidence presented, particularly the stark accuracy disparities illustrated in Figure 1, calls for a reevaluation of model optimization strategies, especially in the use of datasets reflecting real-world diversity. The findings emphasize the necessity for approaches that consider the differential impact of linearization on various subgroups, advocating for a more inclusive and fair machine learning ecosystem.
+
+# References
+
+Ahn, J., Lee, H., Kim, J., and Oh, A. Why knowledge distillation amplifies gender bias and how to mitigate from the perspective of stilbert. Proceedings of the 4th Workshop on Gender Bias in Natural Language Processing (GeBNLP), 2022.  
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+
+# A. Additional Theoretical Results
+
+# A.1. Underrepresented groups have larger gradients.
+
+Let us assume  $a, b \in \mathcal{A}$  such that with  $|S^a| \geq |S^b|$ . Then  $\left\| \boldsymbol{g}_a^\ell \right\| \leq \left\| \boldsymbol{g}_b^\ell \right\|$ .
+
+Proof. Let us assume convergence of the model with parameters  $\phi$  to a (local) minimum. Then, it holds that:
+
+$$
+\begin{array}{l} \nabla \mathcal {L} (\phi ; S) = \sum_ {a \in \mathcal {A}} \frac {| S ^ {a} |}{| S |} \nabla J (\phi ; S ^ {a}) \\ = \frac {| S ^ {a} |}{| D |} \boldsymbol {g} _ {a} ^ {\ell} + \frac {| S ^ {b} |}{| D |} \boldsymbol {g} _ {b} ^ {\ell} = \mathbf {0} \\ \end{array}
+$$
+
+Thus,  $\pmb{g}_a^\ell = -\frac{|S^b|}{|S^a|}\pmb{g}_b$ . Hence  $\left\| \pmb{g}_a^\ell \right\| = \frac{|S^b|}{|S^a|} \left\| \pmb{g}_b^\ell \right\| \leq \left\| \pmb{g}_b^\ell \right\|$ , because  $|S^a| \geq |S^b|$ .
+
+![](images/429ac98f9210844d31ef90fa8022079d70067dad67a9a8f6f624d14b484c841e.jpg)
+
+# A.2. An upper-bound for the residual loss.
+
+Theorem 1. The excessive loss of a group  $a \in \mathcal{A}$  is upper bounded by
+
+$$
+R (a) \leq \left\| \boldsymbol {g} _ {a} ^ {\ell} \right\| \times \| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \| + \frac {1}{2} \lambda \left(\boldsymbol {H} _ {a} ^ {\ell}\right) \times \| \tilde {\boldsymbol {\theta}} - \hat {\boldsymbol {\theta}} \| ^ {2} + O \left(\| \tilde {\boldsymbol {\theta}} - \hat {\boldsymbol {\theta}} \| ^ {3}\right),
+$$
+
+where  $\pmb{g}_a^\ell = \nabla J(\pmb{\theta}; S^a)$  is the vector of gradients associated with the loss function  $\ell$  evaluated at  $\pmb{\theta}$  and computed using group data  $S^a$ ,  $\pmb{H}_a^\ell = \nabla^2 J(\pmb{\theta}; S^a)$  is the Hessian matrix of the loss function  $\ell$ , at the optimal parameters vector  $\pmb{\theta}$ , computed using the group data  $S^a$  (henceforth simply referred to as group hessian), and  $\lambda(\Sigma)$  is the maximum eigenvalue of a matrix  $\Sigma$ .
+
+Proof. Using a second order Taylor expansion around  $\theta$ , the excessive loss  $R(a)$  for a group  $a \in \mathcal{A}$  can be stated as:
+
+$$
+\begin{array}{l} R (a) = J \left(\tilde {\boldsymbol {\theta}}; S ^ {a}\right) - J \left(\boldsymbol {\theta}; S ^ {a}\right) \\ = \left[ J (\boldsymbol {\theta}; S ^ {a}) + (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) ^ {\top} \nabla J (\boldsymbol {\theta}; S ^ {a}) + \frac {1}{2} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) ^ {\top} \boldsymbol {H} _ {a} ^ {\ell} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) + O \left(\| \boldsymbol {\theta} - \tilde {\boldsymbol {\theta}} \| ^ {3}\right) \right] - J (\boldsymbol {\theta}; S ^ {a}) \\ = (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) ^ {\top} \boldsymbol {g} _ {a} ^ {\ell} + \frac {1}{2} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) ^ {\top} \boldsymbol {H} _ {a} ^ {\ell} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) + O \left(\| \boldsymbol {\theta} - \tilde {\boldsymbol {\theta}} \| ^ {3}\right) \\ \end{array}
+$$
+
+The above, follows from the loss  $\ell(\cdot)$  being at least twice differentiable, by assumption. By Cauchy-Schwarz inequality, it follows that
+
+$$
+\left(\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}\right) ^ {\top} \boldsymbol {g} _ {a} ^ {\ell} \leq \left\| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \right\| \times \left\| \boldsymbol {g} _ {a} ^ {\ell} \right\|.
+$$
+
+In addition, due to the property of Rayleigh quotient we have:
+
+$$
+\frac {1}{2} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) ^ {\top} \boldsymbol {H} _ {a} ^ {\ell} (\tilde {\boldsymbol {\theta}} - \boldsymbol {\theta}) \leq \frac {1}{2} \lambda \left(\boldsymbol {H} _ {a} ^ {\ell}\right) \times \| \tilde {\boldsymbol {\theta}} - \boldsymbol {\theta} \| ^ {2}
+$$
+
+# A.3. An upper-bound for the maximum eigenvalue of the group Hessian
+
+Let  $f_{\theta}$  be a binary classifier trained using a binary cross entropy loss. For any group  $a \in \mathcal{A}$ , the maximum eigenvalue of the group Hessian  $\lambda(H_a^\ell)$  can be upper bounded by:
+
+$$
+\lambda \left(\boldsymbol {H} _ {a} ^ {\ell}\right) \leq \frac {1}{| S ^ {a} |} \sum_ {(\boldsymbol {x}, y) \in S ^ {a}} \underbrace {\left(h _ {\boldsymbol {\theta}} (\boldsymbol {x})\right) (1 - h _ {\boldsymbol {\theta}} (\boldsymbol {x}))} _ {\text {P r o x i m i t y t o d e c i s i o n b o u n d a r y}} \times \| \nabla h _ {\boldsymbol {\theta}} (\boldsymbol {x}) \| ^ {2} + \underbrace {| f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y |} _ {\text {A c c u r a c y}} \times \lambda \left(\nabla^ {2} h _ {\boldsymbol {\theta}} (\boldsymbol {x})\right).
+$$
+
+Proof. First notice that an upper bound for the Hessian loss computed on a group  $a \in \mathcal{A}$  can be derived as:
+
+$$
+\lambda \left(\boldsymbol {H} _ {a} ^ {\ell}\right) = \lambda \left(\frac {1}{| S ^ {a} |} \sum_ {(\boldsymbol {x}, y) \in S ^ {a}} \boldsymbol {H} _ {x} ^ {\ell}\right) \leq \frac {1}{| S ^ {a} |} \sum_ {(\boldsymbol {x}, y) \in S ^ {a}} \lambda \left(\boldsymbol {H} _ {x} ^ {\ell}\right) \tag {3}
+$$
+
+where  $H_{\pmb{x}}^{\ell}$  represents the Hessian loss associated with a sample  $\pmb{x} \in S^{a}$  from group  $a$ . The above follows Weily's inequality which states that for any two symmetric matrices  $A$  and  $B$ ,  $\lambda (A + B) \leq \lambda (A) + \lambda (B)$ . Next, we will derive an upper bound on the Hessian loss associated to a sample  $\pmb{x}$ . First, based on the chain rule a closed form expression for the Hessian loss associated to a sample  $\pmb{x}$  can be written as follows:
+
+$$
+\pmb {H} _ {\pmb {x}} ^ {\ell} = \nabla^ {2} \ell \left(f _ {\pmb {\theta}} (\pmb {x}), y\right) \left[ \nabla f _ {\pmb {\theta}} (\pmb {x}) \left(\nabla f _ {\pmb {\theta}} (\pmb {x})\right) ^ {\top} \right] + \nabla \ell \left(f _ {\pmb {\theta}} (\pmb {x}), y\right) \nabla^ {2} f _ {\pmb {\theta}} (\pmb {x}).
+$$
+
+The next follows from that
+
+$$
+\nabla \ell \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}), y\right) = \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y\right),
+$$
+
+$$
+\nabla^ {2} \ell \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}), y\right) = f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \left(1 - f _ {\boldsymbol {\theta}} (\boldsymbol {x})\right).
+$$
+
+Applying the Weily inequality again on the R.H.S. of Equation 12, we obtain:
+
+$$
+\begin{array}{l} \lambda \left(\boldsymbol {H} _ {\boldsymbol {x}} ^ {\ell}\right) \leq f _ {\boldsymbol {\theta}} (\boldsymbol {x}) (1 - f _ {\boldsymbol {\theta}} (\boldsymbol {x})) \times \| \nabla f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \| ^ {2} + \lambda \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y\right) \times \nabla^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \\ \leq f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \left(1 - f _ {\boldsymbol {\theta}} (\boldsymbol {x})\right) \times \left\| \nabla f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \right\| ^ {2} + \left| f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y \right| \lambda \left(\nabla^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x})\right) \tag {4} \\ \end{array}
+$$
+
+The statement of Proposition 4.5 is obtained combining Equation (3) with Equation (4).
+
+![](images/12fb5398ddd1804f5221a553bd71b9137b7b6455e04be8794987295ada37df07.jpg)
+
+# B. Examples
+
+# B.1. On linearized models and the upper-bound in Proposition 4.2: an empirical standpoint.
+
+![](images/75ba5abb330649eb68ecc9f4756f2672de1f6a8529b724d9d265df2f958775fe.jpg)  
+(a) Model with 2 ReLU nodes.  
+Figure 10. The ReLU network in Figure 10a can attain at most 3 linear pieces, while the linearized network in Figure 10b can attain at most 2 linear pieces.
+
+![](images/65f9927e7022804d56a6c119418dc1d0e6e4d0871381bb890c3dbb1aff6fc7ce.jpg)  
+(b) Model with 1 ReLU node and a linearized one.
+
+In general, linearized models may be far away from reaching the bound in Proposition 4.2. It is easy to see how this bound cannot be reached in many cases when one or more ReLU activation are replaced with linear activation functions. For
+
+instance let us consider a ReLU network with a single hidden layer and width  $\omega_{1} = 2$ . According to Proposition 4.2, this network can attain at most 3 linear pieces. Indeed, let us consider bounded inputs  $\pmb{x}\in [-b,b]$  such that  $b\in \mathbb{R}^{+}$ . By properly optimizing weights and biases, the two ReLU nodes can create the orange and blue linear pieces in Figure 10a, that, when linearly combined in the output of the model (cf. green line in Figure 10a), can attain at most 3 linear pieces. On the contrary, if one of the two ReLU nodes is replaced with a linear node, the linear combination of the two nodes which is forwarded to the output, by definition, does not create any new breakpoint, only maintaining the two linear pieces dictated by the presence of the ReLU node. Therefore, the number of linear pieces is reduced to 2, as shown in Figure 10b.
+
+# C. Additional Empirical Results
+
+# C.1. Settings
+
+Each of the results in this paper was produced using an A100 GPUs with 80 GB of GPU memory, up to 100 GB of RAM, and up to 10 Intel(R) Xeon(R) E5-2630 v3 CPUs each clocked at 2.40GHz.
+
+For SNL, we train each base model for 160 epochs, and then perform the SNL finetuning step after ReLU linearization. These values are present in the official implementation of this algorithm provided by the authors of (Cho et al., 2022), and we use this implementation off-the-shelf.
+
+For DeepReDuce, we use the official implementation provided by the authors of (Jha et al., 2021) off-the-shelf which trains each model for 200 epochs.
+
+# C.2. Accuracy drop analysis
+
+For SVHN and CIFAR-10, the classes corresponding to these labels are evident. The class correspondences for UTKFace with age and race labels are as follows.
+
+- UTKFace with age labels: 2-12, 13-17, 18-64, and over 64 years of age correspond to 1, 2, 3, and 4, respectively.  
+- UTKFace with race labels: White, black, Asian, Indian, and other races correspond to 0, 1, 2, 3, and 4, respectively.
+
+In this section, we present more plots for relative accuracy drops in Figure 11 to show the disparate impact of linearizing ReLUs. Observe again how minority groups suffer degradation in accuracy that is higher than for majority groups as the ReLU budget decreases/more ReLUs are linearized.
+
+![](images/97c1b999404f1c5ce6ecf02a57a8411e0c36a5a4281901e0fc8988d478bede76.jpg)  
+Figure 11. SNL: For ResNet18 on CIFAR-10 (left) and SVHN (center left), and for ResNet34 on CIFAR-10 (center right) and SVHN (right)
+
+![](images/0d592c06f5992a973bf1c08bdaef8378ae68ae113c038711fe8e3edab7420c37.jpg)
+
+![](images/606fadf8e15081db03be355a580318a969b17b1961b81e19f7a70f4a17584789.jpg)
+
+![](images/59e21759628a1cb14b6299c45594623c2ed4f734dd3a2e931dd5aba6c5df536c.jpg)
+
+# D. Mitigation
+
+This subsection provides further evidence for the efficacy of the mitigation method discussed in Section 6. The labels in the legend refer to group indices.
+
+# D.1. Mitigation: Gradient Norms and Distance to the Decision Boundary
+
+The plots in Figure 13 and Figure 14 illustrate the differences in the values of gradient norms and (normalized) distances to the decision boundary when linearizing RLUs with and without reduction. As mentioned in Section 6, a reduction in the gradient norms and an increase in the distance to the decision boundary can be observed when using the mitigation method.
+
+![](images/e337043845f0f58264c5db9e61403bcc9d0d4bf634173509348c2f1002fd21c5.jpg)  
+Figure 12. SNL with Mitigation: For ResNet34 trained on UTKFace with age (left) and race (center left) labels and on CIFAR-10 (center right) and ResNet18 trained on CIFAR-10 (right).
+
+![](images/6310623e3fb46cb129b05ebbf6a030b28e7a06a26d779411837e875604812232.jpg)
+
+![](images/d1b446bc05ff5c2a9ed2d13bff1fd610cc35b54c9d3ba3c2134fdc41e889ba67.jpg)
+
+![](images/cc2df9f481d3e50c595109cab42f522f7f8b04727509a4b07dc4c84a26a5bc7d.jpg)
+
+![](images/bd81e8ff2667253771646ca0af77b34e6a71fa719d3425804cd248b25c29aa34.jpg)  
+Figure 13. SNL: (Test) Accuracies for ResNet-18 trained on UTKFace with age labels; left: without mitigation; right: with mitigation.
+
+![](images/a301a990a175a72e033b3a06ce8081e382ab0a4f1c59dbdc5974e8b0ceef537b.jpg)
+
+![](images/3ce43ef2f1383b4be74dce04cc9778f638507a3f62c22c4fd8e629acb01a53ab.jpg)  
+Figure 14. SNL: (Normalized) Distances to Decision Boundary for ResNet-18 trained on UTKFace with age labels; left: without mitigation; right: with mitigation.
+
+![](images/8a8584c3526b8bd032aabae21d18c62be4f26a356ac120d667a8066c6faa8720.jpg)
+
+# D.2. Mitigation: an ablation study of the parameter  $\mu$
+
+![](images/ee150fe99b4ff872a4b2502688164d1d3446f773da9e4247e910452fea561272.jpg)  
+(a) SNL on UTKFace race ResNet18 mitigation with  $\mu = 0.0005$
+
+![](images/3002537c643922e55b8f3d589e5807a8c10cdc08082f2ac526155ac949c7bd72.jpg)  
+(b) SNL on UTKFace race ResNet18 mitigation with  $\mu = 0.005$  
+Figure 15. We report the plots for the ablation study of the parameter  $\mu$  for the SNL on UTKFace (race). We consider  $\mu \in \{0.0005, 0.005, 0.01\}$ . In general we observe that these values of  $\mu$  do not result in significant changes for the performance of the mitigation strategy. For the task at end, we could then say that there exists a neighborhood of values for  $\mu$  where our mitigation consistently benefits the considered linearization techniques.
+
+![](images/22f4b215e7d0b2ffd6bd0b9aee8222a0deace7a835b2a4578e399c5d808a84e6.jpg)  
+(c) SNL on UTKFace race ResNet18 mitigation with  $\mu = 0.01$

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