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2025/MIPT_ Multilevel Informed Prompt Tuning for Robust Molecular Property Prediction/full.md

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+# MIPT: Multilevel Informed Prompt Tuning for Robust Molecular Property Prediction
+
+Yeyun Chen  $^{12}$  Jiangming Shi  $^{12}$
+
+# Abstract
+
+The progress in materials science and drug discovery is impeded by the availability of labeled data and the high costs of manual annotation, driving the need for efficient strategies to capture molecular representations and enable accurate predictions. Pretrained Graph Neural Networks have shown promise in capturing universal molecular representations, but adapting them to task-specific applications remains challenging. In this paper, we propose Multilevel Informed Prompt-Tuning (MIPT), a novel framework for effectively tailoring pretrained models to molecule-related tasks. MIPT utilizes a lightweight, multi-level prompt learning module to capture node-level and graph-level task-specific knowledge, ensuring adaptable and efficient tuning. Additionally, a noise penalty mechanism is introduced to address mismatches between pretrained representations and downstream tasks, reducing irrelevant or noisy information. Experimental results show that MIPT surpasses all baselines, aligning graph space and task space while achieving significant improvements in molecule-related tasks, demonstrating its scalability and versatility for molecular tasks.
+
+# 1. Introduction
+
+Machine learning (ML) has demonstrated significant potential to revolutionize the discovery and development of new materials (Chibani & Coudert, 2020), drugs (Shen & Nicolaou, 2019), and chemical processes (Taskinen & Yliruusi, 2003). However, the rapid growth in these fields is hindered by the scarcity of labeled molecular data and the high cost of manual annotation. These challenges have
+
+$^{1}$ Institute of Artificial Intelligence, Xiamen University, Xiamen Fujian, China  $^{2}$ Shanghai Innovation Institute, Shanghai, China. Correspondence to: Jiangming Shi <jiangming.shi@outlook.com>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+driven interest in strategies that efficiently capture molecular properties and enable accurate predictions with minimal supervision. Pretrained Graph Neural Networks (GNNs), designed to learn universal molecular representations, offer a promising solution. Yet, adapting these representations to task-specific applications remains a key challenge, necessitating efficient and flexible tuning methods. The pretrain-and-finetune paradigm (Hu et al., 2019) has been proposed to address this challenge, but the gap between pre-training objectives and downstream tasks often hinders effective knowledge transfer, leading to suboptimal performance.
+
+Prompt-based learning (Sun et al., 2023b; Li et al., 2024) has recently emerged as a powerful method for bridging this semantic gap. By aligning downstream tasks with pre-trained feature spaces, prompts facilitate knowledge transfer, improving accuracy and efficiency in applications such as molecular property prediction. Graph prompts, in particular, have shown efficacy in narrowing this gap. Some approaches use manually customized prompts (Wang et al., 2024), incorporating task-specific substructures or functional groups, while others explore universal prompts, such as GPF (Fang et al., 2024), which are applicable across all nodes, enhancing task adaptability and generalization while reducing computational costs. Additionally, in-context learning methods like PRODIGY (Huang et al., 2024) leverage label nodes to construct prompt graphs, transforming downstream tasks into link prediction problems and enhancing contextual learning. These approaches collectively provide more flexible and efficient solutions for downstream applications.
+
+Despite these advances, significant challenges remain in applying prompts to pre-trained GNNs. Many existing graph prompts are unable to optimize for specific tasks and exhibit limited scalability. While manually designed prompts (Wang et al., 2024) offer task-specific flexibility, their reliance on predefined templates limits adaptability. Similarly, universal prompts, such as GPF-plus (Fang et al., 2024), struggle to capture the diversity of all nodes, and assigning distinct prompts to each node incurs high computational costs (Sun et al.). Some studies address these issues by introducing label nodes to construct graph prompt (Huang et al., 2024), but this approach can disrupt node depen
+
+dencies and alter graph topology, potentially reducing prediction accuracy. Moreover, the large number of tunable parameters in pre-trained GNNs, combined with the limited availability of labeled molecular data, makes downstream fine-tuning less effective. These limitations highlight the need for adaptable, scalable, and efficient prompt designs to fully leverage the potential of pre-trained GNNs.
+
+On the other hand, while prompts are effective in enhancing generalization, they may inadvertently introduce task-irrelevant noise, which can hinder the model's ability to prioritize critical features, ultimately leading to suboptimal performance in specific tasks. For instance, in graph classification, prompts may overemphasize edge or node details while neglecting the overall graph structure. Similarly, pretrained GNNs, despite capturing broad feature distributions, may fail to align with task-specific requirements, such as mismatches between node-level pre-trained features and subgraph-level features needed downstream, causing overfitting and reduced accuracy. Additionally, downstream tasks often exhibit biases, prioritizing high-probability features while ignoring low-probability tail samples. Effectively bridging the gap between generalized pre-trained knowledge and domain-specific needs presents a critical challenge, necessitating highly adaptive approaches to mitigate noise, address feature mismatches, and align with task-specific biases for enhanced performance.
+
+Based on the above observations, this paper aims to address the following challenges: (i) While prompts are designed to bridge the gap between pre-training and downstream tasks, universal prompts frequently exhibit limited intuitive interpretability and poor scalability. Consequently, effective prompt design necessitates integrating graph-specific information, capturing their adaptability to graph structures, and ensuring efficient scalability. (ii) Although prompts are intended to guide models towards task adaptation, their application may inadvertently introduce task-irrelevant noise due to suboptimal feature modeling, thereby compromising performance. Therefore, it is crucial to effectively extract salient features, mitigate noise interference, and enhance the model's robustness and adaptability to downstream tasks.
+
+In this paper, we propose a novel framework, Multilevel Informed Prompt Tuning (MIPT), to enhance pretrained GNNs for molecule-related tasks. The framework employs a lightweight multi-level prompt learning network to capture task-specific knowledge at both node and graph levels, enabling effective adaptation to diverse scenarios. Furthermore, a noise penalty mechanism is introduced to align distributions and facilitate knowledge transfer across graph spaces. Experiments on public datasets demonstrate that MIPT achieves superior performance in molecular property prediction while requiring fewer trainable parameters.
+
+Our contributions can be summarized as below:
+
+- We introduced a novel framework called Multilevel Informed Prompt Tuning (MIPT), which leverages a lightweight multi-level prompt learning module to effectively capture task-specific knowledge at both the node and graph levels, enabling efficient adaptation of pretrained GNNs to molecular tasks.  
+- We incorporates a noise penalty mechanism to address mismatches between pretrained representations and downstream tasks, effectively reducing irrelevant or noisy information and enhancing task performance.  
+- Experimental results demonstrate that MIPT surpasses baseline model, excelling in aligning graph and task space modalities, while showcasing scalability and versatility across a wide range of molecule-related tasks.
+
+# 2. Related Works
+
+# 2.1. Pre-trained GNNs
+
+Pretraining and fine-tuning are widely used to transfer knowledge from related tasks and enhance model generalization. Pretraining trains on large-scale data via self-supervised or supervised tasks, followed by fine-tuning on smaller labeled datasets. In molecular GNNs, various self-supervised tasks capture chemical rules and patterns at node, subgraph, and graph levels (Xia et al., 2022b). However, pretrained GNNs face challenges in extracting task-relevant knowledge and often suffer from overfitting during fine-tuning (Xia et al., 2022a). Unlike NLP and CV, pretrained GNNs do not consistently improve downstream performance (Sun et al., 2022b), partly due to limited research on selecting effective self-supervised tasks. Most studies adopt only a few tasks for pretraining, resulting in GNN models excelling in specific downstream tasks but lacking consistency (Sun et al., 2022b). Moreover, recent findings suggest self-supervised pretraining sometimes fails to outperform non-pretrained methods (Sun et al., 2022b). Overall, pretrained GNNs offer limited advantages over non-pretrained models. Recent approaches have attempted to improve fine-tuning using regularization (Xuhong et al., 2018) or update constraints (Houlsby et al., 2019; Xia et al., 2022a; Zhang et al., 2022).
+
+# 2.2. Graph Prompt Learning
+
+Despite the advances in pre-trained models, bridging this gap between pre-trained and fine-tuned models has become a critical focus in recent research. GPPT (Sun et al., 2022a) introduced a learnable prompt mechanism for graphs, specifically targeting node classification tasks. While effective in this specific domain, its design lacks generalizability and cannot adapt to other downstream tasks. Building upon this, Graph Prompt (Liu et al., 2023) proposed a novel task-specific learnable prompt, which guides the ReadOut
+
+operation of each downstream task using an appropriate aggregation scheme. However, this approach faces challenges in multitasking scenarios, as differences across tasks make it difficult to identify a universal prompt pattern, ultimately limiting its effectiveness. All in One (Sun et al., 2023a) addressed the issue of multitasking by designing a unified training framework for joint multi-task prompts. While this approach leverages meta-learning to accommodate a large variety of tasks, it is computationally expensive and exhibits limited generalization capabilities across diverse tasks. GPF and its extended version, GPF-plus (Fang et al., 2024), introduced learnable prompts at the node level, aiming for versatility across different downstream tasks. This approach effectively balances adaptability and computational efficiency, presenting a more practical solution for a broad range of task requirements.
+
+# 3. Preliminaries
+
+# 3.1. Problem Formulation
+
+Notations. Let  $\mathcal{G} = (\mathcal{V},\mathcal{E})\in \mathbb{G}$  denotes a graph, where  $\mathcal{V} = \{v_{1},\dots,v_{N}\}$  represents the set of nodes, and  $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$  represents the set of edges.  $\mathcal{G}$  is associated with a node feature matrix  $\mathbf{X}\in \mathbb{R}^{N\times F}$  where  $F$  is the feature dimension, and an adjacency matrix  $\mathbf{A}\in \mathbb{R}^{N\times N}$  where  $\mathbf{A}_{ij} = 1$  if and only if  $(v_{i},v_{j})\in \mathcal{E}$  and  $\mathbf{A}_{ij} = 0$  otherwise.
+
+Problem Definition. Given a pre-trained GNN  $f$ , a learnable projection layer  $g$ , and a downstream task dataset  $D = \{(\mathcal{G}_1, y_1), (\mathcal{G}_2, y_2), \ldots, (\mathcal{G}_m, y_m)\}$ , our objective is to fine-tune the parameters of the prompt  $p_g$  and the projection layer  $g$  to effectively bridge the gap between the pre-trained GNN  $f$  and the downstream task. Specifically, the aim is to maximize the likelihood of accurately predicting the ground truth labels  $y$  associated with the downstream task, formulated as follows:
+
+$$
+\max  _ {p _ {g}, g} P _ {p _ {g}, g} (y \mid \mathcal {G}) \tag {1}
+$$
+
+# 3.2. Molecule Representation Learning.
+
+We adopt the Graph Isomorphism Network (GIN) (Xu et al., 2018) as the backbone for our models, leveraging its strong expressive power and effectiveness in graph-related tasks. In fact, our framework is model-agnostic, enabling seamless integration with various existing molecular prediction model as downstream models. The pre-training process is designed to learn general-purpose graph representations by optimizing auxiliary objectives on large-scale, unlabeled graph data. This process primarily follows two paradigms: self-supervised learning and contrastive learning (Zhou et al., 2024).
+
+Consider a molecule graph  $\mathcal{G} = (\mathcal{V},\mathcal{E})$ . Typically, the predictor  $\rho$  can be expressed as  $f\circ g$ , comprising two compo
+
+nents: a GNN encoder  $f \colon \mathcal{G} \to \mathbb{R}^d$ , which generates molecular representations, and a projection layer  $g : \mathbb{R}^d \to \mathcal{V}$ . In this context, the projection layer serves as a downstream classifier, predicting labels from the generated representations. Specifically, the encoder  $f$  operates in two stages. The first stage utilizes a GNN to learn node-level representations  $h_v^{(k)}$  as formulated as follows:
+
+$$
+\mathbf {h} _ {v} ^ {(k)} = \phi^ {(k)} \left(\mathbf {h} _ {v} ^ {(k - 1)}, \varphi^ {(k)} \left(\left\{\mathbf {h} _ {u} ^ {(k - 1)}: u \in \mathcal {N} (v) \right\}\right)\right) \tag {2}
+$$
+
+where  $\mathbf{h}_v^{(k)}$  represents the node embedding of node  $v$  at the  $k$ -th layer,  $\mathcal{N}(v)$  is the set of neighbors of node  $v$ ,  $\varphi^{(k)}(\cdot)$  is the aggregation function at the  $k$ -th iteration, and  $\phi^{(k)}(\cdot)$  is the combination function at the  $k$ -th iteration. The node embeddings are initialized as  $\mathbf{h}_v^{(0)} = \mathbf{X}_v$ , where  $\mathbf{X}_v$  denotes the input features of node  $v$ .
+
+To ensure effective feature updates, the feature transformation at layer  $k$  can be expressed as:
+
+$$
+\mathbf {h} _ {v} ^ {(k)} = \mathbf {M L P} ^ {(k)} \left(\left(1 + \epsilon^ {(k)}\right) \mathbf {h} _ {v} ^ {(k - 1)} + \sum_ {u \in \mathcal {N} [ v ]} \mathbf {h} _ {u} ^ {(k - 1)}\right) \tag {3}
+$$
+
+where  $\epsilon^{(k)}$  is a learnable parameter,  $\mathbf{MLP}^{(k)}$  represents a multi-layer perceptron applied at the  $k$ -th layer, and  $\mathcal{N}[v]$  includes the neighbors of node  $v$  along with the node itself.
+
+Secondly, at the final layer  $K$ , the GNN aggregates the learned node embeddings using a READOUT function:
+
+$$
+\mathbf {h} _ {v} = \operatorname {R E A D O U T} \left(\left\{\mathbf {h} _ {v} ^ {K}: v \in \mathcal {V} \right\}\right) \tag {4}
+$$
+
+where READOUT can be implemented as a sum, mean, or maximum over all node embeddings in the graph.
+
+# 4. Method
+
+This section provides a thorough explanation of the proposed MIPT framework, detailing its technical design and the motivations behind its development. MIPT is specifically designed to address two primary challenges observed in current prompt-tuning approaches: (i) Universal prompts often lack the necessary flexibility and interpretability to adapt to diverse task-specific requirements, and (ii) prompts can inadvertently introduce task-irrelevant noise, which can significantly degrade overall model performance.
+
+To tackle these issues, we present a comprehensive overview of the framework's core components: the multi-level graph-informed prompts and denoising prompt mechanisms. The overall architecture is illustrated in Figure 1.
+
+# 4.1. Parameter-efficient Tuning for Node Encoder
+
+Efficient Tuning with LoRA Update. To address the gap between pre-training objectives and downstream task requirements, we propose an efficient fine-tuning approach
+
+![](images/177f602077c2cfc5af864b7784ce461d8e028910157993d8da381615f8cd3e74.jpg)
+
+![](images/99e6a975fb348c365de4ebcae69492675998584400b05023786a0ae2e6c25ffc.jpg)  
+Figure 1. Comprehensive overview of the MIPT framework. Our proposed framework synergistically combines efficient node fine-tuning and a unified distribution denoising mechanism, thereby providing an efficient and robust solution for local-global prompt-tuning across varied scenarios.
+
+![](images/5d08cba2a598e7b3b31808096d7d4bb544001edccc211ef22434efee4526072c.jpg)
+
+![](images/0c94325d2564afea4c135f0420b84df444c8dbc4224b2608d3fff92f0a91f0dd.jpg)
+
+that integrates a low-rank adaptation matrix for learning task-specific embeddings. This method prioritizes features pertinent to downstream tasks while preserving the expressiveness of the pre-trained model. For the frozen pre-trained GNN weight matrix  $W_0 \in \mathbb{R}^{d \times k}$ , the parameter update  $\Delta W$  is represented using a low-rank decomposition. The feature update process, as described in Eq. 5, is reformulated as:
+
+$$
+h _ {v} ^ {(k)} = \left(W _ {0} ^ {(k)} + \Delta W ^ {(k)}\right) \sum_ {v} \left(\left(1 + \epsilon^ {(k)}\right) \mathbf {h} _ {v} ^ {(k - 1)} + \sum_ {u \in \mathcal {N} [ v ]} \mathbf {h} _ {u} ^ {(k - 1)}\right) \tag {5}
+$$
+
+here,  $W_{0}$  denotes the weight matrix of the pre-trained MLP layer, and  $\Delta W$  represents the low-rank adaptation. The parameter  $\Delta W$  is further expressed as:
+
+$$
+\Delta W ^ {(k)} = B ^ {(k)} A ^ {(k)} \tag {6}
+$$
+
+where  $A \in \mathbb{R}^{r \times d}$  and  $B \in \mathbb{R}^{d \times r}$  are low-rank matrices with  $r \ll d$ . The low-rank structure of  $\Delta W$  ensures that the additional parameter number is minimal, thus enhancing computational efficiency. These trainable low-rank matrices are injected into the node MLP layer to approximate weight updates effectively.
+
+Proposition 4.1. For a pretrained GNN model  $f$ , with parameters  $\mathbf{W}$ , node features  $\mathbf{X}$ , there satisfies that incorporating LoRA parameters into the linear transformation during node updates does not alter the underlying graph structure.
+
+A detailed proof for proposition 4.1 refers to Appendix A.
+
+# 4.2. Multilevel Graph-Informed Prompt
+
+For downstream prediction tasks, obtaining task-specific graph representations  $h_{g}$  is essential, as graph-level embeddings directly influence model performance. However, existing pooling operations fail to satisfy the expressiveness conditions (Bianchi & Lachi, 2023). To address this, we propose Multilevel Graph-Informed Prompt (MGIP), a novel method that extracts features from node-level embeddings and constructs low-rank graph prompt after aggregation. By encoding domain-specific information directly into  $h_{g}$ , our approach enhances the adaptability and efficiency of graph representations.
+
+Random Node Masking. Let  $h_v^* \in \mathbb{R}^{|V| \times d}$  be the node embeddings with LoRA tuning initialized using Kaiming initialization (He et al., 2016). Subsequently, we employ a random masking method to process  $h_v^*$ , aiming to enhance the model's robustness and its ability to identify critical subgraphs. This method involves three main steps: segmenting, filling, and random masking, ensuring both structural and representational variability are effectively captured. We initialize a binary matrix  $\mathbf{M} \in \{0,1\}^{b \times S_{\max}}$ , where  $b$  is the batch size and  $S_{\max}$  is the maximum number of nodes in the batch. To ensure all samples in the batch share the same input dimensions, we apply a zero-padding operation to the node representations of each sample. The padding is defined as:
+
+$$
+\mathbf {M} _ {i, j} = \left\{ \begin{array}{l l} 1, & \text {i f} j <   s _ {i} \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {7}
+$$
+
+where  $s_i$  represents the size of the  $i$ -th graph. A random
+
+masking matrix  $\mathbf{R} \in \{0,1\}^{b \times S_{\max}}$  is generated, with each element sampled from a Bernoulli distribution:  $\mathbf{R}_{i,j} \sim \text{Bernoulli}(1 - p)$ , where  $p$  is the masking probability (e.g.,  $p = 0.2$ ). The final mask matrix is updated as:  $\mathbf{M} \gets \mathbf{M} \odot \mathbf{R}$  where  $\odot$  denotes element-wise multiplication. The updated mask is applied to the node  $v$  to produce the masked embeddings  $h_v'$ :
+
+$$
+h _ {v} ^ {\prime} = \left\{ \begin{array}{l l} h _ {v} ^ {*}, & \text {i f} \mathbf {M} _ {i, j} = 1 \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {8}
+$$
+
+where  $j$  in  $\mathbf{M}_{i,j}$  is the  $i$ -th graph of the  $j$ -th node. By combining deterministic segmentation with random masking, this approach balances structural consistency with variability, encouraging the model to identify key subgraphs and improving generalization. This process ensures the graph representations retain critical information while being robust to structural noise.
+
+Global-Local Information Maximization. To obtain the global representation of the entire graph, we utilize a readout function  $\Gamma : h_v' \in \mathbb{R}^{|\mathcal{V}| \times d} \to h_g' \in \mathbb{R}^d$  (i.e.,  $h_g' = \Gamma(h_v')$ ,  $h_g = \Gamma(h_v)$ ), which aggregates the local features into a global context. To improve discriminative power, contrastive learning (CL) is introduced to maximize mutual information between original and masked features. The contrastive learning loss function is formulated as:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {c o n t}} = \\ - \frac {1}{N} \sum_ {i = 1} ^ {N} \log \frac {\exp \left(\varphi \left(\mathrm {h} _ {g , i} ^ {\prime} , \mathrm {h} _ {g , i}\right)\right)}{\exp \left(\varphi \left(\mathrm {h} _ {g , i} ^ {\prime} , \mathrm {h} _ {g , i}\right)\right) + \sum_ {j = 1 , i \neq j} ^ {N} \exp \left(\varphi \left(\mathrm {h} _ {g , j} ^ {\prime} , \mathrm {h} _ {g , j}\right)\right)} (9) \\ \varphi (\cdot) = \left(h _ {g, i} ^ {\prime}\right) ^ {\top} \left(h _ {g, i}\right) / \tau (10) \\ \end{array}
+$$
+
+where  $\tau$  is the temperature parameter. CL enhances the correlation between local node representations and global graph representations, improving the multi-level expressiveness of graph embeddings.
+
+Graph Informed Prompt Generation. Then, we propose the initialization of graph prompt is guided by graph masked feature. To capture domain-specific graph features, we introduce a lightweight prompt function, denoted as  $\Phi (\cdot)$ , to provide implicit prompts  $p_{g}$ . Formally, this prompt function, is defined as follows:
+
+$$
+\Phi \left(h _ {g} ^ {\prime}\right) = W _ {B} \left(W _ {A} \cdot h _ {g} ^ {\prime}\right) \tag {11}
+$$
+
+The trainable parameters of this prompt function are denoted as  $W_{A} \in \mathbb{R}^{d' \times r}$  and  $W_{B} \in \mathbb{R}^{r \times d'}$ , where  $r \ll d'$ . The domain-specific prompt is combined with the graph features from pretrained encoder, creating rich information that enables the effective identification of context-specific changes in critical structures. This, in turn, facilitates precise modeling of structural correlations.
+
+Task-specific Projection Head. After that, the refined graph representation,  $h_G = \mathrm{CONCAT}(h_g, p_g)$ , is subsequently passed through a projection head to produce the
+
+final output:
+
+$$
+\hat {y} = g \left(h _ {G}\right) \tag {12}
+$$
+
+where  $g$  is the prediction function. This modification enables downstream models to efficiently leverage the prompts  $p_g$  for adaptation to new data and tasks.
+
+# 4.3. Optimization
+
+In the prompt-tuning phase, we specifically update the parameters of the prompt and downstream by performing a limited training on an unseen dataset. This approach enables downstream models to efficiently adapt to new data.
+
+Training Procedures. To train the model, a task-specific projection head is used.
+
+$$
+\mathcal {L} _ {c l s} ^ {i} = \log P (y ^ {i} \mid g (h _ {G})) \tag {13}
+$$
+
+For graph classification, the loss is typically the binary cross-entropy loss:
+
+$$
+\mathcal {L} _ {c l s} \left(g _ {\theta}\right) = - \sum_ {i = 1} ^ {N} \left(y _ {i} \log \left(\hat {y} _ {i}\right) + \left(1 - y _ {i}\right) \log \left(1 - \hat {y} _ {i}\right)\right) \tag {14}
+$$
+
+where  $\mathbf{N}$  is the number of training samples,  $y_{i}$  is the true label, and  $\hat{y}_i$  is the predicted label of i-th sample.
+
+Noise Penalty Mechanism. Building on Arpit et al. (2017), it has been demonstrated that deep neural networks (DNNs) initially learn simpler and more generalizable patterns during the early stages of training, before gradually overfitting to noisy or degenerate samples. Inspired by this observation, we model the confidence distribution of samples based on their classification loss (Shi et al., 2024). Specifically, we employ a two-component Gaussian Mixture Model (GMM) to represent the loss distribution derived from Eq. (13):
+
+$$
+p \left(\mathcal {L} _ {c l s} \mid \theta\right) = \sum_ {t = 1} ^ {T} \pi_ {t} \Phi \left(\mathcal {L} _ {c l s} \mid t\right) \tag {15}
+$$
+
+where  $h_G$  is pretrained molecular features,  $g(\cdot)$  is downstream task with parameters  $\theta$ , and  $\pi_t$  and  $\Phi(\mathcal{L}_{cls} \mid t)$  represent the mixture coefficient and the probability density of the  $t$ -th Gaussian component, respectively.
+
+The GMM,  $\Phi (\mathcal{L}_{cls}\mid t)\to p$ , is utilized to model the posterior probability, denoted as  $w_{i}(0\leq w_{i}\leq 1)$ , which is computed to quantify the confidence of the  $i$ -th sample:
+
+$$
+w _ {i} = p (t \mid \mathcal {L} _ {c l s} ^ {i}) \tag {16}
+$$
+
+where  $k$  is the Gaussian component with the smaller mean, and  $p\left(t\mid \mathcal{L}_{cls}^i\right)$  represents the posterior probability of  $\mathcal{L}_{cls}^i$  under the  $t$ -th component. The confidence  $\hat{w}_i$  is then updated as follows:
+
+$$
+\hat {w} _ {i} = \left\{ \begin{array}{l l} 1, & \text {i f} w _ {i} > \mu \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {17}
+$$
+
+Here,  $\mu$  is the average of  $w_{i}$ , i.e.  $\mathrm{mean}(w_i)$ . With the confidence  $\hat{w}$  effectively capturing high-relevant representation, we introduce the Noise Prompts Penalty (NPP) loss,  $\mathcal{L}_{NPP}$ , to suppress the influence of redundancy noise features. The loss is defined as:
+
+$$
+\mathcal {L} _ {N P P} = - \frac {1}{N} \sum_ {i = 1} ^ {N} \hat {w} _ {i} \log P (y ^ {i} \mid g (h _ {G})) \tag {18}
+$$
+
+This mechanism leverages  $\hat{w}$  to penalize noisy features, ensuring that the model focuses on clean and reliable representations, thereby improving its robustness and overall performance.
+
+Final Objectives. The overall training objective for the downstream task is illustrated as follows:
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {N P P}} + \mathcal {L} _ {\text {c o n t}} \tag {19}
+$$
+
+The model parameters  $\theta$  are updated by minimizing  $\mathcal{L}_{\text{total}}$  using gradient-based optimization.
+
+Compute the gradient of  $\mathcal{L}_{total}$  with respect to the model parameters:
+
+$$
+\nabla_ {\theta} \mathcal {L} _ {t o t a l} = \nabla_ {\theta} \mathcal {L} _ {N P P} + \nabla_ {\theta} \mathcal {L} _ {c o n t} \tag {20}
+$$
+
+$$
+\theta \leftarrow \theta - \eta \nabla_ {\theta} \mathcal {L} _ {\text {t o t a l}}, \tag {21}
+$$
+
+where  $\eta$  is the learning rate. The parameters of GMM are updated in alternating steps using the Expectation-Maximization (EM) algorithm, with the pseudocode provided in the Appendix 1.
+
+The training process alternates between updating the model parameters  $\theta$  and the GMM parameters until convergence. By progressively refining both components, the model learns to focus on reliable samples, while reducing the impact of noisy prompts, ultimately improving the robustness and accuracy of predictions.
+
+Proposition 4.2. (Robustness of NPP) Given a classification task where prompts  $p_g$  are used to adapt a pretrained model  $f(\cdot)$ , the incorporation of the NPP loss  $\mathcal{L}_{NPP}$  improves robustness by selectively reducing the influence of noisy prompt.
+
+A detailed proof for proposition 4.2 is given in Appendix B.1.
+
+# 5. Experiments
+
+In this section, we present an extensive set of experiments to evaluate the performance of our model. The analysis is structured around addressing the following key research questions:
+
+- RQ1: How does the efficiency of MIPT compare to that of prompt-tuning and end-to-end training approaches?
+
+- RQ2: Does MIPT effective in generalizing pre-trained models to new molecular property prediction datasets and tasks?  
+- RQ3: What impact do the key components of MIPT have on the performance of downstream models?  
+- RQ4: How do hyperparameters influence the performance of MIPT on molecular property prediction?
+
+# 5.1. Experiment Setup
+
+Datasets. We employ eight common datasets from MoleculeNet (Wu et al., 2018) as our benchmark datasets: BBBP, Tox21, ToxCast, SIDER, ClinTox, MUV, HIV and BACE. Random splits and scaffold splits for these datasets are adopted. More dataset and training details are available in the Appendix C.1.
+
+Pretraining strategies. We take into account four pretrained models for comparison: Deep Graph Infomax (InfoMax), Attribute Masking (AttrMasking), Context Prediction (ContextPred), and Edge Prediction (EdgePred).
+
+Baselines. We consider three tuning strategies for comparison is used to evaluate: FT(Fine-Tuning), GPF (Fang et al., 2024), GPF-plus (Fang et al., 2024). In the case of freezing the pre-trained model, our prompt method aims to learn the input graph and reconstruct the downstream task to fit the pre-training strategy.
+
+A variety of baselines are included for an in-depth comparison. Supervised GNN methods include D-MPNN (Yang et al., 2019), MGCN (Lu et al., 2019), and AttentiveFP (Xiong et al., 2019). Meanwhile, pretraining approaches consist of N-gram (Liu et al., 2019), PretrainGNN (Hu et al., 2019), GROVER (Rong et al., 2020), 3D-Infomax (Stark et al., 2022), GraphMVP (Liu et al., 2021), MolCLR (Wang et al., 2022), Uni-Mol (Zhou et al., 2023), and InstructMol (Wu et al.).
+
+Metric. ROC-AUC scores are evaluated on the test set. We report the mean and standard deviation of the results from three random seeds.
+
+# 5.2. Overall Performance (RQ1)
+
+We evaluated the performance of downstream tasks across various pre-training and tuning strategies, with results summarized in Table 1. Our method consistently outperforms all baselines, demonstrating its effectiveness and robustness. It achieves significant improvements over fine-tuning strategies across all datasets. While universal learnable prompts in baseline approaches yield limited gains and lack consistency, our method achieves the highest ROC-AUC (\%) scores on most datasets, as highlighted in bold.
+
+For example, on the BBBP dataset, our method achieves
+
+Table 1. Graph classification (ROC-AUC scores %, higher is better ↑) results across 8 datasets. The best results are highlighted in boldface, while the second-best results are marked with underline.  
+
+<table><tr><td>Pretrained strategy</td><td>Datasets # Molecules # Tasks</td><td>\( BBBP^{\uparrow} \) 2039 1</td><td>Tox21↑ 7831 12</td><td>ToxCast↑ 8576 617</td><td>SIDER↑ 1427 27</td><td>ClinTox↑ 1477 2</td><td>MUV↑ 41127 1</td><td>\( HIV^{\uparrow} \) 93087 17</td><td>\( BACE^{\uparrow} \) 1513 1</td></tr><tr><td rowspan="4">Infomax</td><td>+FT</td><td>67.55(2.06)</td><td>78.57(0.51)</td><td>65.16(0.53)</td><td>63.34(0.45)</td><td>70.06(1.45)</td><td>81.42(2.65)</td><td>77.71(0.45)</td><td>81.32(1.25)</td></tr><tr><td>+GPF</td><td>66.83 (0.86)</td><td>79.09 (0.25)</td><td>66.10 (0.53)</td><td>66.17 (0.81)</td><td>73.56 (3.94)</td><td>80.43 (0.53)</td><td>76.49 (0.18)</td><td>83.60 (1.00)</td></tr><tr><td>+GPF-plus</td><td>67.17 (0.36)</td><td>79.13(0.70)</td><td>66.35(0.37)</td><td>65.62 (0.74)</td><td>75.12(2.45)</td><td>81.33 (1.52)</td><td>77.73(1.14)</td><td>83.67(1.08)</td></tr><tr><td>+Ours</td><td>69.24(0.84)</td><td>80.18(0.25)</td><td>67.10 (0.38)</td><td>66.40(0.36)</td><td>80.02 (1.3)</td><td>82.84(0.58)</td><td>78.77(0.60)</td><td>83.88(0.45)</td></tr><tr><td rowspan="4">AttrMasking</td><td>+FT</td><td>66.33 (0.55)</td><td>78.28 (0.05)</td><td>65.34 (0.42)</td><td>66.77 (1.02)</td><td>74.46 (2.82)</td><td>81.78 (1.95)</td><td>77.90 (0.18)</td><td>80.94 (1.17)</td></tr><tr><td>+GPF</td><td>68.09 (0.38)</td><td>79.04 (0.90)</td><td>66.32 (0.42)</td><td>69.13 (1.16)</td><td>75.06 (1.02)</td><td>82.17 (0.65)</td><td>78.86 (1.42)</td><td>84.33 (0.54)</td></tr><tr><td>+GPF-plus</td><td>67.71 (0.64)</td><td>78.87 (0.31)</td><td>66.58 (0.13)</td><td>68.65 (0.72)</td><td>76.17 (2.98)</td><td>81.12 (1.32)</td><td>78.13 (1.12)</td><td>85.76 (0.36)</td></tr><tr><td>+Ours</td><td>69.56 (0.82)</td><td>80.77 (0.43)</td><td>67.91 (0.38)</td><td>69.15(0.40)</td><td>85.18 (0.12)</td><td>85.01(0.61)</td><td>79.48(0.64)</td><td>83.92 (1.46)</td></tr><tr><td rowspan="4">ContextPred</td><td>+FT</td><td>69.65 (0.87)</td><td>78.29 (0.44)</td><td>66.39 (0.57)</td><td>64.45 (0.60)</td><td>73.71 (1.57)</td><td>82.36 (1.22)</td><td>79.20 (0.51)</td><td>84.66 (0.84)</td></tr><tr><td>+GPF</td><td>68.48 (0.88)</td><td>79.99 (0.24)</td><td>67.92 (0.35)</td><td>66.18 (0.53)</td><td>74.51 (2.72)</td><td>84.34 (0.25)</td><td>78.62 (1.46)</td><td>85.32 (0.41)</td></tr><tr><td>+GPF-plus</td><td>69.15 (0.82)</td><td>80.05 (0.46)</td><td>67.58 (0.54)</td><td>66.94 (0.95)</td><td>75.25 (1.88)</td><td>84.48 (0.78)</td><td>78.40 (0.16)</td><td>85.81 (0.43)</td></tr><tr><td>+ Ours</td><td>73.62 (0.11)</td><td>80.60 (0.27)</td><td>68.81 (0.6)</td><td>68.52 (0.3)</td><td>81.21 (0.86)</td><td>84.96 (1.2)</td><td>81.76 (0.10)</td><td>86.57 (0.49)</td></tr><tr><td rowspan="4">EdgePred</td><td>+FT</td><td>66.56 (3.56)</td><td>78.67 (0.35)</td><td>66.29 (0.45)</td><td>64.35 (0.78)</td><td>69.07 (4.61)</td><td>79.67 (1.70)</td><td>77.44 (0.58)</td><td>80.90 (0.92)</td></tr><tr><td>GPF</td><td>69.57 (0.21)</td><td>79.74 (0.03)</td><td>65.65 (0.30)</td><td>67.20 (0.99)</td><td>69.49(5.17)</td><td>82.86 (0.23)</td><td>77.00 (1.08)</td><td>81.57 (1.08)</td></tr><tr><td>+GPF-plus</td><td>69.06(0.68)</td><td>80.04 (0.06)</td><td>65.94 (0.31)</td><td>67.51(0.59)</td><td>68.80 (2.58)</td><td>83.13(0.42)</td><td>77.00 (0.82)</td><td>81.75(2.09)</td></tr><tr><td>+ Ours</td><td>72.11(0.50)</td><td>81.26(0.80)</td><td>67.05(0.60)</td><td>67.25(0.30)</td><td>77.68(1.10)</td><td>86.30(1.20)</td><td>79.38(0.01)</td><td>81.79(0.11)</td></tr></table>
+
+an ROC-AUC of 73.62 under the ContextPred pretraining strategy, reflecting an  $8.98\%$  improvement over FineTune. Similarly, on ClinTox, it attains 85.18 under AttrMasking, surpassing GPF-plus by  $11.83\%$ . On MUV, despite FT's strong baseline performance, our method reaches 86.30, yielding a  $6.00\%$  improvement. These results highlight its superior performance and generalizability, consistently surpassing alternative approaches across diverse datasets and pre-training strategies. Its strong performance on complex or low-quality datasets further underscores its robustness and adaptability in graph-related tasks.
+
+We also compared our model with other SOTA models for molecular property prediction. As shown in Table 2, our method outperforms SOTA models in 5 out of 8 tasks while using fewer parameters, demonstrating its efficiency and effectiveness.
+
+Further analysis reveals that our approach offers distinct advantages over prompt-tuning models in molecular property prediction. Specifically, the MIPT framework effectively captures multi-level structural features in molecular data. Its novel prompt-learning paradigm facilitates knowledge transfer from pre-training to downstream tasks, mitigating distributional and semantic discrepancies between pre-training and target scenarios.
+
+# 5.3. Additional Experiments
+
+Transfer Performance Analysis (RQ2). We analyzed the training process of the molecular dataset and the GNN model using different tuning methods. Figure 2 (a) shows the testing curves during the tuning phase. From the curve, it is evident that our method exhibits greater fluctuations on
+
+the test dataset compared to other models. This suggests that the model is more sensitive to the data during training, allowing it to flexibly adjust and capture complex patterns in the dataset. The increased fluctuations indicate a more exploratory optimization process, which facilitates finding a better global solution. Additionally, this behavior demonstrates strong adaptability and robustness to complex data distributions and noise. More training curves are provided in Appendix D.
+
+Ablation Study (RQ3). To assess the effect of each component in our method, we conduct an ablation study evaluating LoRA, MGIP, and the loss functions  $(\mathcal{L}_{\mathrm{NPP}}$  and  $\mathcal{L}_{\mathrm{cont}})$ . The results in Table 3 highlight the significance of MGIP over LoRA when used independently, achieving an average improvement of  $6.9\%$  on SIDER and  $1.6\%$  on BACE, demonstrating its superior ability to capture task-specific features. Integrating  $\mathcal{L}_{\mathrm{cont}}$  with LoRA further enhances performance, yielding a  $6.5\%$  gain on SIDER, indicating that contrastive learning effectively aligns global and local representations. Similarly, incorporating  $\mathcal{L}_{\mathrm{NPP}}$  into MGIP boosts ClinTox performance by  $1.2\%$ , underscoring the importance of node-to-graph alignment in refining global representations.
+
+When combining all components, the complete model achieves the best results across all datasets, with improvements of  $2.9\%$  on SIDER,  $1.3\%$  on ClinTox, and  $0.9\%$  on BACE compared to the best individual configurations. These results confirm the complementary nature of the components and the effectiveness of the proposed approach in addressing multi-scale representation challenges in graph learning. Additional ablation results on various datasets are presented in Appendix D Table 6.
+
+Table 2. Performance comparison with the SOTA models on molecular property prediction datasets(ROC-AUC %, higher is better ↑)  
+
+<table><tr><td rowspan="2">Pretraining</td><td>Datasets↑
+# Molecules</td><td>BBBP↑
+2039</td><td>BACE↑
+1513</td><td>ClinTox↑
+1477</td><td>Tox21↑
+7831</td><td>ToxCast↑
+8576</td><td>SIDER↑
+1427</td><td>MUV↑
+93087</td><td>HIV↑
+41127</td></tr><tr><td># Tasks</td><td>1</td><td>1</td><td>2</td><td>12</td><td>617</td><td>27</td><td>17</td><td>1</td></tr><tr><td rowspan="3">X</td><td>D-MPNN</td><td>71.0 (0.3)</td><td>80.9 (0.6)</td><td>90.6 (0.6)</td><td>75.9 (0.7)</td><td>65.5 (0.3)</td><td>57.0 (0.7)</td><td>77.1 (0.5)</td><td>78.6 (1.4)</td></tr><tr><td>Attentive FP</td><td>64.3 (1.8)</td><td>78.4 (0.02)</td><td>84.7 (0.3)</td><td>76.1 (0.5)</td><td>63.7 (0.2)</td><td>60.6 (3.2)</td><td>75.7 (1.4)</td><td>76.6 (1.5)</td></tr><tr><td>MGCN</td><td>65.0 (0.5)</td><td>73.4 (0.8)</td><td>90.5 (0.4)</td><td>74.1 (0.6)</td><td>-</td><td>58.7 (1.9)</td><td>-</td><td>-</td></tr><tr><td rowspan="12">✓</td><td>N-GramRF</td><td>69.7 (0.6)</td><td>77.9 (1.5)</td><td>77.5 (4.0)</td><td>74.3 (0.4)</td><td>-</td><td>66.8 (0.7)</td><td>77.2(0.1)</td><td>76.9(0.7)</td></tr><tr><td>N-GramXGB</td><td>69.1 (0.8)</td><td>79.1 (1.3)</td><td>87.5 (2.7)</td><td>75.8 (0.9)</td><td>-</td><td>65.5 (0.7)</td><td>78.7(0.4)</td><td>74.8(0.2)</td></tr><tr><td>PretrainGNN</td><td>68.7 (1.3)</td><td>84.5 (0.7)</td><td>72.6 (1.5)</td><td>78.1 (0.6)</td><td>65.7 (0.6)</td><td>62.7 (0.8)</td><td>79.9(0.7)</td><td>81.3(2.1)</td></tr><tr><td>GROVERbase</td><td>70.0 (0.1)</td><td>82.6 (0.7)</td><td>81.2 (3.0)</td><td>74.3 (0.1)</td><td>65.4 (0.4)</td><td>64.8 (0.6)</td><td>62.5(0.9)</td><td>67.3(1.8)</td></tr><tr><td>GROVERlarge</td><td>69.5 (0.1)</td><td>81.0 (1.4)</td><td>76.2 (3.7)</td><td>73.5 (0.1)</td><td>65.3 (0.5)</td><td>65.4 (0.1)</td><td>68.2(1.1)</td><td>67.3(1.8)</td></tr><tr><td>3D-Infomax</td><td>69.1 (1.1)</td><td>79.4 (1.9)</td><td>59.4 (3.2)</td><td>74.5 (0.7)</td><td>64.4 (1.0)</td><td>53.3 (3.4)</td><td></td><td></td></tr><tr><td>GraphMVP</td><td>72.4 (1.6)</td><td>81.2 (0.9)</td><td>79.1 (2.8)</td><td>75.9 (0.5)</td><td>63.1 (0.4)</td><td>63.9 (1.2)</td><td>77.0(1.2)</td><td>77.7(0.6)</td></tr><tr><td>MolCLR</td><td>72.2 (2.1)</td><td>82.4 (0.9)</td><td>91.2 (3.5)</td><td>75.0 (0.2)</td><td>-</td><td>58.9 (1.4)</td><td>78.1(0.5)</td><td>79.6(1.9)</td></tr><tr><td>Uni-Mol</td><td>72.9 (0.6)</td><td>85.7 (0.2)</td><td>91.9 (1.8)</td><td>79.6 (0.5)</td><td>69.6 (0.1)</td><td>65.9 (1.3)</td><td>80.8(0.3)</td><td>82.1(1.3)</td></tr><tr><td>GEM</td><td>72.4 (0.4)</td><td>85.6 (1.1)</td><td>90.1 (1.3)</td><td>78.1 (0.1)</td><td>69.2 (0.4)</td><td>67.2 (0.4)</td><td>80.6(0.9)</td><td>81.7(0.5)</td></tr><tr><td>InstructMol</td><td>73.3(0.8)</td><td>85.9(1.3)</td><td>92.5(2.1)</td><td>79.9(0.6)</td><td>70.8(0.4)</td><td>67.4(0.9)</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>73.6(0.1)</td><td>86.6(0.5)</td><td>85.2(0.1)</td><td>80.6 (0.3)</td><td>68.8(0.6)</td><td>68.5 (0.3)</td><td>85.0 (1.2)</td><td>81.8 (0.1)</td></tr></table>
+
+![](images/c434e8ccf61d238368de59cf90ef4fa25ce335e61be991a6f200139de613f6c5.jpg)  
+Figure 2. (a) Test curves for various tuning methods. (b) Impact of different hyperparameters, where the y-axis denotes the ROC-AUC score (\%) and the x-axis represents the LoRA rank of node encoder hyperparameters. (c) Comparison of ROC-AUC (\%) and trainable parameter count between MIPT with varying rank values and full fine-tuning on the Tox21 dataset.
+
+![](images/77a9386df432572eb1480908c55ffda63df810eb8c786f47d0d83aaa82b9b886.jpg)
+
+![](images/ffc3535b85afbde4350721d6a77a223e649d1a1671bbf72dc2e4dc0d824206e1.jpg)
+
+Table 3. Ablation analysis of different configurations on BBBP, ClinTox, BACE, and SIDER datasets.  
+
+<table><tr><td colspan="2">Module</td><td colspan="2">Loss</td><td colspan="4">Datasets</td></tr><tr><td>LoRA</td><td>MGIP</td><td>\( \mathcal{L}_{NPP} \)</td><td>\( \mathcal{L}_{\text{cont}} \)</td><td>bbbp</td><td>ClinTox</td><td>BACE</td><td>Sider</td></tr><tr><td>✓</td><td>-</td><td>-</td><td>✓</td><td>68.72</td><td>67.76</td><td>86.41</td><td>60.75</td></tr><tr><td>-</td><td>✓</td><td>-</td><td>✓</td><td>72.45</td><td>80.19</td><td>85.22</td><td>67.17</td></tr><tr><td>✓</td><td>✓</td><td>-</td><td>✓</td><td>72.88</td><td>81.18</td><td>86.42</td><td>67.76</td></tr><tr><td>✓</td><td>-</td><td>✓</td><td>✓</td><td>69.43</td><td>70.99</td><td>85.51</td><td>61.26</td></tr><tr><td>-</td><td>✓</td><td>✓</td><td>✓</td><td>72.69</td><td>80.27</td><td>85.46</td><td>67.23</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>73.18</td><td>80.70</td><td>85.51</td><td>66.17</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>73.62</td><td>81.21</td><td>86.52</td><td>68.52</td></tr></table>
+
+Hyperparameter Analysis (RQ4). The test curves in Figure 2 (a) show that our method achieves higher ROC-AUC scores and remains stable across training epochs, outperforming FT, GPF, and GPF-plus, demonstrating the effectiveness of low-rank adaptation for generalization and efficiency. Figure 2 (b) compares the performance of four pre-trained models on the Tox21 dataset under different
+
+LoRA ranks (8, 16, 32, 64), showing that lower ranks maintain high performance while reducing parameter overhead, highlighting the efficiency of low-rank adaptation. Figure 2 (c) illustrates the trade-off between trainable parameters and performance, confirming that our approach significantly reduces computational costs without sacrificing accuracy compared to full fine-tuning. These results demonstrate that low-rank prompts provide a scalable and efficient alternative to traditional fine-tuning. Additionally, analysis of label distributions across datasets (Appendix C.1) underscores our method's robustness to noisy data and adaptability to real-world scenarios.
+
+# 6. Conclusion
+
+In this work, we propose Multilevel Informed Prompt Tuning (MIPT), a novel framework for enhancing pre-trained molecular encoders in molecular property prediction tasks. MIPT employs a lightweight, multi-level prompt learning
+
+network and a noise penalty mechanism to bridge the gap between pre-training and downstream tasks, while mitigating noise at both node and graph levels. Experiments on public and real-world datasets demonstrate MIPT's superior performance, highlighting its potential to advance drug discovery and materials science.
+
+# 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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+Yang, K., Swanson, K., Jin, W., Coley, C., Eiden, P., Gao, H., Guzman-Perez, A., Hopper, T., Kelley, B., Mathea, M., et al. Analyzing learned molecular representations for property prediction. Journal of chemical information and modeling, 59(8):3370-3388, 2019.  
+Zhang, J., Xiao, X., Huang, L.-K., Rong, Y., and Bian, Y. Fine-tuning graph neural networks via graph topology induced optimal transport. arXiv preprint arXiv:2203.10453, 2022.  
+Zhou, G., Gao, Z., Ding, Q., Zheng, H., Xu, H., Wei, Z., Zhang, L., and Ke, G. Uni-mol: A universal 3d molecular representation learning framework. 2023.  
+Zhou, G., Wang, Z., Yu, F., Ke, G., Wei, Z., and Gao, Z. S-molsearch: 3d semi-supervised contrastive learning for bioactive molecule search. arXiv preprint arXiv:2409.07462, 2024.
+
+# A. Proof: Effectiveness of LoRA Updates in GNNs
+
+Consider a GNN layer, the feature update at layer  $k + 1$  for node  $v$  is expressed as:
+
+$$
+h _ {v} ^ {(k + 1)} = \sigma \left(\mathbf {W} \cdot h _ {v} ^ {(k)} + \operatorname {A G G} \left(\left\{h _ {u} ^ {(k)}: u \in \mathcal {N} (v) \right\}\right)\right), \tag {22}
+$$
+
+where  $h^{(k)} \in \mathbb{R}^{n \times d}$  represents the node embeddings at layer  $k$ , and  $W \in \mathbb{R}^{d \times d}$  is a trainable weight matrix. The function  $\sigma(\cdot)$  denotes a non-linearity such as ReLU.
+
+Let  $M \in \mathbb{R}^{n \times d}$  stack all aggregated messages, and let  $\delta \in \mathbb{R}^{n \times d}$  be the gradient of the loss wr.t. outputs. Then:
+
+$$
+g = \nabla_ {W} \mathcal {L} = \delta^ {\top} M \tag {23}
+$$
+
+Since  $M$  results from summing over neighbors in sparse graphs, it is typically low-rank or approximately so. Thus,  $g$  is approximately low-rank. From the Eckart-Young-Mirsky theorem, the best rank- $r$  approximation of  $g$  under Frobenius norm is:
+
+$$
+\Delta W ^ {*} = \sum_ {i = 1} ^ {r} \sigma_ {i} u _ {i} v _ {i} ^ {\top}, \tag {24}
+$$
+
+where  $g = U\sigma V^{\top}$  is the SVD of gradient, and  $\sigma_{i}$  are the top  $r$  singular values. This can be written as  $\Delta W = AB$  with:
+
+$$
+A = U _ {r} \sigma_ {r} ^ {1 / 2} \in \mathbb {R} ^ {d \times r}, \quad B = \sigma_ {r} ^ {1 / 2} V _ {r} ^ {\top} \in \mathbb {R} ^ {r \times d}, \tag {25}
+$$
+
+which is precisely the LoRA parameterization. Following PAC-Bayes or norm-based generalization analysis:
+
+$$
+\mathbb {E} _ {\mathcal {D}} \left[ \mathcal {L} _ {\text {t e s t}} \right] \leq \mathbb {E} _ {\mathcal {D}} \left[ \mathcal {L} _ {\text {t r a i n}} \right] + O \left(\frac {\| \Delta W \| _ {F} ^ {2}}{n}\right), \tag {26}
+$$
+
+where  $n$  is the number of training samples and  $\| \cdot \|_F$  denotes the Frobenius norm. Since  $\| \Delta W \|_F^2 = \| AB \|_F^2 \ll \| G \|_F^2$ , the low-rank constraint imposed by LoRA acts as an implicit regularizer, effectively reducing model complexity and improving generalization performance.
+
+Then, the updated node feature equation becomes:
+
+$$
+\left. h _ {v} ^ {(k + 1)} = \sigma \left(\left(\mathbf {W} _ {0} + \mathbf {A B}\right) \cdot h _ {v} ^ {(k)} + \operatorname {A G G} \left(\left\{h _ {u} ^ {(k)}: u \in \mathcal {N} (v) \right\}\right)\right). \right. \tag {27}
+$$
+
+The key point is that the neighborhood  $\mathcal{N}(v)$  is solely determined by the static graph  $G = (V, E)$ , which can be described via the adjacency matrix  $\mathbf{A}_{\mathrm{adj}}$ :
+
+$$
+A _ {a d j} (u, v) = \left\{ \begin{array}{l l} 1, & \text {i f} u \in \mathcal {N} (v) \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {28}
+$$
+
+and does not depend on the model parameters.  $LoRA$  modifies only the linear transformation term and does not alter the adjacency matrix or neighborhood relations. Therefore,  $\mathcal{N}(v)$  are unaffected by the addition of AB.
+
+# B. Proof: Robustness of NPP
+
+# B.1. Derivation of Posterior Probability  $p(t \mid \mathcal{L}_{\mathrm{cls}}^i)$
+
+Let  $\mathcal{L}_{\mathrm{cls}} = \{\mathcal{L}_{\mathrm{cls}}^{1},\mathcal{L}_{\mathrm{cls}}^{2},\ldots ,\mathcal{L}_{\mathrm{cls}}^{N}\}$  represent the observed classification loss values, and assume that each sample  $\mathcal{L}_{\mathrm{cls}}^i$  follows a Gaussian Mixture Model (GMM) distribution. For a GMM with  $T$  components, the parameters include:
+
+-  $\pi_t$ : The mixture coefficient for the  $t$ -th component.  
+-  $\mu_t$ : The mean of the  $t$ -th Gaussian component.  
+-  $\sigma_t^2$ : The variance of the  $t$ -th Gaussian component.
+
+The posterior probability  $p(t \mid \mathcal{L}_{\mathrm{cls}}^i)$  can be computed using Bayes' rule:
+
+$$
+p (t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}) = \frac {p \left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} \mid t\right) \cdot \pi_ {t}}{\sum_ {j = 1} ^ {T} p \left(\mathcal {L} _ {\mathrm {c l s}} ^ {j} \mid j\right) \cdot \pi_ {j}}. \tag {29}
+$$
+
+For the  $k$ -th Gaussian component, the likelihood term is given by:
+
+$$
+p \left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} \mid t\right) = \frac {1}{\sqrt {2 \pi \sigma_ {t} ^ {2}}} \exp \left(- \frac {\left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} - \mu_ {t}\right) ^ {2}}{2 \sigma_ {t} ^ {2}}\right). \tag {30}
+$$
+
+Combining the likelihood and the prior, the posterior probability becomes:
+
+$$
+p \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right) = \frac {\pi_ {t} \cdot \frac {1}{\sqrt {2 \pi \sigma_ {t} ^ {2}}} \exp \left(- \frac {\left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} - \mu_ {t}\right) ^ {2}}{2 \sigma_ {t} ^ {2}}\right)}{\sum_ {j = 1} ^ {T} \pi_ {j} \cdot \frac {1}{\sqrt {2 \pi \sigma_ {j} ^ {2}}} \exp \left(- \frac {\left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} - \mu_ {j}\right) ^ {2}}{2 \sigma_ {j} ^ {2}}\right)}. \tag {31}
+$$
+
+# B.2. EM Updates
+
+# E-STEP: COMPUTE POSTERIOR PROBABILITIES
+
+It aims to estimate the posterior distributions of the loss, i.e.  $P(\mathcal{L}_{\mathrm{cls}}^i \mid \mu, \sigma)$ , by using the current estimated parameters  $\theta_{old} : \{\mu_{(old)}, \sigma^2\}$ . Hence, we compute the posterior probabilities  $p(t \mid \mathcal{L}_{\mathrm{cls}}^i)$  using the current estimates of  $\pi_t^{(n)}$ ,  $\mu_t^{(n)}$ , and  $\sigma_t^{(n)}$ :
+
+$$
+p ^ {(n)} (t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}) = \frac {\pi_ {t} ^ {(n)} \cdot p \left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} \mid \mu_ {t} ^ {(n)} , \sigma_ {t} ^ {(n)}\right)}{\sum_ {j = 1} ^ {T} \pi_ {j} ^ {(n)} \cdot p \left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} \mid \mu_ {j} ^ {(n)} , \sigma_ {j} ^ {(n)}\right)}. \tag {32}
+$$
+
+# M-STEP:PARAMETERUPDATES
+
+Using the posterior probabilities  $p^{(n)}(t \mid \mathcal{L}_{\mathrm{cls}}^i)$ , we update the parameters as follows:
+
+- Mixture Coefficient:
+
+$$
+\pi_ {t} ^ {(n + 1)} = \frac {\sum_ {i = 1} ^ {N} p ^ {(n)} \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right)}{N}. \tag {33}
+$$
+
+- Mean:
+
+$$
+\mu_ {t} ^ {(n + 1)} = \frac {\sum_ {i = 1} ^ {N} p ^ {(n)} \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right) \cdot \mathcal {L} _ {\mathrm {c l s}} ^ {i}}{\sum_ {i = 1} ^ {N} p ^ {(n)} \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right)}. \tag {34}
+$$
+
+- Variance:
+
+$$
+\sigma_ {t} ^ {2 (n + 1)} = \frac {\sum_ {i = 1} ^ {N} p ^ {(n)} \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right) \cdot \left(\mathcal {L} _ {\mathrm {c l s}} ^ {i} - \mu_ {t} ^ {(n + 1)}\right) ^ {2}}{\sum_ {i = 1} ^ {N} p ^ {(n)} \left(t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}\right)}. \tag {35}
+$$
+
+# B.3. Proof of NPP Robustness
+
+The Noise Prompt Penalty (NPP) framework leverages the posterior probability  $p(t \mid \mathcal{L}_{\mathrm{cls}}^i)$  to compute the weight  $w_i$  for each sample:
+
+$$
+w _ {i} = p (t \mid \mathcal {L} _ {\mathrm {c l s}} ^ {i}). \tag {36}
+$$
+
+By thresholding the weights  $w_{i}$  based on the mean value, we define:
+
+$$
+\hat {w} _ {i} = \left\{ \begin{array}{l l} 1, & \text {i f} w _ {i} > \operatorname {m e a n} \left(w _ {i}\right), \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {37}
+$$
+
+This filtering ensures that low-confidence (likely noisy) samples are excluded from the loss computation. The final NPP loss
+
+Input: Downstream task model  $g_{\theta}$ , labeled data  $\mathcal{D}$ , classification loss  $\mathcal{L}_{cls}$ , noise prompts penalty loss  $\mathcal{L}_{NPP}$ , learning rate  $\eta$ , noise weight  $\lambda$ , prompt-related parameters  $p_g$ , LoRA parameters  $\delta$ . Initialize model parameters  $\theta$ , GMM parameters  $\pi_t, \mu_t, \sigma_t$ .
+
+Algorithm 1 Noise Penalty Sample Training Procedure  
+Output: Optimized model parameters  $\theta$  
+1: for  $t = 1,2$    
+2: for epoch  $\coloneqq 1$  :max epoch do   
+3: for each sample  $(G_{i},y_{i})\in \mathcal{D}$  do   
+4: Compute prediction:  $\hat{y}_i\gets g_\theta (G_i)$    
+5: Compute classification loss using Eq. 14;   
+6: end for   
+7: GMM Modeling:  $w_{i}\gets \mathrm{GMM}(\mathcal{L}_{cls}^{i})$    
+8: while not converged or max iterations reached do   
+9: Update  $w_{i}$  using EM algorithm   
+10: end while   
+11: Update confidence  $\hat{w}_i$  using Eq. 17   
+12: Compute noise penalty loss using Eq. 13;   
+13: for iteration from 1 to  $T$  do   
+14:  $\mathcal{L}_{total} = \mathcal{L}_{cont} + \mathcal{L}_{Npp};$  optimize the model parameters  $\theta$  by  $\mathcal{L}_{total}$    
+15: end for   
+16: end for   
+17: Return  $\theta ,p_g,\delta$
+
+is computed as:
+
+$$
+\mathcal {L} _ {\mathrm {N P P}} = - \frac {1}{N} \sum_ {i = 1} ^ {N} \hat {w} _ {i} \log P \left(y ^ {i} \mid g \left(h _ {G, i}\right)\right). \tag {38}
+$$
+
+This framework achieves robustness by:
+
+- Utilizing the EM algorithm to iteratively refine the estimation of GMM parameters  $(\pi_t, \mu_t, \sigma_t^2)$ , leading to improved separation of noise and trustworthy samples.  
+- Filtering out low-confidence samples  $(\hat{w}_i = 0)$ , ensuring that only high-confidence samples contribute to the model optimization.
+
+The robustness of NPP is thus guaranteed by the iterative EM updates and selective loss computation, effectively mitigating the impact of noisy samples. Pseudocode is presented in Algorithm 1.
+
+# C. More Experiment Settings.
+
+# C.1. Details of Datasets.
+
+Pre-training datasets. The dataset includes 2 million unlabeled molecules drawn from the ZINC15 database, used for node-level self-supervised pre-training. For graph-level multi-task supervised pre-training, the preprocessed ChEMBL dataset is utilized, consisting of 456K molecules spanning 1310 distinct biochemical assays.
+
+Downstream Task. To evaluate model performance, we utilized eight binary graph classification datasets from MoleculeNet (Wu et al., 2018), described as follows:
+
+- BBBP (Martins et al., 2012): Evaluates blood-brain barrier penetration based on membrane permeability.  
+- Tox21 (tox): Provides toxicity data for 12 biological targets, including nuclear receptors and stress response pathways.
+
+- **ToxCast** (Richard et al., 2016): Contains toxicological measurements derived from over 600 in vitro high-throughput screening assays.  
+- SIDER: A comprehensive database of marketed drugs and their associated adverse drug reactions (ADR), categorized into 27 system organ classes.  
+- ClinTox: Includes qualitative data distinguishing FDA-approved drugs from those that failed clinical trials due to toxicity.  
+- MUV: A subset of PubChem BioAssay data, specifically curated using a refined nearest-neighbor analysis for validating virtual screening techniques.  
+- HIV: Contains experimental data assessing the inhibitory efficacy of compounds against HIV replication.  
+- BACE (Subramanian et al., 2016): Provides qualitative binding data for inhibitors targeting human  $\beta$ -secretase 1.
+
+Table 4. Dataset statistics.  
+
+<table><tr><td>Dataset</td><td>BBBP</td><td>Tox21</td><td>ToxCast</td><td>SIDER</td><td>Clintox</td><td>MUV</td><td>HIV</td><td>BACE</td></tr><tr><td># Molecule</td><td>2039</td><td>7831</td><td>8575</td><td>1427</td><td>1478</td><td>93127</td><td>41127</td><td>1513</td></tr><tr><td># Property</td><td>1</td><td>12</td><td>617</td><td>27</td><td>2</td><td>17</td><td>1</td><td>1</td></tr><tr><td>% Positive Label</td><td>76.44</td><td>6.24</td><td>12.60</td><td>56.76</td><td>50.61</td><td>0.31</td><td>96.49</td><td>45.67</td></tr><tr><td>% Negative Label</td><td>23.56</td><td>76.71</td><td>72.43</td><td>43.24</td><td>49.39</td><td>15.76</td><td>1.18</td><td>54.33</td></tr><tr><td>% Unknown Label</td><td>0</td><td>17.05</td><td>14.97</td><td>0</td><td>0</td><td>84.21</td><td>2.33</td><td>0</td></tr></table>
+
+# C.2. Details of pre-training strategies.
+
+We utilize five widely recognized strategies for pre-training GNN models, detailed as follows:
+
+- Deep Graph Infomax: Initially proposed by Velickovic et al. (Velicković et al., 2018), this method acquires expressive representations for graphs or nodes by maximizing mutual information between graph-level and substructure-level representations across various granularities.  
+- Edge Prediction: A common graph reconstruction task employed by many models, such as GAE (Kipf & Welling, 2016), where the goal is to predict the existence of edges between pairs of nodes.  
+- Attribute Masking: Introduced by Hu et al. (Hu et al., 2019), this technique involves masking node/edge attributes and then using GNNs to predict these attributes based on the neighboring structure.  
+- Context Prediction: Also proposed by Hu et al. (Hu et al., 2019), this approach uses subgraphs to forecast their surrounding graph structures, aiming to map nodes in similar structural contexts to nearby embeddings.
+
+In our model pre-training phase, we adhere to the methodologies described by Hu et al. (2019) for the tasks of Infomax, EdgePred, AttrMasking, and ContextPred. Subsequently, we engage in supervised graph-level property prediction to further boost the efficacy of our pre-trained models.
+
+Training Details. We perform five rounds of experiments with different random seeds for each experimental setting and report the average results. The projection head  $\theta$  is selected from a range of [1, 2, 3]-layer MLPs with equal widths. The hyper-parameter  $k$  of GPF-plus is chosen from the range [5,10,20]. Further details on the hyper-parameter settings can be found in the appendix.
+
+Implementation Details. All experiments were conducted on a high-performance computing server equipped with an NVIDIA 3090 GPU (24 GB memory). The implementation was based on Python 3.9, PyTorch 1.12, and the torc-geometric library. For the GNN architecture, we utilized Graph Isomorphism Network (GIN), configured with a hidden dimension of 300, 3 graph convolutional layers, ReLU activation, and batch normalization. The optimizer was Adam with a learning rate of 0.001, dropout rate is 0.5, the mask probability is 0.2. Pretraining tasks included node-level attribute prediction and
+
+![](images/6866da6ca1d6ec00a29c328dfb20c2a681afd44661b5549b18c4295e42aed381.jpg)  
+Figure 3. Test ROC-AUC of molecule property prediction using different tuning strategies with different pre-training strategies. Each point represents a particular individual downstream task from (Fang et al., 2024) (left) and our method (right).
+
+![](images/9cb8123f983a52b1eae0a7c147b23b344f27b70f4503273af70b58030ac3e82e.jpg)
+
+graph-wise classification, with data augmentation strategies such as node masking, edge sampling, and feature perturbation applied during training. The experiments were conducted on MoleculeNet datasets, running for 100 epochs with a batch size of 32. Performance was evaluated using ROC-AUC, and each experiment was repeated five times to ensure reproducibility, with the mean and standard deviation reported.
+
+# D. Additional Experiments
+
+# D.1. More comparison experiments
+
+Figure 3 compares the test ROC-AUC performance of molecular property prediction across different tuning strategies with various pre-training approaches. Each point represents the performance of an individual downstream task, with GPF shown on the left and our method on the right. The red diagonal line indicates parity, where points above the line represent improved performance compared to fine-tuning.
+
+The left plot reveals that the method from GPF exhibits negative transfer in several tasks, as some points fall below the diagonal, indicating a degradation in performance due to ineffective knowledge transfer. In contrast, the right plot demonstrates that our proposed Instruct Prompt effectively mitigates negative transfer, resulting in a more consistent improvement across tasks. This highlights the robustness of our approach in enhancing molecular representation learning and improving generalization across diverse downstream tasks.
+
+# D.2. More training cases.
+
+We provide more training curves in Figure 4 as a supplement to Section 5.3. Our method achieves a considerable performance improvement over other approaches during training. We note that on some smaller datasets, such as Clintox and BBBP, the training curves exhibit greater fluctuations. This behavior is hypothesized to stem from the increased number of parameters in our model coupled with the constrained data availability. Crucially, despite these fluctuations, overfitting is not observed, suggesting that the model remains robust. In stark contrast, training on larger datasets yields much smoother performance curves, with fluctuations being significantly less prominent.
+
+Table 5. The number of tunable parameters for different tuning strategies. * is the frozen parameters.  
+
+<table><tr><td></td><td>FT</td><td>GPF</td><td>GPF-plus</td><td>Ours</td></tr><tr><td>Size of GNN(Encoder)</td><td>1.86M</td><td>1.86M*</td><td>1.86M*</td><td>1.86M*</td></tr><tr><td>Size of Prompt</td><td>0</td><td>0.3K</td><td>3-12K</td><td>0.115M</td></tr><tr><td>Size of Graph Linear Layer</td><td>0.6K-0.18M</td><td>0.6K-0.18M</td><td>0.6K-0.18M</td><td>0.6K-0.18M</td></tr><tr><td>Size of Total Tunable Part</td><td>1.92M-2.14M</td><td>0.9K-0.21M</td><td>3.6K-0.192M</td><td>0.175M-0.295M</td></tr><tr><td># Params(%)</td><td>100</td><td>0.046-0.98</td><td>0.187-8.97</td><td>9.1-13.78</td></tr></table>
+
+![](images/f6105c713d3d5307b60adbe8cbc249a22c83ae52b61cfa7c353da3d7f1f98e28.jpg)
+
+![](images/b6f1ebf50463dbb0d5f58ac7426c07a6f7a8affd619e6b78b25b4b4b8f87f8be.jpg)
+
+![](images/ce43dcd4b1a36ee430ecd52bda59c562f4ac49278e4fd28540917783cb8d4c40.jpg)  
+Figure 4. Test curves of different tuning strategies on pretrained GINs. Solid and dashed lines indicate training and validation curves, respectively.
+
+![](images/d47a9e7481e9d71af3bc160fd6f7213576c1bb714ce9eec027fd37f9b0332a8f.jpg)
+
+Table 6. Ablation analysis of different configurations on Tox21, Toxcast, HIV, MUV, BBBP, Clintox, BACE and SIDER datasets.  
+
+<table><tr><td colspan="2">MODULE</td><td colspan="2">LOSS</td><td colspan="8">DATASETS</td></tr><tr><td>LORA</td><td>MGIP</td><td>\( \mathcal{L}_{NPP} \)</td><td>\( \mathcal{L}_{CONT} \)</td><td>TOX21</td><td>TOXCAST</td><td>HIV</td><td>MUV</td><td>BBBP</td><td>CLINTOX</td><td>BACE</td><td>SIDER</td></tr><tr><td>✓</td><td>-</td><td>-</td><td>✓</td><td>75.31</td><td>64.34</td><td>75.95</td><td>82.44</td><td>68.72</td><td>67.76</td><td>86.41</td><td>60.75</td></tr><tr><td>-</td><td>✓</td><td>-</td><td>✓</td><td>80.57</td><td>68.78</td><td>79.97</td><td>84.47</td><td>72.45</td><td>80.19</td><td>85.22</td><td>67.17</td></tr><tr><td>✓</td><td>✓</td><td>-</td><td>✓</td><td>80.19</td><td>67.70</td><td>78.50</td><td>83.07</td><td>72.88</td><td>81.18</td><td>86.42</td><td>67.76</td></tr><tr><td>✓</td><td>-</td><td>✓</td><td>✓</td><td>76.40</td><td>68.55</td><td>75.03</td><td>79.48</td><td>69.43</td><td>70.99</td><td>85.51</td><td>61.26</td></tr><tr><td>-</td><td>✓</td><td>✓</td><td>✓</td><td>80.29</td><td>68.27</td><td>80.42</td><td>84.71</td><td>72.69</td><td>80.27</td><td>85.46</td><td>67.23</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>80.45</td><td>68.52</td><td>79.84</td><td>82.45</td><td>73.18</td><td>80.70</td><td>85.51</td><td>66.17</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>80.60</td><td>68.81</td><td>81.76</td><td>84.96</td><td>73.62</td><td>81.21</td><td>86.52</td><td>68.52</td></tr></table>
+
+# D.3. More ablation study
+
+Here, Table 6 shows the comprehensive ablation results on eight datasets. A comparison of the parameters of the different high-efficiency fine-tuning methods is shown in Table 5. From the experimental results, we can conclude that our method can achieve the optimal performance with fewer parameters, showing superiority.
+
+# E. Limitations and Future Work
+
+As discussed in Section 5, while our method significantly outperforms state-of-the-art baselines, it has certain limitations in few-shot learning. This is primarily due to the constraints of GMM modeling, which requires at least two samples per class for effective learning, limiting its applicability in few-shot scenarios. Additionally, GMM introduces some computational overhead, though its resource consumption remains modest. Furthermore, our method exhibits higher standard deviations across multiple runs with different ranks, likely due to model uncertainty in handling diverse graph structures. Addressing the integration of node-level and graph-level features remains an open challenge for future research.

2025/MIPT_ Multilevel Informed Prompt Tuning for Robust Molecular Property Prediction/images.zip

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2025/MIRROR_ Make Your Object-Level Multi-View Generation More Consistent with Training-Free Rectification/9418165a-02db-46f2-93ca-76a3b807ff88_origin.pdf

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2025/MIRROR_ Make Your Object-Level Multi-View Generation More Consistent with Training-Free Rectification/full.md

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+# MIRROR: Make Your Object-Level Multi-View Generation More Consistent with Training-Free Rectification
+
+![](images/f142524b5b9c41dfd51239bd40b23854063128e7e78ff036d4d1288cb3c5ee9e.jpg)  
+Figure 1: Generated multi-view images by applying MIRROR on SyncDreamer (Liu et al., 2023b) and VideoMV (Zuo et al., 2024) for rectification. MIRROR effectively improves the consistency of generated results in both image-based and text-based multi-view generation models.
+
+# Abstract
+
+Multi-view Diffusion has greatly advanced the development of 3D content creation by generating multiple images from distinct views, achieving remarkable photorealistic results. However, existing works are still vulnerable to inconsistent 3D geometric structures (commonly known as Janus Problem) and severe artifacts. In this paper, we introduce MIRROR, a versatile plug-and-play method that rectifies such inconsistencies in a training-free manner, enabling the acquisition of high-fidelity, realistic structures without compromising diversity. Our key idea focuses on tracing the motion trajectory of physical points across adjacent viewpoints, enabling rectifications based on neighboring observations of the same region.
+
+*Equal contribution  ${}^{1}$  Department of Mathematical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Shijingshan District, Beijing, China. Correspondence to: Congying Han <hancy@ucas.ac.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada, PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Technically, MIRROR comprises two core modules: Trajectory Tracking Module (TTM) for pixel-wise trajectory tracking that labels identical points across views, and Feature Rectification Module (FRM) for explicitly adjustment of each pixel embedding on noisy synthesized images by minimizing the distance to corresponding block features in neighboring views, thereby achieving consistent outputs. Extensive evaluations demonstrate that MIRROR, through seamless integration with a variety of off-the-shelf object-level multi-view diffusion models, achieves efficient improvements in consistency and fidelity, thereby functioning as a practical rectification tool.
+
+# 1. Introduction
+
+Recently, 3D asset generation has become one of the most promising fields in computer vision, and demonstrated extensive application in artworks and creation. Inspired by the success in text-to-image generation (Rombach et al., 2022), text-to-3D generation models have increasingly attracted significant attention. Regarding the scarcity of 3D data, the
+
+seminal works leverages 2D diffusion models as priors to optimize 3D representations using score distillation (Poole et al., 2022; Liu et al., 2023a; Wang et al., 2023; Lin et al., 2023; Qian et al., 2023; Wang et al., 2024b). Unfortunately, these methods suffer from serious inconsistencies, most notably the Janus Problem and content drifting, due to the lack of 3D prior knowledge. To address these issues, MVDream (Shi et al., 2023b) was the first to introduce the multi-view diffusion model, leveraging joint training on 2D and 3D data to simultaneously generate four orthogonal views. Building upon it, multi-view diffusion models such as SyncDreamer (Liu et al., 2023b), MVD-Fusion (Hu et al., 2024) and VideoMV (Zuo et al., 2024) have been proposed to employ strategies for spatial and temporal alignment to enhance multi-view consistency, thereby enabling the simultaneous generation of dense views at the object level.
+
+Despite notable progress in multi-view diffusion models, they continue to struggle with issues of poor quality and inconsistencies across different perspectives as shown in Fig. 1 and Fig. 5. These issues primarily stem from the extensive resources required for training, the incompleteness of training datasets, and inaccurate correspondences between 3D spatial points and their projections across views. Consequently, it is crucial to delve deeper into efficient 3D asset generation method that accurately captures the real correspondences. To this end, we devise an training-free method, namely MIRROR, to amend the multiple images produced by multi-view diffusion models during sampling process, thereby providing consistent results to perform high-quality 3D reconstruction. We showcase remarkable rectification results in Fig. 1.
+
+The core of MIRROR lies in modifying the current viewpoint based on the content of adjacent viewpoints, thereby enforcing neighbor-view consistency and naturally deriving whole-view coherence. To achieve effective rectification, we approach this from the following two aspects: (1) What should we focus on in neighbor views for rectification? Compared to image-level adjustments, we opt for point-to-point corrections, ensuring that the features of each physical point remain consistent across different viewpoints. The main challenge of our approach lies in effectively tracking the image coordinates of the same physical point across neighbors. To tackle this, Trajectory Tracking Module (TTM) is proposed to successfully track the motion trajectory of physical points as viewpoints change. This strategy enables precise adjustments for each point, selectively targeting relevant features while disregarding irrelevant ones, which is a key advantage of TTM over the epipolar correspondence methods (Ye et al., 2024; Kant et al., 2024; Zhou & Tulsiani, 2023; Li et al., 2024b). (2) How to perform rectification effectively and efficiently? We observe that cross-view attentions (Shi et al., 2023b; Li et al., 2024b) inherently struggle to enforce 3D consistency, as they lack
+
+![](images/0b18e8e5284427ccff4af1177bc8016269bf713e7ab67ec61fb874e74ae9ac67.jpg)  
+Figure 2: High-level Overview of MIRROR. This is a plug-and-play rectification technique that can be seamlessly incorporated into multi-view diffusion models to enhance consistency and fidelity without any training or fine-tuning.
+
+explicit geometric constraints and rely solely on implicit learning through network weights. Given the limited availability of 3D data, such methods are prone to becoming trapped in local optima and often exhibit poor generalization. Although CoSER (Li et al., 2024a) attempts to address this by injecting local spatial correspondence information from neighboring views into the cross-view attention to promote multi-view consistency, its effectiveness remains limited due to the absence of depth information—which results in inaccurate view correspondences—and the intrinsic challenges in optimizing attention layers as discussed earlier. Therefore, unlike previous approaches that rely on fine-tuning within the attention mechanism, we propose Feature Rectification Module (FRM), which, without requiring any training or fine-tuning, performs feature rectification directly on the predicted images during inference, thereby achieving enhanced performance in a more streamlined and efficient manner. To this end, we compute the similarity between each pixel's feature in the current view and its counterpart in adjacent views during sampling, producing a gradient map that guides denoising. Additionally, to mitigate potential biases and maintain regional continuity, we extend the reference pixel point into a block.
+
+Technically, we propose a two-stage inference strategy. In the first stage, we generate a set of images using the multiview diffusion model to extract depth information for each view. Based on transformations in camera angles, in TTM, we introduce a trajectory tracking operator guided by depth maps to track motion trajectories of 3D physical points from the current view to adjacent views at the fixed elevation. This enables precise localization of the corresponding points in neighboring views. We then utilize DDIM Inversion (Song et al., 2020) to progressively add noise to the generated images, transitioning them to intermediate noisy states for the subsequent denoising process. During each sampling step of the second-stage inference, we apply a corrective gradient map based on pixel-level similarity in FRM to refine the denoising direction of the multi-view diffusion model, resulting in images with improved quality and consistency. To address background redundancy caused by surrounding blocks, we implement a dual-anchor feature fusion strategy, which fuses relevant features from adjacent views. This not only identifies meaningful features but also significantly improves computational efficiency.
+
+Furthermore, gradient-based guidance, as discussed in the prior work (Dhariwal & Nichol, 2021), is known to be computationally intensive due to the gradient computation requirements for large neural networks. To mitigate this, we provide a theoretical analysis demonstrating that omitting the UNet Jacobian term in the diffusion model during gradient-based denoising guidance introduces negligible error. Experimentally, we validate that this optimization achieves a twofold acceleration in the rectification process without compromising the quality of the generated images. An overview of MIRROR is depicted in Fig. 2.
+
+In summary, our contributions are as follows:
+
+- We innovatively introduce MIRROR, an efficient, training-free, plug-and-play method that enhances multi-view consistency in 3D asset generation by directly rectifying latent noisy images across views during the sampling process.  
+- We present TTM based on depth information to precisely ascertain the corresponding positions of points across distinct views, and FRM to eliminate ambiguity by enforcing consistency in the representation of the same physical point across different viewpoints.  
+- Extensive experiments demonstrate that MIRROR consistently enhances the performance of various multi-view generator, both quantitatively and qualitatively.
+
+# 2. Related Work
+
+# 2.1.3D Generation
+
+Due to the scarcity of 3D data, numerous studies have turned to high-quality 2D diffusion models for 3D generation tasks. A fundamental technique in this domain, Score Distillation Sampling (SDS), introduced by DreamFusion (Poole et al., 2022), utilizes pre-trained 2D models as priors to optimize 3D representations, enabling zero-shot text-to-3D generation without 3D data. Inspired by it, several works adopt this pipeline to optimize a neural radiance field (Mildenhall et al., 2021) to generate 3D assets (Lin et al., 2023; Liu et al., 2023a; Qian et al., 2023; Wang et al., 2024b; Zhu et al., 2023; Huang et al., 2023). Furthermore, DreamGaussian (Tang et al., 2023) and GaussianDreamer (Yi et al., 2024) accelerate the optimization process by applying Gaussian Splitting techniques (Kerbl et al., 2023) in Score Distillation Sampling. However, these 2D-lifting methods exhibit critical drawbacks, such as severe multi-view inconsistencies and slow generation speeds, often requiring tens of thousands of iterations to produce a single 3D asset. In response to these challenges, recent research on multi-view diffusion models has emerged as a promising direction in 3D generation, wherein the model is capable of generating multiple views of an object in a single inference process.
+
+# 2.2. Multi-View Diffusion Methods
+
+Recent advancements have extended 2D diffusion models to generate multi-view images for reconstruction. Notable developments include MVDream (Shi et al., 2023b), Viewset Diffusion (Szymanowicz et al., 2023), SyncDreamer (Liu et al., 2023b). However, these methods face challenges like texture ambiguity and high computational costs. To address these, Wonder3D (Long et al., 2024) introduces a cross-domain model for generating normal maps, while Zero123++ (Shi et al., 2023a) and One-2-3-45 (Liu et al., 2024) improve texture quality and accelerate 3D reconstruction. Despite these innovations, quality issues persist due to reliance on limited synthetic data. Video generative models, with their temporal modules ensuring frame consistency, are increasingly favored for multi-view generation over image diffusion models. SV3D (Voleti et al., 2024) and IM-3D (Melas-Kyriazi et al., 2024) enhance 3D generation by leveraging advanced video diffusion models to optimize multiview synthesis for improved output quality. And VideoMV (Zuo et al., 2024) also develops a consistent multi-view generation model using video generative models, further improving multi-view consistency with a 3D Aware Denoising Sampling technique. While these methods improve consistency, they lack explicit geometric constraints, entirely relying on learned networks. With limited 3D data, they are prone to local optima, and issues like Janus Problem and content drifting remain at inference. Building on these efforts, we propose MIRROR, an efficient, training-free rectification technique specifically designed to enhance 3D consistency in these dense multi-view diffusion models.
+
+# 3. Preliminaries
+
+# 3.1. Diffusion Models
+
+Diffusion Models (DMs) (Ho et al., 2020) learn a target distribution  $p_{\theta}(x_0)$ , by progressively denoising a standard Gaussian distribution. This process is mathematically represented as:  $p_{\theta}(x_0) = \int p_{\theta}(x_{0:T}) dx_{1:T}$ , where  $x_{1:T}$  denotes intermediate noisy samples. In the forward process, DMs iteratively add Gaussian noise  $\varepsilon$  to the clean data  $x_0$ , controlled by a pre-defined variance schedule, formulated as:
+
+$$
+x _ {t} = \sqrt {\bar {\alpha} _ {t}} x _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} \varepsilon , \varepsilon \sim \mathcal {N} (0, I). \tag {1}
+$$
+
+Conversely, in the reverse process, DMs employ a denoiser  $\varepsilon_{\theta}$ , parameterized as a UNet, to predict the noise added at each denoising time step  $t$ .
+
+Furthermore, to reduce the computational demands associated with synthesizing high-resolution images, Latent Diffusion Models (LDMs) (Rombach et al., 2022) implement a pre-trained encoder  $\mathcal{E}$  of VQGAN to compress image  $x_0\in \mathbb{R}^{H\times W\times C}$  into a low-dimensional latent feature,  $z_{0} = \mathcal{E}(x_{0})$  , where  $z_{0}\in \mathbb{R}^{h\times w\times c}$  . In addition, LDMs uti
+
+![](images/4f2cc5dd56a839c983b09986b44c04873ac6e696645a4d03922cedb7507a5be5.jpg)  
+Figure 3: Pipeline of MIRROR. We employ a two-stage sampling strategy. In stage 1, given a single image or a text prompt as input, we employ a base multi-view diffusion model to sample a set of images. In stage 2, we first apply DDIM Inversion to convert generated images into the noisy state, and take it as the initial noise for the second stage. Then, Trajectory Tracking Module (TTM) is employed to capture the motion trajectory of physical points from the current view to the adjacent views using the depth maps derived from generated images in stage 1. For Feature Rectification Module (FRM), the block features are fused based on dual anchors,  $R_{pos}(\boldsymbol{u})$  and  $R_{neg}(\boldsymbol{u})$ . Then, we explicitly enforce consistency for the same physical point across neighbor views by calculating the distance between the feature of point  $\boldsymbol{u}$  and the corresponding block features in adjacent views, and compute the overall gradient map  $\nabla \mathcal{L}_t$  to guide the denoising process for rectification.
+
+![](images/314f9d5d2508bdf962009d1b795be0f936cd87a5639f2f3a4caf56168ccf3db2.jpg)
+
+lize deterministic DDIM sampling (Song et al., 2020) to efficiently transform random noise  $z_{T}$  into clean data  $z_{0}$  through a sequence of discrete time steps from  $t = T$  to  $t = 1$ , which is formulated as:
+
+$$
+\begin{array}{l} z _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\frac {z _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \varepsilon_ {\theta} (z _ {t} , t)}{\sqrt {\bar {\alpha} _ {t}}}\right) \tag {2} \\ + \sqrt {1 - \bar {\alpha} _ {t - 1}} \varepsilon_ {\theta} (z _ {t}, t). \\ \end{array}
+$$
+
+This sequence ultimately results in the recovery of the latent clean state  $z_0$ , which is then processed by the VQGAN decoder  $\mathcal{D}$  to reconstruct a high-fidelity image  $x_0 = \mathcal{D}(z_0)$ .
+
+Evolved from classical LDMs (Rombach et al., 2022), Multi-View Diffusion Models (MV-DMs) have broadened their focus from single images to handling sequences of multi-view images, essentially treating them like videos. Specifically, consider an image sequence  $x_0$  represented in  $\mathbb{R}^{F \times H \times W \times 3}$ , where  $F$  denotes the number of frames. MV-DMs encode this video into a latent space, represented as  $z_0 = \mathcal{E}(x_0)$ , where  $z_0 \in \mathbb{R}^{F \times h \times w \times c}$ .
+
+# 3.2. DDIM Inversion
+
+In contrast to DDIM, DDIM inversion acts as a forward process, where clean data  $z_{0}$  progressively transitions into a noisy state  $z_{T}$ , effectively reversing the original sampling
+
+sequence Eq. (2) into:
+
+$$
+\begin{array}{l} z _ {t + 1} = \sqrt {\bar {\alpha} _ {t + 1}} \left(\frac {z _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \varepsilon_ {\theta} (z _ {t} , t)}{\sqrt {\bar {\alpha} _ {t}}}\right) \tag {3} \\ + \sqrt {1 - \bar {\alpha} _ {t + 1}} \varepsilon_ {\theta} (z _ {t}, t). \\ \end{array}
+$$
+
+# 4. Methods
+
+In this section, we propose MIRROR, a training-free modifier specifically designed to enhance consistency across images generated by multi-view diffusion models. In Sec. 4.1, we systematically outline our approach. Then, we introduce Trajectory Tracking Module (TTM) that tracks point transformation correspondences between adjacent views (Sec. 4.2) and proposes a Feature Rectification Module (FRM) that integrates 3D-aware feature information to rectify inconsistencies during the sampling process (Sec. 4.3).
+
+# 4.1. Framework of MIRROR
+
+Given a pre-trained multi-view diffusion, we adopt a two-stage sampling strategy to rectify generated images. In the first stage, the base multi-view model generates a set of images  $\hat{x}_0$ , from which the depth map  $Z_{absolute}'$  is derived via monocular depth estimation. Subsequently, DDIM Inversion is then applied to convert clean images into noisy state  $z_T$ , serving as the initial noise for the second stage of the sampling process. At each denoising step  $t$ , TTM
+
+tracks point correspondences of the predicted latent images,  $\hat{z}_0 = \hat{z}_0(z_t;t)$ , across adjacent views with  $Z_{absolute}^{\prime}$ . Given this, FRM integrates 3D-aware feature information according to the correspondences from TTM, and computes a pixel-level rectification loss gradient  $\nabla \mathcal{L}_t$  to refine the denoising direction of the predicted noise  $\varepsilon_{\theta}$ . Finally,  $z_{t - 1}$  is generated through DDIM sampling with the rectified noise  $\hat{\varepsilon}_{\theta}$ , enhancing the quality and consistency of the final output. The pipeline of MIRROR is illustrated in Fig. 3, with the corresponding pseudocode provided in Algorithm 1.
+
+# 4.2. Trajectory Tracking Module
+
+Pixel-Wise Trajectory Tracking Transformation. In theory, there exists a pixel-wise correspondence between two images of an object projected from different viewpoints. We observe that large viewpoint gaps under arbitrary camera trajectories tend to reduce cross-view correlation, resulting in increased reconstruction errors and limited improvement. In contrast, fixing the elevation angle strengthens inter-view correspondence and yields better results by providing more related information. Therefore, we adopt a standardized setting with a fixed elevation angle. In this setup, we consider  $F$  views captured by a camera rotating horizontally around the object in a standardized manner. The circular camera trajectories ensure stable and uniform coverage of the viewing space, thereby enhancing view correlation and improving the quality of 3D reconstruction. Furthermore, sampling across multiple elevation settings helps recover occluded regions that are invisible from a single elevation. Specifically, this configuration maintains a constant elevation angle for the images while distributing the azimuth angles uniformly across  $360^{\circ}$ . Thus, let  $\alpha$  represent the change in azimuth angle between each pair of adjacent views.
+
+Next, we establish the pixel-wise correspondence on the imaging plane between two differing viewpoints. As the camera angle variation is confined solely to the azimuth direction, this motion can be modeled as the imaging plane rotating around the object along the elevation axis. Consequently, the  $y$ -coordinate (height) remains constant, while the  $x$ -coordinate (width) undergoes a rotation transformation related to the  $z$ -coordinate (depth). Let  $V$  be the function space  $L^2(\mathbb{R}^2, \mathbb{R})$ , the specific transformation formula defined on  $V$  is outlined in Definition 4.1.
+
+Definition 4.1. The trajectory tracking operator  $\mathcal{T}_{\alpha}:V\to V$  , defined by an azimuthal rotation angle  $\alpha \in \mathbb{R}$  , is formulated as  $\mathcal{T}_{\alpha}(Z(\pmb {u})) = Z_{\alpha}(\pmb{u}_{\alpha})$  , where  $\pmb {u} = (x,y)$ $u_{\alpha} =$ $(x_{\alpha},y_{\alpha})$  . The coordinates transformation is given by
+
+$$
+\left\{ \begin{array}{l} x _ {\alpha} = \left(x - \frac {W}{2}\right) \cos \alpha + \frac {W}{2} - z \sin \alpha , \\ y _ {\alpha} = y, \end{array} \right. \tag {4}
+$$
+
+where  $z$  represents the depth (distance from the camera center to the object), and  $W$  is the width of the image.
+
+Depth Prior Estimation. Since images directly output by multi-view diffusion models typically lack depth information, we utilize a monocular depth estimation model  $\mathcal{H}$  to estimate this crucial detail  $z$  in Eq. (4). However, given the potential for considerable bias in depth estimates derived from noisy images, we feed the clean images  $\hat{x}_0$  that have been denoised by the multi-view diffusion model into  $\mathcal{H}$ , enabling us to accurately assess depth information for each viewpoint, ultimately leading to the creation of a relative normalized depth map  $\mathcal{Z}_{\text{relative}} := \mathcal{H}(\hat{x}_0) \in \mathbb{R}^{F \times H \times W}$ . Although the depth is estimated from a monocular model, the depth scale factor  $r$  remains consistent across views, as each baseline multi-view diffusion model fixes the camera distance under circular poses during training. This consistency enables the transformation of relative depth into absolute depth for cross-view alignment, formulated as  $\mathcal{Z}_{\text{absolute}} = r \cdot \mathcal{Z}_{\text{relative}}$ .
+
+Furthermore, we establish that as the denoising step  $t$  decreases, the trajectory tracking of the predicted images  $\hat{x}_0(x_t,t) = \frac{1}{\sqrt{\overline{\alpha}_t}} (x_t - \sqrt{1 - \overline{\alpha}_t}\varepsilon_\theta (x_t,t))$  at time step  $t$  converges to that of the real images  $x_0$ , which is detailed in the following proposition.
+
+Proposition 4.2. (Convergence Analysis). The trajectory tracking error of  $\mathcal{T}_{\alpha}$  is upper-bounded by the error induced by the diffusion model, which is given by
+
+$$
+\left\| \mathcal {T} _ {\alpha} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \mathcal {T} _ {\alpha} \left(x _ {0}\right) \right\| \leq O \left(\left\| \hat {x} _ {0} \left(x _ {t}, t\right) - x _ {0} \right\|\right), \tag {5}
+$$
+
+where  $||\cdot ||$  is the  $L_{2}$  norm and  $O$  denotes a term of the same infinitesimal order. Moreover, it converges to zero as  $t\to 0$
+
+The proof is provided in Appendix B.1. Next, to leverage TTM during sampling for guiding the denoising direction via pixel-level correspondence, the depth map must be interpolated to match the resolution of the latent space, resulting in:  $\mathcal{Z}_{absolute}^{\prime} = \mathcal{B}(\mathcal{Z}_{absolute},h,w)$ , where  $\mathcal{B}$  represents the bilinear interpolation function. And the visualization results of trajectory tracking based on depth information are provided in Appendix G. Unlike methods (Ye et al., 2024; Kant et al., 2024; Zhou & Tulsiani, 2023; Li et al., 2024b) that integrate epipolar geometry into multi-view attention, MIRROR uniquely employs a more precise and efficient point-to-point correspondence tracking strategy. As shown in Fig. 4, we demonstrate that epipolar correspondence establishes an inaccurate relationship, thus introduces irrelevant background features, leading to overly smooth rectification that lacks detail and the Janus problem. In contrast, MIRROR-rectified images maintain higher fidelity and visual quality.
+
+# 4.3. Feature Rectification Module
+
+Feature Scrutiny. To maintain the continuity of pixel-wise feature information and in consideration of potential biases
+
+![](images/eacdf5f572c21b46a4ac869a1821da776286672db354386a250a75bf9649c773.jpg)  
+Figure 4: Comparison of the rectification results between the epipolar correspondence method and MIRROR. For better presentation, the background of each image is removed.
+
+in depth information estimation, we adopt a more robust approach, which extracts the most relevant corresponding features from adjacent views, thereby delivering a more precise direction for consistency rectification. Specifically, for each spatial point  $\pmb{u}$  in the current view, we select not only its corresponding point  $u_{\alpha}$ , but also the surrounding  $3\times 3$  block in adjacent views, making a total of nine points, denoted as  $B_{u_{\alpha}}$ , as the feature interaction block for point  $\pmb{u}$ . This method is figuratively described as the "looking forward and back" feature scrutiny approach, emphasizing a comprehensive examination of the contextual area.
+
+Feature Information Fusion. As mentioned in the introduction, we directly utilize the predicted map  $\hat{z}_0 = \hat{z}_0(z_t; t)$ :
+
+$$
+\hat {z} _ {0} = \frac {z _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \varepsilon_ {\theta} (z _ {t} , t)}{\sqrt {\bar {\alpha} _ {t}}}, \hat {z} _ {0} \in \mathbb {R} ^ {F \times h \times w \times c}, \tag {6}
+$$
+
+from each reverse step  $t$  as feature tensors, rather than extracting features from the intermediate layers of the UNet network. Our approach offers a simpler and more effective method of feature extraction, providing superior pixel-level alignment at the same time. Since the interaction block  $B_{\mathbf{u}_{\alpha}}$  may contain invalid background information, we employ a feature information fusion method based on both positive and negative anchors. The target correction point  $\mathbf{u}$  of the current view is selected as the positive anchor  $R_{pos}(\mathbf{u})$ , while the mean of the four corners serves as the negative anchor  $R_{neg}(\mathbf{u})$ . We calculate the cosine similarity between points in the feature interaction block  $B_{\mathbf{u}_{\alpha}}$  and dual anchors,  $R_{pos}(\mathbf{u})$  and  $R_{neg}(\mathbf{u})$ . Subsequently, compute the weighted map  $W$ , where higher values indicate closer distance to the positive anchor  $R_{pos}(\mathbf{u})$  and greater distance from the negative anchor  $R_{neg}(\mathbf{u})$ :
+
+$$
+W = m ^ {+} \operatorname {s o f t m a x} \left(\mathcal {C} ^ {+} (\boldsymbol {u}, \boldsymbol {v})\right) + m ^ {-} \operatorname {s o f t m a x} \left(\mathcal {C} ^ {-} (\boldsymbol {u}, \boldsymbol {v})\right), \tag {7}
+$$
+
+for all  $\pmb{v} \in B_{\pmb{u}_{\alpha}}$ , where  $m^{\pm}$  denote the combined coefficients satisfying  $m^{+} + m^{-} = 1$ , and  $\mathcal{C}^{\pm}(\pmb{u}, \pmb{v})$  refer to
+
+$$
+\begin{array}{l} \mathcal {C} ^ {+} (\boldsymbol {u}, \boldsymbol {v}) = \text {C o s i n e S i m i l a r i t y} \left(R _ {\text {p o s}} (\boldsymbol {u}), \boldsymbol {v}\right), \\ \mathcal {C} ^ {-} (\boldsymbol {u}, \boldsymbol {v}) = \text {C o s i n e S i m i l a r i t y} \left(R _ {\text {p o s}} (\boldsymbol {v})\right) \end{array} \tag {8}
+$$
+
+$$
+\mathcal {C} ^ {-} (\boldsymbol {u}, \boldsymbol {v}) = 1 - \text {C o s i n e S i m i l a r i t y} \left(R _ {n e g} (\boldsymbol {u}), \boldsymbol {v}\right).
+$$
+
+Given the weighted map  $W$ , the feature interaction block
+
+can be fused as a mapping  $\mathcal{M}:V\to V$  , where  $\mathcal{M}$  is defined by
+
+$$
+\mathcal {M} \left(Z _ {\alpha} \left(\boldsymbol {u} _ {\alpha}\right)\right) \triangleq \sum_ {\boldsymbol {v} \in B _ {\boldsymbol {u} _ {\alpha}}} W _ {\boldsymbol {v}} \times Z _ {\alpha} (\boldsymbol {v}). \tag {9}
+$$
+
+This approach is strategically designed to distance the fused feature points from irrelevant background and enhance their relevance to the target correction point  $\mathbf{u}$ .
+
+Rectification Formulation. At first, We can define the adjacent operator  $\mathcal{F}_{\alpha}$  and use its  $L_{2}$ -norm to measure the difference between two different views as follows.
+
+Definition 4.3. The operator  $\mathcal{F}_{\alpha}:V\to V$  is referred to as the adjacent operator, and is defined as
+
+$$
+\mathcal {F} _ {\alpha} \triangleq \mathcal {I} - \mathcal {M} \circ \mathcal {T} _ {\alpha},
+$$
+
+where  $\mathcal{I}$  denotes the identity operator on space  $V$ , and  $\mathcal{M}$  and  $\mathcal{T}_{\alpha}$  is defined in Eq. (9) and Definition 4.1, respectively, with  $\circ$  indicating the composition of operators. Note that the composition operator  $\mathcal{M} \circ \mathcal{T}$  is a group homomorphism (detailed in Appendix B.4). Then, the correction loss function of the spatial point  $\pmb{u}$  is given by:
+
+$$
+\mathcal {L} _ {\alpha} (\boldsymbol {u}) \triangleq \| \mathcal {F} _ {\alpha} (Z (\boldsymbol {u})) \| _ {2} ^ {2} = \| (\mathcal {I} - \mathcal {M} \circ \mathcal {T} _ {\alpha}) (Z (\boldsymbol {u})) \| _ {2} ^ {2}. \tag {10}
+$$
+
+Specifically, for each view  $i$ , we apply  $\mathcal{L}_{\alpha}$  to the feature map  $\hat{z}_0^{(i)}(\boldsymbol{u};t)$  at each point  $\boldsymbol{u}$ , derived from Eq. (6), for time step  $t$ . We use one preceding and one subsequent view as reference neighboring views for adjustments. The loss function is formulated as:
+
+$$
+\mathcal {L} _ {i} (\boldsymbol {u}; t) = \sum_ {j \in \mathrm {A} _ {i}} \omega_ {j} \| \hat {z} _ {0} ^ {(i)} (\boldsymbol {u}; t) - \mathcal {M} ^ {(j)} \circ \mathcal {T} _ {\alpha} \left(\hat {z} _ {0} ^ {(i)} (\boldsymbol {u}; t)\right) \| _ {2} ^ {2}, \tag {11}
+$$
+
+where  $A_{i} = [i - 1,i + 1]\backslash \{i\}$  denotes the set of adjacent views for the  $i$  -th view,  $\{\omega_j\}$  are the weighted coefficients that  $\sum \omega_{j} = 1$  , and  $\mathcal{M}^{(j)}$  represents the feature interaction block for the  $j$  -th view. Now, we can calculate the gradient of the loss  $\mathcal{L}_i(\pmb {u};t)$  with respect to  $z_{t}^{(i)}(\pmb {u})$  at each spatial point  $\pmb{u}$  across all views to get a gradient map denoted as  $\nabla_{z_t}\mathcal{L}_t\in \mathbb{R}^{F\times h\times w\times c}$  . We then use the gradient map to rectify the denoising direction (detailed in Appendix B.2) at each time step  $t$  , formulated as:
+
+$$
+\hat {\varepsilon} _ {\theta} \left(z _ {t}, t\right) \leftarrow \varepsilon_ {\theta} \left(z _ {t}, t\right) + s (t) \sqrt {1 - \bar {\alpha} _ {t}} \nabla_ {z _ {t}} \mathcal {L} _ {t}, \tag {12}
+$$
+
+where  $s(t)$  is the rectification scale with respect to  $t$ . Due to the high computational cost and time when calculating the UNet Jacobian  $\frac{\partial \varepsilon_{\theta}}{\partial z_t}$  during gradient computation, we theoretically (detailed in Appendix B.3) and experimentally (detailed in Appendix D) show that neglecting the UNet Jacobian term causes negligible changes in rectification results. The following theorem provides a specific explanation.
+
+![](images/4e4763e64b4b5e48b2a27bd2320c7eb09b423b7b47f6ce4ee353ad4e3543e9ee.jpg)  
+Figure 5: Qualitative Results. The images at the top are multi-view images generated by each baseline, while those at the bottom showcase the consistency rectification achieved through MIRROR. For easier observation, some areas with Janus Problem or content drifting are marked with red circles. For visual consistency, we replace the gray background of the generated images from text prompts with a pristine white one.
+
+Theorem 4.4. In the denoising process, the rectification error induced by neglecting the UNet Jacobian term, has an upper bound controlled by a constant  $\gamma$ , formulated as
+
+$$
+\left\| \nabla_ {z _ {t}} \mathcal {L} _ {t} - \frac {1}{\sqrt {\bar {\alpha} _ {t}}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} \right\| \leq \gamma , \tag {13}
+$$
+
+and this error converges to zero as  $t \to 0$ .
+
+Therefore, we can rewrite the rectification formula Eq. (12):
+
+$$
+\hat {\varepsilon} _ {\theta} \left(z _ {t}, t\right) \leftarrow \varepsilon_ {\theta} \left(z _ {t}, t\right) + s (t) \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t}, \tag {14}
+$$
+
+where  $\nabla_{\hat{z}_0}\mathcal{L}_t$  denotes the gradient map with respect to  $\hat{z}_0$ . We demonstrate that Eq. (14) achieves nearly a twofold speedup compared to Eq. (12) (Tab. 2), enabling faster rectification without sacrificing the quality of the generation. The rectified noise  $\hat{\varepsilon}_{\theta}$  is then subsequently fed into Eq. (2) to carry out the sampling procedure utilized in standard DDIM (Song et al., 2020), but with the rectified noise predictions.
+
+# 5. Experiments
+
+# 5.1. Experimental Setup
+
+Baselines. In this section, we conduct experimental evaluations based on image-based and text-based multi-view generation tasks. For image-based tasks, we choose SyncDreamer (Liu et al., 2023b), MVD-Fusion (Hu et al., 2024), and VideoMV (Zuo et al., 2024) as our baselines. For text-based tasks, we adopt VideoMV as baseline method. Each method provides a pretrained multi-view model for generating multiple views from a single image or a text prompt.
+
+Metrics. Building on these baselines, we quantitatively evaluate performance using metrics including PSNR, SSIM (Wang et al., 2004), and LPIPS (Zhang et al., 2018). We also report Clip-Score for both image-based and text-based generation tasks to evaluate the alignment between generated images and the corresponding input.
+
+![](images/130c7fce465d561ffdf0b6c923b041c976e65b012a30fb49dbd742972d656e23.jpg)  
+Figure 6: Ablation Study on the Two-Stage Rectification Process. Results generated using baseline methods often exhibit severe Janus Problem. Attempts to rectify these directly at stage 1, as well as outcomes from a two-stage baseline process, have proven inadequate in resolving these issues. In contrast, our method significantly reduces these inconsistencies.
+
+Implementation Details. We implement MIRROR on top of the baseline models discussed earlier in this section. This integration of the rectification process occurs exclusively during the inference phase, thereby obviating the need for any supplementary training or fine-tuning. Given that TTM relies on depth guidance, we employ the state-of-the-art monocular depth estimation model, Depth-Anything-V2-Small (Yang et al., 2024), to generate depth information. We also provide an ablation study on alternative depth guidance methods, as detailed in Appendix E, and demonstrate that MIRROR's performance improves with the advancement of depth estimation models. Following baselines, we use NeuS (Wang et al., 2021) for 3D reconstruction. Although calculating the UNet Jacobian in Eq. (12) is time-consuming and significantly increases inference time, our method, as described in Eq. (14), shows that by omitting this term and integrating MIRROR into the baseline models, the inference time is considerably reduced. As shown in Tab. 2, our two-stage sampling process is completed in just a few minutes. Additional experimental details are provided in Appendix C.
+
+# 5.2. Comparison with Baseline Methods
+
+We implement MIRROR across diverse open-source multiview diffusion models and carry out both quantitative and qualitative comparisons.
+
+Qualitative Analysis. For image-based generation, Fig. 5 clearly illustrate that although baseline models can produce plausible views from angles near the input perspective, they exhibit inconsistencies (Janus Problem and content drift) in lateral and rear views, along with undesirable indentations, and misalignments. Fig. 5 shows that text-based generation of VideoMV also suffers from low-quality outcomes with undesirable artifacts and inconsistencies, such as an 'elephant' with multiple trunks and missing legs, and a 'chameleon' without a head. By comparison, incorporating MIRROR into these baselines successfully resolves the prominent artifacts and multi-face issues encountered in baselines, as observed in Fig. 1 and Fig. 5, which not only
+
+Table 1: Quantitative Comparison with Baselines. Image-based and text-based generation results are evaluated on GSO (Downs et al., 2022) and  $\mathrm{T}^3$ Bench (He et al., 2023) datasets, respectively.  
+
+<table><tr><td>Method</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>ClipS ↑</td></tr><tr><td>SyncDreamer</td><td>18.75</td><td>0.739</td><td>0.198</td><td>76.60</td></tr><tr><td>+MIRROR (ours)</td><td>19.37</td><td>0.794</td><td>0.172</td><td>78.28</td></tr><tr><td>MVD-Fusion</td><td>18.90</td><td>0.785</td><td>0.172</td><td>68.66</td></tr><tr><td>+MIRROR (ours)</td><td>20.15</td><td>0.801</td><td>0.168</td><td>71.52</td></tr><tr><td>VideoMV (image-based)</td><td>17.64</td><td>0.755</td><td>0.197</td><td>71.77</td></tr><tr><td>+MIRROR (ours)</td><td>18.26</td><td>0.793</td><td>0.176</td><td>73.30</td></tr><tr><td>VideoMV (text-based)</td><td>21.67</td><td>0.814</td><td>0.206</td><td>26.96</td></tr><tr><td>+MIRROR (ours)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+enhances consistency but also significantly improves quality and geometric fidelity, clearly surpassing all baseline results. More results are provided in Appendix K.
+
+Quantitative Analysis. The quantitative results in Tab. 1 are in strong concordance with the visualization outcomes. Incorporating MIRROR significantly boosts 3D consistency-related metrics (PSNR, SSIM, and LPIPS) over all baseline methods, clearly demonstrating notable enhancements in consistency and quality for multi-view generation. Moreover, we calculate the average CLIP score between generated images and the input, and we observed a marked increase in it, indicating stronger alignment between the generated content and the input. Moreover, 3D reconstruction metrics are also reported in Appendix C.4.
+
+Inference Time. Our experiments are conducted using a single NVIDIA L40S GPU, with the two-stage sampling process completing in just a few minutes. Notably, incorporating MIRROR into the baseline models does not result in a substantial increase in computational load or processing time. We test the inference time of the baseline methods and the methods with MIRROR applied, as shown in Tab. 2. Although the computation of the UNet Jacobian is time-
+
+Table 2: Inference Time of Baseline Models and Their MIRROR-Enhanced Versions. The time multiples relative to the baseline are shown in parentheses.  
+
+<table><tr><td>Models</td><td>Baseline</td><td>MIRROR (w. UNet Jacobian)</td><td>MIRROR (ours) (w/o UNet Jacobian)</td></tr><tr><td>SyncDreamer</td><td>16 s</td><td>91 s (×5.7)</td><td>43 s (×2.7)</td></tr><tr><td>MVD-Fusion</td><td>20 s</td><td>112 s (×5.6)</td><td>52 s (×2.6)</td></tr><tr><td>VideoMV (image-based)</td><td>22 s</td><td>103 s (×4.7)</td><td>64 s (×2.9)</td></tr><tr><td>VideoMV (text-based)</td><td>34 s</td><td>154 s (×4.5)</td><td>98 s (×2.9)</td></tr></table>
+
+![](images/db78742aa5d25a592609432339d8472bf4fc27efd307c7824527a7148e848c30.jpg)  
+Figure 7: Ablation Study on Depth Guidance. TTM without depth guidance exhibits errors in correspondence, leading to unsatisfactory and inconsistent results.
+
+![](images/1cfcb04a807fd10c1ea60f4db1999bd55aa2a108f97b11efbf520352c791622d.jpg)  
+Figure 8: Ablation Study on Dual-Anchors-Based Fusion. Alternative methods yield suboptimal results, while the dual-anchor-based method (ours) markedly reduces background interference and improves the continuity of local features.
+
+consuming (w. UNet Jacobian), increasing the inference time by 4.5 to 5.7 times compared to the baselines, our method demonstrates that the UNet Jacobian term can be omitted, leading to a significant reduction in inference time, with only a 2.6 to 2.9 times increase (w/o UNet Jacobian). This computational cost is still minimal when compared to training-based methods, enabling the generation of high-quality dense images within minutes.
+
+# 5.3. Ablation Study
+
+Two-Stage Rectification Process. Fig. 6 highlights that replacing our two-stage rectification process with a single-stage correction (the second column) fails to resolve the Janus problem evident in baselines, as inaccuracies in depth extraction lead to deformed 'horse' and 'elephant' models with missing or unrealistic features. And relying solely on a two-stage denoising process without rectification (the third column) still leads to inconsistencies. Conversely, MIR
+
+ROR utilizes depth information from the initial denoising stage for more accurate tracking and rectification in the subsequent stage, which significantly improves multi-view consistency and model fidelity, as shown in the final column.
+
+Depth Guidance. Within TTM, our approach leverages depth information to obtain more accurate spatial correspondences between different viewpoints. However, in the absence of depth guidance (i.e.,  $z = 0$  in Eq. (4)), it fails to precisely establish correspondences between adjacent views, leading to unsatisfactory results, as depicted in the second column of Fig. 7, with the missing head in the first row and content drift problem of 'red scarf' in the second row.
+
+Dual-Anorh- Based Fusion. To validate our method's effectiveness and generalizability, we conducte ablation studies on multi-view generation against white and gray backgrounds. Our findings, detailed in Fig. 8, show that moving away from a dual-anchor approach to using average pooling or single-anchor feature fusion leads to various issues. Specifically, omitting the anchor-based approach results in excessive saturation and discontinuity in details, while using only positive anchors improve local feature precision but is prone to background interference. Sole reliance on negative anchors minimizes distractions but lacks focus on essential target characteristics. Conversely, our integrated approach, which combines positive and negative anchors, substantially enhances relevant detail attention and consistency, thereby outperforming other methods. More ablation study results are provided in Appendix J.
+
+# 6. Conclusions
+
+In this paper, we introduce MIRROR, a novel plug-and-play technique that effectively resolves inconsistencies in 3D geometry and artifacts in diffusion-based multi-view generation without training. Using trajectory tracking and feature rectification modules, MIRROR successfully enhances the photorealism of synthesized images across views. Generalization evaluations confirm MIRROR's compatibility with various multi-view diffusion models, making it a universal correction tool that markedly improves generation consistency and fidelity, thereby delivering higher-quality multi-view images for 3D asset reconstruction.
+
+# Acknowledgements
+
+This paper is supported by National Key R&D (Research and Development) Program of China (2021YFA1000403), the National Natural Science Foundation of China (Nos. 12431012, U23B2012, 12471308) and Beijing Natural Science Foundation (1254050).
+
+# Impact Statement
+
+This project aims to provide an effective, training-free tool for improving the generative quality of multi-view diffusion models. Our approach enhances the 3D consistency of multi-view images while simultaneously improving the overall quality of the generated content. However, similar to other generative technologies, there exists a potential for misuse. Malicious entities could combine this method with targeted generative models to produce misleading or harmful 3D content. Addressing these ethical considerations is critical, and future research in 3D generation must prioritize understanding and mitigating these risks to ensure the responsible application of such technologies.
+
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+
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+
+# A. Appendix Overview
+
+In the appendix, we provide the following supplementary material to support the main text:
+
+- Derivation and Proofs (Appendix B);  
+Additional Implementation Details (Appendix C);  
+- Convergence and Stability Analysis of MIRROR (Appendix D);  
+- Depth Guidance (Appendix E);  
+- Discussion with Multi-View Depth Estimation (Appendix F);  
+Correspondence Visualization of Trajectory Tracking Transformation (Appendix G);  
+- Discussion with Point Tracking Method (Appendix H);  
+- Limitations and Future Work (Appendix I);  
+- More Ablation Studies (Appendix J);  
+- More Results (Appendix K);  
+- Failure Cases (Appendix L).
+
+# B. Derivation and Proofs
+
+# B.1. Proof of Proposition 4.2
+
+Proof. Based on Definition 4.1 of TTM, the trajectory tracking error between the predicted image  $\hat{x}_0(x_t,t)$  and the true image  $x_0$  can be scaled as:
+
+$$
+\left\| \mathcal {T} _ {\alpha} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \mathcal {T} _ {\alpha} \left(x _ {0}\right) \right\| ^ {2} \leq \left\| \boldsymbol {u} _ {\alpha} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \boldsymbol {u} _ {\alpha} \left(x _ {0}\right) \right\| ^ {2} \leq r ^ {2} \left\| \mathcal {H} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \mathcal {H} \left(x _ {0}\right) \right\| ^ {2} \tag {15}
+$$
+
+Using the first-order Taylor expansion, we obtain:
+
+$$
+\mathcal {H} \left(x _ {0}\right) = \mathcal {H} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) + \nabla \mathcal {H} \left(x _ {0}\right) \left(x _ {0} - \hat {x} _ {0} \left(x _ {t}, t\right)\right) + o \left(\left\| x _ {0} - \hat {x} _ {0} \left(x _ {t}, t\right) \right\|\right), \tag {16}
+$$
+
+where  $o$  represents a higher-order infinitesimal. And  $\mathcal{H}$  is a pretrained ViT network with a continuous, bounded gradient that outputs scale-consistent absolute depth, showing that the depth-level error is of the same order as the pixel-level error:
+
+$$
+\left\| \mathcal {H} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \mathcal {H} \left(x _ {0}\right) \right\| \approx \left\| \nabla \mathcal {H} \left(x _ {0}\right) \left(x _ {0} - \hat {x} _ {0} \left(x _ {t}, t\right)\right) \right\| \simeq O \left(\left\| x _ {0} - \hat {x} _ {0} \left(x _ {t}, t\right) \right\|\right). \tag {17}
+$$
+
+Then, substituting Eq. (17) into Eq. (15) results in the following inequality:
+
+$$
+\left\| \mathcal {T} _ {\alpha} \left(\hat {x} _ {0} \left(x _ {t}, t\right)\right) - \mathcal {T} _ {\alpha} \left(x _ {0}\right) \right\| ^ {2} \leq O \left(\left\| \hat {x} _ {0} \left(x _ {t}, t\right) - x _ {0} \right\| ^ {2}\right). \tag {18}
+$$
+
+Moreover, note that the right side of the inequality Eq. (18) is equivalent to the Kullback-Leibler (KL) divergence between forward process posteriors  $q$  and reverse process posteriors estimation  $p_{\theta}$  of diffusion process:
+
+$$
+O \left(\| \hat {x} _ {0} \left(x _ {t}, t\right) - x _ {0} \| ^ {2}\right) \simeq \mathrm {K L} \left(q \left(x _ {t - 1} \mid x _ {t}, x _ {0}\right) \| p _ {\theta} \left(x _ {t - 1} \mid x _ {t}\right)\right), \tag {19}
+$$
+
+since both terms represent the training objective of the diffusion model. Therefore, in the inference process, the trajectory tracking error is controlled by the diffusion model, which has a finite upper bound. Besides, as  $t$  decreases, the predicted image  $\hat{x}_0(x_t,t)$  approaches the clean image  $x_0$ , causing the error to converge to zero.
+
+# B.2. The Derivation of the Rectification Formula Eq. (12)
+
+According to Bayes' Theorem, the objective function can be decomposed into the following two terms:
+
+$$
+\nabla_ {z _ {t}} \log p _ {\theta} (z _ {t} | c) = \nabla_ {z _ {t}} \log p _ {\theta} (z _ {t}) + \nabla_ {z _ {t}} \log p _ {\theta} (c | z _ {t}), \tag {20}
+$$
+
+where  $c$  denotes the azimuth angles of the cameras in the multi-view images. Furthermore, since  $p_{\theta}$  is a Gaussian distribution, the transformation relationship between the score function and the denoiser of the diffusion model can be easily derived, i.e.,
+
+$$
+\nabla_ {z _ {t}} \log p _ {\theta} (z _ {t}) = - \frac {1}{\sqrt {1 - \bar {\alpha} _ {t}}} \varepsilon_ {\theta} (z _ {t}, t). \tag {21}
+$$
+
+According to Eq. (20) and Eq. (21), we can derive the following expression:
+
+$$
+\varepsilon_ {\theta} \left(z _ {t}, t, c\right) = \varepsilon_ {\theta} \left(z _ {t}, t\right) - \sqrt {1 - \bar {\alpha} _ {t}} \nabla_ {z _ {t}} \log p _ {\theta} (c | z _ {t}). \tag {22}
+$$
+
+By substituting the exponential form of the loss function,  $\exp(-\mathcal{L}_t)$ , into Eq. (22), the rectification formula of the denoising direction is derived as follows:
+
+$$
+\hat {\varepsilon} _ {\theta} \left(z _ {t}, t, c\right) \leftarrow \varepsilon_ {\theta} \left(z _ {t}, t\right) + s (t) \sqrt {1 - \bar {\alpha} _ {t}} \nabla_ {z _ {t}} \mathcal {L} _ {t}, \tag {23}
+$$
+
+where  $s(t)$  is the rectification scale with respect to  $t$ . For brevity,  $\hat{\varepsilon}_{\theta}(z_t,t,c)$  is denoted as  $\hat{\varepsilon}_{\theta}(z_t,t)$  in the main paper.
+
+# B.3. Proof of Theorem 4.4
+
+Proof. First, by applying the chain rule, the gradient  $\nabla_{z_t}\mathcal{L}_t$  can be expanded as follows:
+
+$$
+\nabla_ {z _ {t}} \mathcal {L} _ {t} = \frac {\partial \hat {z} _ {0}}{\partial z _ {t}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} = \frac {1}{\sqrt {\bar {\alpha} _ {t}}} \left(1 - \sqrt {1 - \bar {\alpha} _ {t}} \frac {\partial \varepsilon_ {\theta}}{\partial z _ {t}}\right) \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t}, \tag {24}
+$$
+
+with  $\frac{\partial\varepsilon_{\theta}}{\partial z_t}$  being the UNet Jacobian of the diffusion model. When the UNet Jacobian term is neglected, the gradient error between the two terms can be expressed as:
+
+$$
+\left\| \nabla_ {z _ {t}} \mathcal {L} _ {t} - \frac {1}{\sqrt {\bar {\alpha} _ {t}}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} \right\| = \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} \left\| \frac {\partial \varepsilon_ {\theta}}{\partial z _ {t}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} \right\|. \tag {25}
+$$
+
+Note that as  $t$  decreases, the noise  $\varepsilon_{\theta}(z_t,t)$  becomes increasingly independent of the data distribution, causing the UNet Jacobian term to converge to zero as  $t\to 0$ . Then,  $\| \frac{\partial\varepsilon_{\theta}}{\partial z_t}\nabla_{\hat{z}_0}\mathcal{L}_t\| \to 0$  as  $t\to 0$ . Therefore, there exists a constant  $\gamma$  such that  $\| \frac{\partial\varepsilon_{\theta}}{\partial z_t}\nabla_{\hat{z}_0}\mathcal{L}_t\| \leq \gamma$ , for all  $t$ . Moreover, since the coefficient  $\frac{\sqrt{1 - \overline{\alpha}_t}}{\sqrt{\overline{\alpha}_t}}\in [0,1]$ , the gradient error of Eq. (25) has a finite upper bound.
+
+Furthermore, after individually rectifying the noise estimation based on the two gradients in Eq. (12) and Eq. (14), respectively, the error of  $z_{t-1}$  obtained through sampling process also converges to zero as  $t$  decreases. The detailed derivation process is as follows:
+
+$$
+\begin{array}{l} \left\| z _ {t - 1} ^ {(z _ {t})} - z _ {t - 1} ^ {(\hat {z} _ {0})} \right\| = \left\| \left(\sqrt {\overline {{\alpha}} _ {t - 1}} \left(\frac {z _ {t} - \sqrt {1 - \overline {{\alpha}} _ {t}} \hat {\varepsilon} _ {\theta} ^ {(z _ {t})}}{\sqrt {\overline {{\alpha}} _ {t}}}\right) + \sqrt {1 - \overline {{\alpha}} _ {t - 1}} \hat {\varepsilon} _ {\theta} ^ {(z _ {t})}\right) \right. \\ - \left(\sqrt {\overline {{\alpha}} _ {t - 1}} \left(\frac {z _ {t} - \sqrt {1 - \overline {{\alpha}} _ {t}} \hat {\varepsilon} _ {\theta} ^ {(\hat {z} _ {0})}}{\sqrt {\overline {{\alpha}} _ {t}}}\right) + \sqrt {1 - \overline {{\alpha}} _ {t - 1}} \hat {\varepsilon} _ {\theta} ^ {(\hat {z} _ {0})}\right) \Bigg \| \\ = \left| \frac {\sqrt {\overline {{\alpha}} _ {t - 1}} \sqrt {1 - \overline {{\alpha}} _ {t}}}{\sqrt {\overline {{\alpha}} _ {t}}} - \sqrt {1 - \overline {{\alpha}} _ {t - 1}} \right| \cdot \left\| \hat {\varepsilon} _ {\theta} ^ {(z _ {t})} - \hat {\varepsilon} _ {\theta} ^ {(\hat {z} _ {0})} \right\| \\ = s (t) \left| \frac {\sqrt {\bar {\alpha} _ {t - 1}} \left(1 - \bar {\alpha} _ {t}\right)}{\sqrt {\bar {\alpha} _ {t}}} - \sqrt {\left(1 - \bar {\alpha} _ {t - 1}\right) \left(1 - \bar {\alpha} _ {t}\right)} \right| \cdot \left\| \nabla_ {z _ {t}} \mathcal {L} _ {t} - \frac {1}{\sqrt {\bar {\alpha} _ {t}}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} \right\| \\ = s (t) \frac {\sqrt {1 - \overline {{\alpha}} _ {t}}}{\sqrt {\overline {{\alpha}} _ {t}}} \left| \frac {\sqrt {\overline {{\alpha}} _ {t - 1}} (1 - \overline {{\alpha}} _ {t})}{\sqrt {\overline {{\alpha}} _ {t}}} - \sqrt {(1 - \overline {{\alpha}} _ {t - 1}) (1 - \overline {{\alpha}} _ {t})} \right| \cdot \left\| \frac {\partial \varepsilon_ {\theta}}{\partial z _ {t}} \nabla_ {\hat {z} _ {0}} \mathcal {L} _ {t} \right\| \\ \leq \gamma \cdot \mathcal {A} (t), \tag {26} \\ \end{array}
+$$
+
+where the superscripts  $(z_{t})$  and  $(\hat{z}_0)$  correspond to the two gradient correction methods in Eq. (12) and Eq. (14), respectively. And the coefficient  $\mathcal{A}(t) = s(t)\frac{\sqrt{1 - \overline{\alpha}_t}}{\sqrt{\overline{\alpha}_t}}\left|\frac{\sqrt{\overline{\alpha}_{t-1}}(1 - \overline{\alpha}_t)}{\sqrt{\overline{\alpha}_t}} - \sqrt{(1 - \overline{\alpha}_{t-1})(1 - \overline{\alpha}_t)}\right| \to 0$  as time step  $t \to 0$ .
+
+Algorithm 1 Framework of MIRROR  
+Input: a single image or a text prompt  $c$   
+Pretrained baseline model:  $\mathcal{BM}$   
+Parameter: rectification scale  $s(t)$  with respect to  $t$   
+Output: multi-view consistent images  $\hat{x}_0$   
+1:  $z_T \gets$  sample from  $\mathcal{N}(0, I)$   
+2:  $z_t \gets z_T$ .  
+3: for  $t = T, T - 1, \dots, 1$  do  
+4:  $\varepsilon_\theta \gets \mathcal{BM}(z_t, c, t)$ .  
+5:  $z_{t-1} \gets DDIM(z_t, \varepsilon_\theta)$   
+6: end for  
+7:  $\hat{x}_0 \gets \mathcal{D}(z_0)$ .  
+8:  $\mathcal{Z}'_{absolute} \gets \text{DEPTH ESTIMATION}(\hat{x}_0)$ .  
+9:  $z_T \gets DDIM_INVERSION(z_0, \mathcal{BM}, T, c)$ .  
+10:  $z_t \gets z_T$ .  
+11: for  $t = T, T - 1, \dots, 1$  do  
+12:  $\varepsilon_\theta \gets \mathcal{BM}(z_t, c, t)$ .  
+13:  $\mathcal{L}_t \gets \| \mathcal{F}_\alpha(z_t, \mathcal{Z}'_{absolute}) \|_2^2$ .  
+14:  $\hat{\varepsilon}_\theta \gets \varepsilon_\theta + s(t) \frac{\sqrt{1 - \overline{\alpha}_t}}{\sqrt{\overline{\alpha}_t}} \nabla_{\hat{z}_0} \mathcal{L}_t$ .  
+15:  $z_{t-1} \gets DDIM(z_t, \hat{\varepsilon}_\theta)$   
+16: end for  
+17:  $\hat{x}_0 \gets \mathcal{D}(z_0)$   
+18: return  $\hat{x}_0$
+
+# B.4. Group Homomorphism Property of Operator  $\mathcal{M} \circ \mathcal{T}_{\alpha}$
+
+To prove this property, we first establish the following lemma.
+
+Lemma B.1. The trajectory tracking operator  $\mathcal{T}_{\alpha}$  is a group homomorphism. Specifically, for any  $\alpha_{1},\alpha_{2}\in \mathbb{R}$ , it holds that:  $\mathcal{T}_{\alpha_1}\circ \mathcal{T}_{\alpha_2} = \mathcal{T}_{\alpha_1 + \alpha_2}$ .
+
+Proof. It is important to note that  $\mathcal{T}_{\alpha}$  represents a rotation operator of angle  $\alpha$  around the  $y$ -axis within the special orthogonal group  $SO(3)$ , which can be expressed using the matrix exponential of the Lie algebra element associated with the rotation group. Specifically, it is given by:
+
+$$
+\mathcal {T} _ {\alpha} = \exp (\alpha A), A = \left( \begin{array}{c c c} 0 & 0 & - 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right). \tag {27}
+$$
+
+By leveraging the properties of the matrix exponential, we observe that the composition of two such rotation operators results in a new rotation operator. That is, the combination of two rotations is given by:
+
+$$
+\mathcal {T} _ {\alpha_ {1}} \circ \mathcal {T} _ {\alpha_ {2}} = \exp (\alpha_ {1} A) \cdot \exp (\alpha_ {2} A) = \exp ((\alpha_ {1} + \alpha_ {2}) A) = \mathcal {T} _ {\alpha_ {1} + \alpha_ {2}}. \tag {28}
+$$
+
+Therefore, the rotation operators  $\mathcal{T}_{\alpha}$  are closed under the operation of parameter addition, demonstrating the group property of  $SO(3)$  with respect to the rotation angle.
+
+Proposition B.2. For any  $\alpha_{1},\alpha_{2}\in \mathbb{R}$ , the composition of two operators  $\mathcal{M}\circ \mathcal{T}_{\alpha_1}$  and  $\mathcal{M}\circ \mathcal{T}_{\alpha_2}$  adheres to the following operational relation:
+
+$$
+\left(\mathcal {M} \circ \mathcal {T} _ {\alpha_ {1}}\right) \circ \left(\mathcal {M} \circ \mathcal {T} _ {\alpha_ {2}}\right) = \mathcal {M} \circ \mathcal {T} _ {\alpha_ {1} + \alpha_ {2}}. \tag {29}
+$$
+
+In other words,  $\mathcal{M} \circ \mathcal{T}_{\alpha}$  is a group homomorphism.
+
+Now, we proceed to prove Proposition B.2.
+
+![](images/e8ea5d4ae3756b22fbab8bfd8e3e31f8f8268c721a3b29a5413d581f8fcf9e0b.jpg)  
+A pig wearing a backpack, 3D asset.  
+Input  
+Figure 9: Ablation Study on the Step  $T$  in DDIM Inversion. When  $T$  is too small, noise disrupts the sampling; too large, it increases inference time. Therefore, after balancing inference time and generation quality, we set  $T = 50$ .
+
+Proof. It is important to observe that, since the operator  $\mathcal{T}_{\alpha}$  is solely dependent on the coordinates of the spatial points and is independent of the feature values, it can be deduced that
+
+$$
+\left(\mathcal {M} \circ \mathcal {T} _ {\alpha_ {1}}\right) \circ \left(\mathcal {M} \circ \mathcal {T} _ {\alpha_ {2}}\right) = \mathcal {M} \circ \left(\mathcal {T} _ {\alpha_ {1}} \circ \mathcal {T} _ {\alpha_ {2}}\right) = \mathcal {M} \circ \mathcal {T} _ {\alpha_ {1} + \alpha_ {2}}, \tag {30}
+$$
+
+where the second equality is derived from Lemma B.1.
+
+# C. Additional Implementation Details
+
+# C.1. Feature Rectification Module
+
+In Algorithm 1, we systematically present the pseudocode for the entire rectification algorithm. Below are the parameter selection strategies for several key parameters used in FRM. First, we have determined the weights for the fusion of positive and negative anchors at  $(m^{+} = 0.5, m^{-} = 0.5)$  based on empirical observations. Fig. 8 shows the  $(m^{+} = 1, m^{-} = 0)$  would cause grayish or bright tones and  $(m^{+} = 0, m^{-} = 1)$  may fail to focus on the current point, leading to poor results. To maintain general applicability, we chose  $(m^{+} = 0.5, m^{-} = 0.5)$  as the robust option. Second, we define the rectification scale  $s(t)$  to match the consistency gradient magnitude with  $\varepsilon_{\theta}$  of each baseline. As geometric prototypes primarily develop during the initial sampling stages of the inference process, with subsequent stages dedicated to refining detailed textures, it is essential to focus on the early phases since we aim to adjust geometric shapes to address inconsistencies such as Janus Problem. In the later stages, we intentionally decrease the correction scale to prevent overcorrection. Besides, based on experimental observations, the first five denoising steps contain excessive noise, and applying the correction algorithm during this stage leads to redundant information. Therefore, we start applying our correction algorithm after the fifth step. Technically, we define the rectification scale  $s(t)$  as a gradient function dependent on  $t$ . Its specific form is given by Eq. (31) for SyncDreamer and MVD-Fusion, and by Eq. (32) for VideoMV, and can be further adapted to accommodate different models.
+
+$$
+s (t) = \left\{ \begin{array}{l l} 0. 0 0 2, & t \in (4 0, 4 5 ], \\ 0. 0 0 8, & t \in (3 0, 4 0 ], \\ 0. 0 0 5, & t \in (2 0, 3 0 ], \\ 0. 0 0 3, & t \in (1 0, 2 0 ], \\ 0. 0 0 1, & t \in [ 1, 1 0 ]. \end{array} \right. \tag {31}
+$$
+
+$$
+s (t) = \left\{ \begin{array}{l l} 0. 0 0 5, & t \in (4 0, 4 5 ], \\ 0. 0 2 4, & t \in (3 0, 4 0 ], \\ 0. 0 1 5, & t \in (2 0, 3 0 ], \\ 0. 0 0 8, & t \in (1 0, 2 0 ], \\ 0. 0 0 3, & t \in [ 1, 1 0 ]. \end{array} \right. \tag {32}
+$$
+
+Furthermore, adjustments per model are feasible. Third, the weights  $\{\omega_{j}\}$  in the rectification formulation of the adjacent views, both preceding and succeeding, are assigned randomly, as the feature information provided by these two neighboring views is considered equally important. Additionally, we acknowledge that camera parameter noise is unavoidable, but its impact is minimal since the baselines were trained with standard parameters. To alleviate this, we employed block-based trajectory tracking, using a  $3\times 3$  block to fuse the corresponding features. While we experimented with larger windows, it introduced excessive redundant information, ultimately affecting performance. Additionally, increasing the window size further increased the computational load. As a trade-off between computation time and generation quality, we selected the  $3\times 3$  block size.
+
+# C.2. The Step of DDIM Inversion
+
+The number of DDIM sampling and inversion steps is both set to  $T = 50$ . The number of DDIM sampling steps in the two-stage process is kept the same as the baselines. And we have tried with different  $T$  for DDIM Inversion. As shown in Fig. 9, when  $T$  is too small, the added noise becomes excessively coarse, negatively impacting the denoising process in the second stage. Conversely, when  $T$  is too large, it significantly increases the overall inference time. Therefore, we have settled on  $T = 50$ , which strikes a balance between maintaining high quality and ensuring computational efficiency, consistent with the inference process steps.
+
+# C.3. Evaluation Datasets
+
+For image-based generation, we evaluated 100 objects from Google Scanned Objects (GSO) (Downs et al., 2022) dataset, using the 16, 16, and 24 views provided by SyncDreamer, MVD-Fusion, and VideoMV, respectively. Following text-based generation in VideoMV, we used 100 single-object prompts from  $\mathrm{T}^3$  Bench (He et al., 2023), sampling half of the equidistant views (12 views) for neural field reconstruction, employing them as pseudo ground truth for quantitative comparisons.
+
+# C.4.3D Reconstruction
+
+We follow the baseline SyncDreamer (Liu et al., 2023b) to utilize NeuS (Wang et al., 2021) for 3D reconstruction. Specifically, the training process for one object consists of 10,000 steps with a warm-up of 100 steps and loss weights  $\lambda_{rgb} = 0.5$ ,  $\lambda_{mask} = 1.0$ ,  $\lambda_{eikonal} = 0.1$ . We provide the commonly used Chamfer Distance (CD) and Volume IoU between ground-truth shapes and reconstructed shapes in Tab. 3 to further validate the effectiveness of our method in improving the quality and consistency of both multi-view images and 3D reconstruction.
+
+Table 3: Quantitative Comparison with Baselines on 3D Reconstruction Metrics. Image-based and text-based generation results are evaluated on GSO and  $\mathrm{T}^3$ Bench datasets, respectively.  
+
+<table><tr><td>Method</td><td colspan="2">Chamfer Distance ↓ Volume IoU ↑</td></tr><tr><td>SyncDreamer</td><td>0.042</td><td>0.514</td></tr><tr><td>+MIRROR (ours)</td><td>0.039</td><td>0.530</td></tr><tr><td>MVD-Fusion</td><td>0.033</td><td>0.602</td></tr><tr><td>+MIRROR (ours)</td><td>0.029</td><td>0.656</td></tr><tr><td>VideoMV (image-based)</td><td>0.031</td><td>0.643</td></tr><tr><td>+MIRROR (ours)</td><td>0.026</td><td>0.667</td></tr><tr><td>VideoMV (text-based)</td><td>0.046</td><td>0.538</td></tr><tr><td>+MIRROR (ours)</td><td>0.028</td><td>0.626</td></tr></table>
+
+# D. Convergence and Stability Analysis of MIRROR
+
+We experimentally validate the convergence and stability of our algorithm on baseline methods. Firstly, as shown in Fig. 10 (a), we observe that as  $t$  decreases, the predicted noise  $\hat{\varepsilon}_{\theta}$  rectified by MIRROR steadily decreases. Second, Fig. 10 (b) and (c) demonstrates that both the loss gradient error of Eq. (25) and the error about  $z_{t-1}$  during one-step DDIM sampling of Eq. (26), which are caused by neglecting the UNet Jacobian term, gradually converge to zero as  $t$  decreases. Furthermore, Fig. 11 demonstrates that both rectification formulas effectively produce consistent multi-view images. Therefore, by combining the results in Fig. 10 and Fig. 11, we can conclude that the error introduced by omitting the UNet Jacobian term is negligible. Therefore, using Eq. (14) as the rectification formula is a more efficient and reasonable strategy.
+
+# E. Depth Guidance
+
+We evaluated the performance of the current state-of-the-art monocular depth estimation method, Depth-Anything-V2 (Yang et al., 2024) and an alternative method, MiDaS (Ranftl et al., 2020). Experimental results demonstrate that Depth-Anything-V2 is the optimal choice for depth estimation in the TTM framework.
+
+![](images/7605be9da146be7b61bf3cd060c1e1571eb8d41ec149852cd6aa65f7fd830a8b.jpg)  
+(a)
+
+![](images/01a758d09aadca1399632a7dce3b1dba204f4907b5025a8d1e394b4572cf1fad.jpg)  
+(b)
+
+![](images/2301a420f3ad9fef9aecc2622bec4642235ffb44e39bd0c4309988b552d5573d.jpg)  
+(c)
+
+![](images/84e50f85201f8783715316c852008918bbe6a2b311070e21b6b70896c253bb48.jpg)  
+Figure 10: Convergence and Stability Analysis of MIRROR. (a) The trend of predicted noise  $\hat{\varepsilon}_{\theta}$  variation during the inference process with MIRROR; (b) The convergence of gradient errors in Eq. (25); (c) The convergence of one-step DDIM sampling errors in Eq. (26) when omitting the UNet Jacobian term during the inference process.  
+Figure 11: Comparison of generated results with (w.) and without (w/o) considering the UNet Jacobian term, generated from VideoMV (Zuo et al., 2024) rectified with MIRROR. Both rectification formulas are capable of generating multi-view images with strong consistency.
+
+First, we evaluate the error of Eq. (5) for depth estimation in TTM using three image-based baseline models, VideoMV (Zuo et al., 2024), SyncDreamer (Liu et al., 2023b) and MVD-Fusion (Hu et al., 2024), with Depth-Anything-V2 on GSO dataset. Fig. 12 (a) demonstrates that during the denoising process, the trajectory tracking error is minimal with the depth guidance from Depth-Anything-V2 (DA2), and as  $t$  decreases from  $T$  to 0, it gradually converges to zero. Besides, Fig. 12 (b)-(d) show the comparison of depth estimation errors for each baseline model using DA2 (blue) or MiDaS (orange). In particular, in Fig. 12 (d), for MVD-Fusion, we also utilize the depth values predicted by its pretrained model to facilitate TTM, which serves as the reference standard (green). It is noteworthy that the depth estimation error of DA2 is the smallest among all the baseline methods.
+
+Moreover, we also evaluate the qualitative results (Fig. 13) and quantitative metrics (Tab. 4) on VideoMV (Zuo et al., 2024) rectified with MIRROR. using different monocular depth estimation models for guidance. We found that both the absence of depth estimation and the use of the alternative MiDaS model introduced significant errors to TTM, resulting in 3D inconsistency in the generated multi-view images. In contrast, when using DA2 for depth estimation, the generated results were more consistent and achieved the best performance across all quantitative metrics.
+
+# F. Discussion with Multi-View Depth Estimation
+
+We replaced Depth-Anything-V2 (DA2) with the multi-view depth estimator, DUSt3R (Wang et al., 2024a). As shown in Tab. 5, DUSt3R does not significantly increase inference time but requires more memory, whereas DA2 incurs minimal overhead. In low-memory environments (e.g., a single NVIDIA 3090 GPU), DUSt3R fails to run. While DUSt3R slightly
+
+![](images/c7509ae1aec566b5506b7692e5e9a325fa2789328be7d2774cf2551c11e9d4bb.jpg)  
+(a)
+
+![](images/74e1dcdf167ab2689a1b78ca8476640940a65d31b302825efe5982c588fd8f6a.jpg)  
+(b)
+
+![](images/aa30492f96b0b622cb1ff5d287381d9166300023fa6642e6b2c0aeb63bae93a1.jpg)  
+(c)
+
+![](images/bf393ee9de98a95b4bb6103bfdc109a29e8e0ad39fff73aab4e472d9ab032de9.jpg)  
+(d)  
+Figure 12: Convergence Analysis of Depth Guidance Models. (a) shows that the trajectory tracking error is minimal with Depth-Anything-V2 (DA2) guidance, converging to zero as  $t$  decreases. (b)-(d) compare the depth estimation errors of each baseline model using Depth-Anything-V2 (DA2, blue) and MiDaS (orange). In (d), for MVD-Fusion, we additionally use the depth values predicted by its pretrained model to facilitate TTM, which is set as the reference standard (green).
+
+![](images/82d037319bceb530a539e7afb266afd8d840aaa21b76c7c9cbe9ff8b6f7beeff.jpg)  
+Figure 13: Ablation Study on Depth Guidance Models: MiDaS vs. Depth-Anything-V2.
+
+improves PSNR and LPIPS, SSIM and CLIP Score remain comparable to DA2. Visual results in Fig. 14 show marginal improvement, but in some cases, DUSt3R underperforms DA2.
+
+In summary, multi-view depth provides modest gains but with memory trade-offs. As depth estimation is independent of our core contribution, these results—along with those in Appendix E—show that MIRROR can benefit from ongoing advances in depth estimation.
+
+Table 4: Quantitative Ablation Study on Different Depth Guidance Methods.  
+
+<table><tr><td>Ablation Study</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>ClipS ↑</td></tr><tr><td>Baseline</td><td>21.67</td><td>0.814</td><td>0.206</td><td>26.96</td></tr><tr><td>w/o Depth Guidance</td><td>16.13</td><td>0.757</td><td>0.369</td><td>26.51</td></tr><tr><td>MiDaS</td><td>22.41</td><td>0.863</td><td>0.258</td><td>25.79</td></tr><tr><td>Depth-Anything-V2 (ours)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+![](images/2a5779ab9d6b90616b3edf4255dcddcfa454a6dfbc312ebbfa127c5c93717082.jpg)  
+Input  
+A sleek stainless steel teapot, 3D asset.  
+A bear dressed as a lumberjack, 3D asset.  
+Baseline  
+Figure 14: Qualitative Comparison with multi-view depth estimation model DUSt3R (Wang et al., 2024a). The first row shows that depth estimated with DUSt3R leads to more realistic generation results, while our DA2-based approach also produces consistent and reasonable outputs. The second row presents a failure case of DUSt3R-based correction, where our method successfully resolves the hand inconsistency in the baseline. We use the text-based model VideoMV for evaluation, with details consistent with Appendix C.3.
+
+![](images/1b9be4e867d61f64f9c01b5189a0c48724214dc948d57ff3b66dd93051cafe5a.jpg)  
+DUSt3R
+
+![](images/f3f17164ff79b8c02189c415fd573aa66ff9d70c806c7a10d28f37091c559656.jpg)  
+Ours
+
+# G. Correspondence Visualization of Trajectory Tracking Transformation
+
+In both image-based and text-based generation tasks, we employ the TTM method to track the correspondence of 3D physical points across adjacent views, as illustrated in Fig. 15. TTM without depth guidance ( $z = 0$  in Eq. (4)) exhibits substantial correspondence errors. In contrast, the depth-guided TTM method achieves more precise correspondences, thereby enhancing the outcomes of multi-view generation tasks.
+
+# H. Discussion with the Point Tracking Method
+
+We incorporate the recent point tracking SOTA method, CoTracker3 (Karaev et al., 2024), as a substitute for TTM for comparison. Tab. 6 shows that while CoTracker3 improves the baseline's generation quality to some extent, the gains are less significant than those achieved with TTM. Moreover, Fig. 16 shows CoTracker3 fails to resolve multi-face artifacts, exhibiting noticeable inconsistencies such as multiple legs and misaligned heads, which TTM effectively mitigates. Both quantitative and qualitative results demonstrate that TTM outperforms point tracking methods like CoTracker3 in addressing multi-view inconsistency.
+
+Table 5: Quantitative Comparison with multi-view depth estimation model DUSt3R (Wang et al., 2024a). Our experiments are conducted using a single NVIDIA L40S GPU, with VideoMV (Zuo et al., 2024) serving as the baseline. There is a trade-off between memory usage and performance between the two methods.  
+
+<table><tr><td>Ablation Study</td><td>Time ↓</td><td>Memory Usage ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>ClipS ↑</td></tr><tr><td>Baseline</td><td>34 s</td><td>25.00 GB</td><td>21.67</td><td>0.814</td><td>0.206</td><td>26.96</td></tr><tr><td>DUSt3R</td><td>107 s (×3.15)</td><td>27.04 GB (+2.04)</td><td>25.09</td><td>0.865</td><td>0.109</td><td>30.82</td></tr><tr><td>DA2 (ours)</td><td>98 s (×2.89)</td><td>25.08 GB (+0.08)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+![](images/5e1f1b9308840a3123dd3631baacba9e962d032179f6ac3e147223fb2efe4f66.jpg)  
+Figure 15: Correspondence visualization of TTM on both image-based and text-based generation. The upper rows illustrate TTM without depth guidance, whereas the lower rows represent the depth-guided TTM. The depth-guided approach more precisely identifies the correspondence of spatial points between adjacent views. For greater accuracy, readers are encouraged to zoom in on the picture. The areas with the Janus Problem or content drifting are marked with red circles.
+
+Table 6: Quantitative Comparison of TTM and the point tracking method, CoTracker3 (Karaev et al., 2024). Notably, while CoTracker3 provides slight improvements over the baseline across all metrics, our method achieves more significant gains.  
+
+<table><tr><td>Ablation Study</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>ClipS ↑</td></tr><tr><td>Baseline</td><td>21.67</td><td>0.814</td><td>0.206</td><td>26.96</td></tr><tr><td>CoTracker3</td><td>23.01</td><td>0.871</td><td>0.124</td><td>28.35</td></tr><tr><td>TTM (ours)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+![](images/1cd2c58a214a3674e71e81248970cc9771d1e4a8dca68be93a8bc8ab37aeeab0.jpg)  
+Figure 16: Qualitative Comparison of TTM and the point tracking method, CoTracker3 (Karaev et al., 2024). We use the text-based model VideoMV (Zuo et al., 2024) for evaluation, with details consistent with Appendix C.3. The second column, based on CoTracker3, exhibits noticeable inconsistencies such as multiple legs and misaligned heads. In contrast, our method effectively corrects these issues.
+
+# I. Limitations and Future Work
+
+To ensure uniform multi-view coverage, all baseline models in our paper adopt a fixed elevation setting, and MIRROR is similarly designed under this constraint. While this facilitates consistent geometric modeling and simplifies the correspondence estimation across views, it inevitably limits the applicability of the method in scenarios requiring images from diverse or freely varying viewpoints. This constraint becomes more pronounced in real-world applications where arbitrary camera poses are often necessary. To address this, future work will explore integrating more powerful priors from recent advances in free-viewpoint multi-view diffusion models, thereby extending our trajectory tracking approach to support flexible viewpoint transformations and enhancing the generality of the method.
+
+In addition, our current task setting focuses exclusively on object-level generation. Although this scope allows for fine-grained consistency and controllability, it leaves out a broader class of problems centered around scene-level 3D generation—an increasingly important direction in the field. Nevertheless, the core principles of MIRROR remain relevant. We believe our approach provides valuable insights that can help bridge the gap toward scalable scene-level generation. To enable MIRROR to handle such tasks, future work could incorporate layout-aware priors or explicit scene maps to better account for larger spatial contexts and inter-object relationships.
+
+Ultimately, we view MIRROR as a foundational step toward training-free rectification methods for more complex tasks such as the generation of real-world environments, dynamic scenes, and deformable objects, where geometric consistency and view correspondence remain essential yet challenging components.
+
+# J. More Ablation Studies
+
+We conducted quantitative ablation studies on stages and dual anchors for the text-based generation task (VideoMV) using 50 prompts from  $\mathrm{T}^3$  Bench, comparing our method with alternative approaches, as shown in Tab. 7 and Tab. 8. In Tab. 7, the first three rows correspond to ablation experiments on the two-stage strategy, while the last row shows the metrics for
+
+the results generated using our two-stage rectification. And in Tab. 8, the first three rows are ablation experiments on the dual-anchor points, and the last row shows the metrics of our dual anchors design. The results of both tables demonstrate that our method outperforms others across multiple metrics.
+
+Table 7: Ablation Study Results on Stages.  
+
+<table><tr><td>Ablation Study</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>ClipS↑</td></tr><tr><td>Stage 1 w/o Rec</td><td>21.67</td><td>0.814</td><td>0.206</td><td>26.96</td></tr><tr><td>Stage 1 w. Rec</td><td>12.03</td><td>0.668</td><td>0.469</td><td>23.35</td></tr><tr><td>Stage 2 w/o Rec</td><td>23.90</td><td>0.884</td><td>0.117</td><td>24.77</td></tr><tr><td>Stage 2 w. Rec (ours)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+Table 8: Ablation Study Results on Dual Anchors.  
+
+<table><tr><td>Ablation Study</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>ClipS ↑</td></tr><tr><td>No Anchors</td><td>24.56</td><td>0.782</td><td>0.165</td><td>27.36</td></tr><tr><td>Only Positive Anchors</td><td>23.27</td><td>0.877</td><td>0.140</td><td>28.15</td></tr><tr><td>Only Negative Anchors</td><td>24.13</td><td>0.854</td><td>0.122</td><td>28.69</td></tr><tr><td>Dual Anchors (ours)</td><td>24.82</td><td>0.898</td><td>0.115</td><td>32.78</td></tr></table>
+
+# K. More Results
+
+We present additional results demonstrating the application of MIRROR to rectify baseline models. SyncDreamer (Liu et al., 2023b) and MVD-Fusion (Hu et al., 2024) can generate images from 16 distinct perspectives simultaneously, whereas VideoMV (Zuo et al., 2024) is capable of producing 24 views in a single session. The reduced angle between adjacent frames of VideoMV facilitates a more precise identification of view correspondences, thereby enhancing the accuracy of the refinements. For image-based and text-based generation tasks, please refer to Fig. 17-19 and Fig. 20, respectively. Fig. 17 illustrates the generated results with SyncDreamer as the baseline. In the first two examples, the baseline model exhibits significant issues with content drift. And the subsequent two examples display problematic geometric shapes. Fig. 18 displays the outcomes generated with MVD-Fusion as the baseline. All cases suffer from severe multi-view inconsistency in the baseline. In Fig. 19, the first example exhibits the color drift issue on the 'train' of the baseline model. In the 'teapot' and 'lunch bag', the results generated by the baseline model show inconsistencies across multiple views in terms of shape. And in the last example, it exhibits issues with multiple faces. Fig. 20, which utilizes VideoMV as the baseline model, presents generated results based on text prompts that similarly struggle with a pronounced Janus Problem and low fidelity. After refinement with MIRROR, the generated results significantly surpass all the baseline models. These figures illustrate that our method markedly improves multi-view consistency through incremental refinements during the sampling process, resulting in enhanced quality and superior accuracy.
+
+# L. Failure Cases
+
+Some failure cases are presented in Fig. 21. When the baseline model fails to generate a reasonable geometric structure, is severely flawed, or is completely inconsistent with the prompt, our method struggles to correct such fundamental errors, as the expert priors we rely on are inaccurate. However, by changing the seed to generate well-formed geometric structures from the baseline, this limitation can be mitigated to the greatest extent.
+
+![](images/d1ea718b77299f124af7fe5f10fd97fa6f76f4871d8109f38d6a084306acc417.jpg)  
+Figure 17: Generated multi-view images by applying MIRROR on SyncDreamer for image-based diffusion rectification. The areas with the Janus Problem or content drifting are marked with red circles.
+
+![](images/2aae4a0abea4639c88080386d773b4071adc7d3a159b08de838802ea6f21620c.jpg)  
+Figure 18: Generated multi-view images by applying MIRROR on MVD-Fusion for image-based diffusion rectification. The areas with the Janus Problem or content drifting are marked with red circles.
+
+![](images/2a1c95f09cefcb12309b31a86a56bff9528b40a5474ac877eaf8b22bca3a42f2.jpg)  
+Figure 19: Generated multi-view images by applying MIRROR on VideoMV for image-based diffusion rectification. The areas with the Janus Problem or content drifting are marked with red circles.
+
+![](images/64895aba83e5f5c0eb9ee88f9179abe99eebadef52fc6aa66135a414b6277377.jpg)  
+Figure 20: Generated multi-view images by applying MIRROR on VideoMV for text-based diffusion rectification.
+
+![](images/e08eb2812bebaea23944d5b7b0e937cb04a6c55e8505031e7026ee716af84e39.jpg)  
+Input
+
+![](images/a5b4e5631e1d7ff07ddbeaeca31297dc363fc05027fef08d91a701dbd88d710c.jpg)  
+Baseline
+
+![](images/c00a54224d249d7449e62aec904bffbbffd010e75f9d2b5b312e138557375175.jpg)
+
+![](images/fe88ac79434f57926f8b890acac14bb3333c7d71c2d939ae9512decf66ef87bc.jpg)  
++MIRROR (Ours)
+
+![](images/c69a357f73007b6e6a0fb9838b18f461c1a000cfaca18964e94f042db5827fe4.jpg)
+
+![](images/3b16f6ea10ba16280d2667c23d729b6fac5d9d4b61d1988f5b4a8e52adcf4471.jpg)  
+A white wedding dress on a hanger, 3D asset.
+
+![](images/d894cf56a742bea0e606e298a119f14d5c700d7af5a3ea3f09161fbd26a1afea.jpg)  
+Figure 21: Failure Cases for both image-based (SyncDreamer) and text-based (VideoMV) models.

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2025/ML$^2$-GCL_ Manifold Learning Inspired Lightweight Graph Contrastive Learning/full.md

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+# $\mathbf{ML}^2$ -GCL: Manifold Learning Inspired Lightweight Graph Contrastive Learning
+
+Jianqing Liang<sup>1</sup> Zhiqiang Li<sup>1</sup> Xinkai Wei<sup>1</sup> Yuan Liu<sup>1</sup> Zhiqiang Wang<sup>1</sup>
+
+# Abstract
+
+Graph contrastive learning has attracted great interest as a dominant and promising self-supervised representation learning approach in recent years. While existing works follow the basic principle of pulling positive pairs closer and pushing negative pairs far away, they still suffer from several critical problems, such as the underlying semantic disturbance brought by augmentation strategies, the failure of GCN in capturing long-range dependence, rigidity and inefficiency of node sampling techniques. To address these issues, we propose Manifold Learning Inspired Lightweight Graph Contrastive Learning (ML²-GCL), which inherits the merits of both manifold learning and GCN. ML²-GCL avoids the potential risks of semantic disturbance with only one single view. It achieves global nonlinear structure recovery from locally linear fits, which can make up for the defects of GCN. The most amazing advantage is about the lightweight due to its closed-form solution of positive pairs weights and removal of pairwise distances calculation. Theoretical analysis proves the existence of the optimal closed-form solution. Extensive empirical results on various benchmarks and evaluation protocols demonstrate effectiveness and lightweight of ML²-GCL. We release the code at https://github.com/a-hou/ML2-GCL.
+
+# 1. Introduction
+
+Graph representation learning, which aims to learn appropriate low-dimensional representations for graph structured
+
+*Equal contribution  ${}^{1}$  Key Laboratory of Computational Intelligence and Chinese Information Processing of Ministry of Education, School of Computer and Information Technology, Shanxi University, Taiyuan 030006, Shanxi, China. Correspondence to: Jianqing Liang <liangjq@sxu.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+data to facilitate various downstream tasks (Cai et al., 2018), has shown wide applications in social network analysis (Pal et al., 2020), molecular property prediction (Liang et al., 2024) and point clouds (Du et al., 2024). Unlike traditional Euclidean data, unstructured features and complex relations in graphs pose unique challenges to representation learning. While a tremendous number of works have been developed (Kipf & Welling, 2017; Velickovic et al., 2018), the requirement of task-dependent labels limits their applicability in practical scenarios. Inspired by the success of contrastive learning in the visual domain, graph contrastive learning (GCL) has demonstrated great potential and yielded promising performance (Velickovic et al., 2019; Peng et al., 2020; Hassani & Khasahmadi, 2020; Zhu et al., 2020).
+
+GCL pulls positive pairs closer and pushes negative pairs far away without labels to learn node embeddings for downstream tasks. Despite comparable performance to supervised learning methods, some critical challenges of augmentation strategies and node sampling approaches still remain to be solved. The commonly used random augmentation strategies such as node dropping (You et al., 2020) and edge perturbation (Qiu et al., 2020; Zhang et al., 2023), may destroy the structural integrity and semantic consistency of graphs. Current node sampling approaches mainly rely on the similarity of structure relations and node embeddings (Lee et al., 2022; Shen et al., 2023; Liang et al., 2025), or probability models, especially clustering algorithms or mixed models, to estimate the probability of positive pairs and negative pairs (Li et al., 2023; Dong et al., 2024; Sun et al., 2024). Such a simple and straightforward strategy may lead to high computational cost, and thus is not applicable to large-scale applications. Moreover, they also ignore the importance of different nodes, among which there exist significant differences in social networks. We are highly motivated to develop an efficient and effective GCL method significantly differs from existing approaches.
+
+Manifold learning, which can capture both the local structure and global topology through modeling the geometric characteristics of data in low dimensional manifold space, provides a new perspective for graph contrastive learning. However, existing GCL methods have not fully exploited the potential of manifold learning yet, especially the lack
+
+![](images/6bb144e58393390e5567753be48b7bb98836a7eac2d7a550aec53de68c92b74e.jpg)  
+Figure 1. The overview of  $\mathrm{ML}^2$ -GCL. With the original node feature  $\mathbf{X}$  and the adjacency matrix  $\mathbf{A}$ , we compute the positive pairs weight matrix  $\mathbf{W}^p$  with a closed-form solution of a manifold learning inspired node representation optimization problem. Then, we calculate the negative pairs weight matrix  $\mathbf{W}^n$  and apply GCN as a graph encoder and obtain the node embedding  $\mathbf{Z}$ . Last, we compute the normalized loss function with regard to the positive pairs weight matrix  $\mathbf{W}^p$ , negative pairs weight matrix  $\mathbf{W}^n$  as well as the node embedding  $\mathbf{Z}$  and then update the graph encoder.
+
+of theoretical guidance in node sampling. Inspired by LLE (Roweis & Saul, 2000), coherent structure of graphs leads to strong correlations between inputs, such as between neighboring nodes, generating observations that lie on or close to a smooth low-dimensional manifold.
+
+In this paper, we propose Manifold Learning Inspired Lightweight Graph Contrastive Learning  $(\mathrm{ML}^2$ -GCL). Figure 1 gives the overview of  $\mathrm{ML}^2$ -GCL. Figure 2 shows the comparison of GRACE (Zhu et al., 2020), NCLA (Shen et al., 2023) and  $\mathrm{ML}^2$ -GCL. Different from previous GCL methods,  $\mathrm{ML}^2$ -GCL eliminates the need to estimate pairwise distances between node embeddings. It does not need to generate augmented views, thereby avoiding the underlying semantic disturbance. In addition,  $\mathrm{ML}^2$ -GCL recovers global nonlinear structure from locally linear fits, which can better distinguish the importance of neighboring nodes. Our contributions are listed as follows.
+
+- To the best of our knowledge, it is the first exploration to marry manifold learning with graph contrastive learning, which inherits the merits of both manifold learning and GCN.  
+- We propose  $\mathrm{ML}^2$ -GCL solely relying on the original
+
+graph with GCN to maintain semantic consistency and achieve lightweight.
+
+- We design a novel contrastive loss function with the closed-form solution of anchor node reconstruction combination weights, which can better distinguish the importance of different nodes.
+
+# 2. Related Work
+
+# 2.1. Graph Contrastive Learning
+
+Current GCL methods can be categorized into three mainstream paradigms, i.e., DGI (Velickovic et al., 2019), InfoNCE (Oord et al., 2018) and BGRL framework (Thakoor et al., 2022). DGI (Velickovic et al., 2019) aggregates the features of all nodes in graphs to obtain a global feature, and then maximizes the mutual information between the global feature and the node features. Based on this framework, MVGRL (Hassani & Khasahmadi, 2020) adopts graph diffusion and subgraph sampling strategies to advance the development of unsupervised graph contrastive learning. Despite the outstanding performance, the global features fail to preserve node-level embedding information adequately.
+
+![](images/e9736e530008a75394411ca24cdd5001ca4ba5b4acf1f01ce757007a5ca05c65.jpg)  
+Figure 2.  $\mathbf{ML}^2$ -GCL versus GRACE and NCLA.  $\mathbf{ML}^2$ -GCL learns positive and negative pairs weights with the input graphs. GRACE removes edges and masks features to generate views of graphs. NCLA generates  $M$  learnable augmented views with adaptive topology by multi-head GAT.
+
+GRACE (Zhu et al., 2020) first utilizes the InfoNCE loss (Oord et al., 2018) to maximize the mutual information between positive pairs and minimize that of negative pairs under two augmented views. Subsequently, a series of methods have been proposed and applied to node classification (You et al., 2020; 2021; Xia et al., 2022b; Shen et al., 2023; Liang et al., 2025) and graph classification (Tan et al., 2021; Yin et al., 2022; Liu et al., 2023) tasks. While these methods have achieved significant success, generation of multiple views results in substantial computational overhead. Inspired by BYOL (Grill et al., 2020), BGRL (Thakoor et al., 2022) uses the random augmentation strategy to generate two views and regards the corresponding augmented nodes as positive pairs. Despite the improvement in graph representation learning and computational efficiency, random augmentation may disrupt the underlying semantics of graphs. AFGRL (Lee et al., 2022) generates two views with encoders and employs the  $k$ -means algorithm for positive pair sampling. The random initialization of the  $k$ -means algorithm may influence the performance.
+
+# 2.2. Graph Augmentation
+
+Graph augmentation strategies include node dropping (You et al., 2020), edge perturbation (Qiu et al., 2020; Zhang et al., 2023), attribute masking (Zhu et al., 2021; Zhang et al., 2022) and subgraph extraction (Hassani & Khasahmadi, 2020). GRACE (Zhu et al., 2020) uses random topology perturbation and node attribute masking to generate two augmented views. GCA (Zhu et al., 2021) proposes an adap
+
+tive augmentation strategy that integrates both structural and attribute information. CI-GCL (Tan et al., 2021) proposes a community-invariant GCL framework to preserve the community structure of graphs. Similarly, HomoGCL (Li et al., 2023) employs community detection algorithms to assess the importance of edges and node features. GCS (Wei et al., 2023) leverages both structural and semantic information to achieve adaptive graph augmentation. Although these methods have achieved significant success, existing augmentation strategies often exhibit poor adaptability across diverse graphs, which is due to the potential risks of augmentation strategies to disturb the underlying semantics. To avoid the above-mentioned issues, some works (Lee et al., 2022; Mo et al., 2022; Xia et al., 2022a; Liang et al., 2025) apply graph encoders to generate multiple views.
+
+# 2.3. Node Sampling
+
+Existing works on node sampling can be divided into two types, i.e., methods with graph structure relations and probabilistic models. Earlier works with graph structure relations (Zhu et al., 2020; You et al., 2020; Zhang et al., 2023) randomly perturb graphs to generate augmented views. They regard the corresponding augmented nodes as positive pairs and all the other nodes of augmented views as negative pairs. Therefore, each anchor point has only one positive pair. AFGRL (Lee et al., 2022) constructs multiple positive pairs from three aspects, including  $k$ -means clustering sets,  $k$ -NN sets and neighboring nodes. NCLA (Shen et al., 2023) directly utilizes the neighbors of anchor points as positive
+
+pairs. GTCA (Liang et al., 2025) utilizes the node representation with GNN and Transformer as graph encoders and topological property, and thus shows excellent performance. While these methods have made some progress, multiple views may generate a large number of pairs, inevitably cause high computational complexity. HomoGCL (Li et al., 2023) regards  $k$ -means as a special case of the Gaussian Mixture Model to estimate the probability of positive pairs. NF-N2N (Dong et al., 2024) uses mutual information to measure the topological similarity between anchor nodes and their neighbors for positive pair sampling. SIGNA (Sun et al., 2024) randomly selects neighboring nodes as positive pairs. These methods may suffer from issues related to randomness and high complexity due to an excessive number of parameters.
+
+# 3. Method
+
+In this section, we first present the preliminaries and notations about GCL. Then, we propose ML $^2$ -GCL. Finally, we give the theoretical analysis and complexity analysis.
+
+# 3.1. Preliminaries and Notations
+
+Let  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  be an input graph, where  $\mathcal{V} = \{v_{1},\dots ,v_{N}\}$ ,  $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$  are the node set and the edge set, respectively. We denote the embedding matrix and the adjacency matrix as  $X\in \mathbb{R}^{N\times F}$  and  $A\in \{0,1\}^{N\times N}$  where  $\pmb {x}_i\in \mathbb{R}^F$  is the node feature of  $v_{i}$ , and  $A_{ij} = 1$  iff  $(v_{i},v_{j})\in \mathcal{E}$ . The goal of GCL is to learn a graph encoder  $f_{\theta}(X,A)\in \mathbb{R}^{N\times D}$  and get node embeddings in low dimensional space without label information, i.e.,  $D\ll F$ , which can be directly applied to downstream tasks.
+
+Existing GCL methods typically generate positive/negative pairs with augmentation strategies. Take one of the most popular GCL methods GRACE (Zhu et al., 2020) as an example, it generates 2 augmented views  $\tilde{\mathcal{G}}_1 = (\tilde{X}_1,\tilde{A}_1)$  and  $\tilde{\mathcal{G}}_2 = (\tilde{X}_2,\tilde{A}_2)$ . Then, it encodes the 2 views with the same GNN encoder to get node embeddings  $Z = f_{\theta}(\tilde{X}_1,\tilde{A}_1)$  and  $Z^{\prime} = f_{\theta}(\tilde{X}_{2},\tilde{A}_{2})$ . Finally, the loss function defined by the InfoNCE loss (Oord et al., 2018) is as follows
+
+$$
+\mathcal {L} = \frac {1}{2 N} \sum_ {i = 1} ^ {N} \left(\ell \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {i} ^ {\prime}\right) + \ell \left(\boldsymbol {z} _ {i} ^ {\prime}, \boldsymbol {z} _ {i}\right)\right) \tag {1}
+$$
+
+where
+
+$$
+\begin{array}{l} \ell \left(z _ {i}, z _ {i} ^ {\prime}\right) = \\ - \log \underbrace {\frac {\exp \left(s \left(z _ {i} , z _ {i} ^ {\prime}\right) / \tau\right)}{\exp \left(s \left(z _ {i} , z _ {i} ^ {\prime}\right) / \tau\right)} + \underbrace {\sum_ {j \neq i} \exp \left(s \left(z _ {i} , z _ {j} ^ {\prime}\right) / \tau\right)} _ {\text {p o s i t i v e p a i r}} + \underbrace {\sum_ {j \neq i} \exp \left(s \left(z _ {i} , z _ {j}\right) / \tau\right)} _ {\text {i n t e r - v i e w n e g a t i v e p a i r s}}} _ {(2)} \\ \end{array}
+$$
+
+where  $s(\cdot, \cdot)$  is the similarity function and  $\tau$  is a temperature parameter.
+
+# 3.2.  $\mathbf{ML}^2$  -GCL
+
+Neighboring Sampling As Figures 1-2 present,  $\mathrm{ML}^2$ -GCL is based on simple geometric intuitions. Suppose the anchor point and its  $k$ -hop neighboring nodes are sampled from the same underlying manifold. Assume there are sufficient nodes, i.e., the manifold is well-sampled, we expect each node and its neighbors to lie on or close to a locally linear patch of the manifold. We characterize the local geometry of these nodes by linear coefficients that reconstruct each anchor point from its positive pairs. Then, the optimization objective is as follows
+
+$$
+\begin{array}{l} \min  _ {w _ {i j} ^ {p}} \sum_ {i = 1} ^ {N} \left\| \boldsymbol {x} _ {i} - \sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \boldsymbol {x} _ {j} \right\| _ {2} ^ {2} + \lambda \sum_ {i = 1} ^ {N} \sum_ {v _ {j} \in \mathcal {P} _ {i}} \left(w _ {i j} ^ {p}\right) ^ {2} \\ s. t. w _ {i j} ^ {p} \geq 0, i = 1, 2, \dots , N, v _ {j} \in \mathcal {P} _ {i} \tag {3} \\ \sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} = 1, i = 1, 2, \dots , N \\ \end{array}
+$$
+
+where  $\mathcal{P}_i$  is the positive pair set of  $v_{i}$ . It consists of  $k$ -hop neighboring nodes of  $v_{i}$ .  $w_{ij}^{p}$  is the reconstruction weight to be solved, which can also be regarded as the probability that  $v_{i}$  and  $v_{j}$  are positive pairs.  $\lambda$  is a positive hyperparameter to balance the reconstruction error and the regularizer.
+
+For simplicity, Equation 3 can be converted to the problem as following vector form
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {w} _ {i} ^ {p}} \sum_ {i = 1} ^ {N} \left(\left| \left| \boldsymbol {x} _ {i} - \sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \boldsymbol {x} _ {j} \right| \right| _ {2} ^ {2} + \lambda \left| \left| \boldsymbol {w} _ {i} ^ {p} \right| \right| _ {2} ^ {2}\right) \\ s. t. w _ {i j} ^ {p} \geq 0, i = 1, 2, \dots , N, v _ {j} \in \mathcal {P} _ {i} \tag {4} \\ \sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} = 1, i = 1, 2, \dots , N \\ \end{array}
+$$
+
+Figure 1 shows the equivalent matrix form with  $w_{ik}^{p} = 0, v_{k} \in \mathcal{N}_{i}$ . The optimization problem above is a typical quadratic programming problem that is easy to solve. Refer to Sec 3.3 for detailed theoretical analysis.
+
+Then, for a given anchor point  $v_{i}$ , we assign all of the negative pairs the same weight as
+
+$$
+w _ {i k} ^ {n} = \frac {1}{\left| \mathcal {N} _ {i} \right|}, v _ {k} \in \mathcal {N} _ {i} \tag {5}
+$$
+
+where  $\mathcal{N}_i$  is the negative pair set of  $v_{i}$  . Similarly, we set  $w_{ij}^n = 0,v_j\in \mathcal{P}_i$
+
+Graph Encoders We apply GCN (Kipf & Welling, 2017) as graph encoder. GCN uses a number of graph convolution
+
+layers to aggregate information from neighbors and then updates each layer with the equation as follows
+
+$$
+\boldsymbol {Z} ^ {(l + 1)} = \sigma (\hat {\boldsymbol {A}} \boldsymbol {Z} ^ {(l)} \boldsymbol {W} ^ {(l)}) \tag {6}
+$$
+
+where  $Z^{(l)}$  is the feature matrix at layer  $l$ ,  $Z^{(0)} = X$ ,  $\hat{A}$  is the normalized adjacency matrix with self-loops,  $W^{(l)}$  denotes the learnable weight matrix of layer  $l$  and  $\sigma$  is a non-linear activation function, i.e., ReLU.
+
+Overview As a lower bound of the Mutual Information, InfoNCE (Oord et al., 2018) is the most commonly used and popular loss function in GCL. In recent years, a large number of related works have sprung up (Zhu et al., 2020; 2021; Xia et al., 2022b; Zhang et al., 2023; Guo et al., 2023; Yu et al., 2024). In spite of some success, the exploited augmentation strategy may bring potential risks of the underlying semantics disturbance. Moreover, only one positive pair for an anchor point can inevitably exacerbate the sampling bias problem. To remedy these deficiencies, we construct multiple positive pairs for a given anchor point with its  $k$ -hop neighboring nodes and original node embeddings  $\mathbf{X}$  as Figures 1-2 show. Then, we use GCN as graph encoder to get the projected node embeddings  $\mathbf{Z}$ . In the end, we define the training objective for each anchor point  $v_{i}$  as follows
+
+$$
+\begin{array}{l} \ell \left(\boldsymbol {z} _ {i}\right) = \\ - \log \frac {\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp (s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right) / \tau)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp (s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right) / \tau) + \sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} \exp (s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}\right) / \tau)} \tag {7} \\ \end{array}
+$$
+
+where  $\tau \in [0,1]$  is a tunable temperature parameter to adjust the degree of attention to the hard negative samples.
+
+Finally, we can define the following overall loss
+
+$$
+\mathcal {L} = \frac {1}{N} \sum_ {i = 1} ^ {N} \ell \left(\boldsymbol {z} _ {i}\right) \tag {8}
+$$
+
+# 3.3. Theoretical Analysis
+
+We propose the manifold learning inspired positive sampling strategy for GCL from the node neighbor sets. To our best knowledge, this is the first exploration to marry manifold learning with graph contrastive learning. It is superior to the similarity/distance-based approaches and probability models that previous GCL models adopt. The details of the proofs are provided in the Appendix A.
+
+In the following, we will give the closed form solution of  $\pmb{w}^p$  without constraints and with constraints, respectively.
+
+Proposition 1. Equation 4 without constraints has an optimal closed solution  $\pmb{w}_i^p = (\pmb{X}_i\pmb{X}_i^T + \lambda \pmb{I})^{-1}\pmb{X}_i\pmb{x}_i$ , where  $\pmb{w}_i^p \in \mathbb{R}^{|\mathcal{P}_i|}$  is the reconstruction weight vector of  $v_i$ ,
+
+$\pmb{X}_i \in \mathbb{R}^{|\mathcal{P}_i| \times F}$  is the positive pairs embedding matrix of  $v_i$  and  $\pmb{x}_i \in \mathbb{R}^F$  is the embedding vector of  $v_i$ .
+
+Proposition 2. The optimal closed solution of Equation 4 can be derived with non-negativity and normalization constraints on solution of Proposition 1. Specifically, we may input the weights into the ReLU activation function, i.e.,  $\boldsymbol{w}_i^p = \mathrm{ReLU}(\boldsymbol{w}_i^p)$  and normalize the weights, i.e.,  $\boldsymbol{w}_i^p = \boldsymbol{w}_i^p / \sum_{j=1}^{N} w_{ij}^p$ .
+
+In fact, the essence of GCL is instance discrimination as a pretext task for self-supervised learning (Wang et al., 2022; Liu et al., 2021). However, learning semantics relies heavily on intra-class data distribution, as shown in (Wang et al., 2022). It is limited by the one-to-one positive sampling strategy, and  $\mathbf{ML}^2$ -GCL may improve it to form a new task that we refer to as personalized graph embedding task, which has less reliance on the data distribution.
+
+Theorem 1. Suppose for a given anchor point  $v_{i}$  in a graph, other nodes can be divided into its positive pairs or negative pairs. The training objective is equivalent to graph embedding as given:
+
+$$
+\ell \Longleftrightarrow \frac {\sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}\right)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right)} \tag {9}
+$$
+
+We present the whole process of our ML $^2$ -GCL method in Algorithm 1.
+
+# Algorithm 1 ML²-GCL
+
+Input: Graph  $\mathcal{G} = (\mathcal{V},\mathcal{E})$
+
+Output: Node embeddings  $Z$
+
+1: for i in 1 to  $N$  do  
+2: Construct positive pair set  $\mathcal{P}_i$  of  $v_{i}$  with its  $k$ -hop neighboring nodes;  
+3: Construct negative pair set  $\mathcal{N}_i$  of  $v_{i}$  with other nodes;  
+4: end for  
+5: Compute positive pairs weight matrix  $\mathbf{W}^p$  as in Propositions 1-2;  
+6: Compute negative pairs weight matrix  $\mathbf{W}^n$  as in Equation 5;  
+7: Initialize GNN encoder parameters  $\{\pmb{W},\pmb{b}\}$  
+8: while not converge do  
+9: Obtain GNN embeddings  $Z = f_{\theta}(\mathcal{G})$  
+10: Do forward pass, compute  $\mathcal{L}$  as in Equations 7-8;  
+11: Do backward propagation with  $\mathcal{L}$  
+12: end while  
+13: return  $Z$  for downstream tasks;
+
+# 3.4. Complexity Analysis
+
+Consider a graph with  $N$  nodes,  $M$  edges and simple encoders which compute embeddings in time and space
+
+Table 1. Node classification accuracy (%) on 6 datasets. Best results are colored: first, second, third. Models with stars adopt graph augmentation strategy. Models in italic type use at least 2 views.  
+
+<table><tr><td>Model</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Amazon-Photo</td><td>Amazon-Computers</td><td>Wiki-CS</td></tr><tr><td>GCN (Kipf &amp; Welling, 2017)</td><td>79.6 ± 1.8</td><td>66.0 ± 1.2</td><td>79.0 ± 2.5</td><td>86.3 ± 1.6</td><td>76.4 ± 1.8</td><td>67.3 ± 1.5</td></tr><tr><td>GAT (Velickovic et al., 2018)</td><td>81.2 ± 1.6</td><td>68.9 ± 1.8</td><td>78.5 ± 1.8</td><td>86.5 ± 2.1</td><td>77.9 ± 1.8</td><td>68.6 ± 1.9</td></tr><tr><td>CGPN (Wan et al., 2021b)</td><td>74.0 ± 1.7</td><td>63.7 ± 1.6</td><td>73.3 ± 2.5</td><td>84.1 ± 1.5</td><td>74.7 ± 1.3</td><td>66.1 ± 2.1</td></tr><tr><td>CG3 (Wan et al., 2021a)</td><td>80.6 ± 1.6</td><td>70.9 ± 1.5</td><td>78.9 ± 2.6</td><td>89.4 ± 1.9</td><td>77.8 ± 1.7</td><td>68.0 ± 1.5</td></tr><tr><td>DGI* (Velickovic et al., 2019)</td><td>82.1 ± 1.3</td><td>71.6 ± 1.2</td><td>78.3 ± 2.4</td><td>83.5 ± 1.2</td><td>78.8 ± 1.1</td><td>69.1 ± 1.4</td></tr><tr><td>GMI (Peng et al., 2020)</td><td>79.4 ± 1.2</td><td>66.9 ± 2.2</td><td>76.8 ± 2.3</td><td>86.7 ± 1.5</td><td>76.1 ± 1.2</td><td>67.8 ± 1.8</td></tr><tr><td>MVGRL* (Hassani &amp; Khasahmadi, 2020)</td><td>82.4 ± 1.5</td><td>71.1 ± 1.4</td><td>79.5 ± 2.2</td><td>89.7 ± 1.2</td><td>78.7 ± 1.7</td><td>69.2 ± 1.2</td></tr><tr><td>GRACE* (Zhu et al., 2020)</td><td>79.6 ± 1.4</td><td>67.0 ± 1.7</td><td>74.6 ± 3.5</td><td>87.9 ± 1.4</td><td>76.8 ± 1.7</td><td>67.8 ± 1.4</td></tr><tr><td>GCA* (Zhu et al., 2021)</td><td>79.0 ± 1.4</td><td>65.6 ± 2.4</td><td>81.5 ± 2.5</td><td>87.0 ± 1.9</td><td>76.9 ± 1.4</td><td>67.6 ± 1.3</td></tr><tr><td>SUGRL (Mo et al., 2022)</td><td>81.3 ± 1.2</td><td>71.0 ± 1.8</td><td>80.5 ± 1.6</td><td>90.5 ± 1.9</td><td>78.2 ± 1.2</td><td>68.7 ± 1.1</td></tr><tr><td>AFGRL (Lee et al., 2022)</td><td>78.6 ± 1.3</td><td>70.8 ± 2.1</td><td>76.4 ± 2.5</td><td>89.2 ± 1.1</td><td>77.7 ± 1.1</td><td>68.0 ± 1.7</td></tr><tr><td>NCLA(Shen et al., 2023)</td><td>82.2 ± 1.6</td><td>71.7 ± 0.9</td><td>82.0 ± 1.4</td><td>90.2 ± 1.3</td><td>79.8 ± 1.5</td><td>70.3 ± 1.7</td></tr><tr><td>GTCA (Liang et al., 2025)</td><td>82.5 ± 1.3</td><td>69.7 ± 1.7</td><td>79.8 ± 1.3</td><td>90.5 ± 1.2</td><td>79.2 ± 1.4</td><td>69.7 ± 1.5</td></tr><tr><td>ML2-GCL</td><td>83.0 ± 1.3</td><td>71.7 ± 1.0</td><td>82.1 ± 1.8</td><td>90.7 ± 1.6</td><td>83.0 ± 2.1</td><td>73.3 ± 1.7</td></tr></table>
+
+$O(M + N)$ . This property is satisfied by most popular GNN architectures, such as GCN (Kipf & Welling, 2017) and GAT (Velickovic et al., 2018). In each update step,  $\mathrm{ML}^2$ -GCL performs 1 encoder computation and backpropagates the learning signal once plus a prediction step, while GRACE performs 2 encoder computations (once for each augmentation) and backpropagates the learning signal twice (once for each augmentation), plus a prediction step. We assume the backward pass to be approximately as costly as the forward pass. We ignore the cost of augmentation computation in this analysis. Therefore, the total time and space complexity per update step for  $\mathrm{ML}^2$ -GCL is  $2C_{\text{encoder}}(M + N) + 2C_{\text{prediction}}N + CN^2$ , compared to  $4C_{\text{encoder}}(M + N) + 4C_{\text{prediction}}N + 4CN^2$  for GRACE, where  $C$  are constants dependent on architecture of different components. Appendix D gives further details.
+
+# 4. Experiments
+
+In this section, we conduct a series of experiments to demonstrate the superiority of  $\mathrm{ML}^2$ -GCL. First, we briefly describe the datasets. Then, we evaluate the empirical performance across various graph datasets on node classification and link prediction tasks. In the end, we present the ablation study, hyperparameters analysis and embeddings visualization results. We implement all experiments on the platform with PyTorch 2.0.1 and PyTorch Geometric 2.6.1 on NVIDIA 3090 GPUs with 24GB memory.
+
+# 4.1. Datasets
+
+We conduct experiments on 6 widely-used datasets including Cora, Citeseer, Pubmed, Amazon-Photo, AmazonComputers and Wiki-CS. For node classification task, we split Cora, Citeseer and Pubmed for the training, validation
+
+and testing following (Yang et al., 2016), and all the other datasets following (Liu et al., 2020). For link prediction task, we follow the experimental setup of GCA (Zhu et al., 2021). For each dataset, we conduct 20 random splits of training/validation/test, and report the average performance of all algorithms on the same random splits. The statistics are summarized in Appendix B.
+
+# 4.2. Node Classification
+
+Node classification is one of the common tasks in graph neural networks. The goal is to predict the class of each node according to its characteristics and context information in the graph structure. In this task, the graph structure and node embeddings are jointly used as inputs, and the relations between nodes and local or global topological information are captured through graph neural networks.
+
+To evaluate node classification, we compare  $\mathbf{ML}^2$ -GCL with 13 state-of-the-art methods including 2 semi-supervised GNNs, i.e., GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018), 2 semi-supervised GCL methods, i.e., CGPN (Wan et al., 2021b), CG3 (Wan et al., 2021a), and 9 self-supervised GCL methods, i.e., DGI (Velickovic et al., 2019), GMI (Peng et al., 2020), MVGRL (Hassani & Khasahmadi, 2020), GRACE (Zhu et al., 2020), GCA (Zhu et al., 2021), SUGRL (Mo et al., 2022), AFGRL (Lee et al., 2022), NCLA (Shen et al., 2023) and GTCA (Liang et al., 2025). For chosen hyper-parameters see Appendix C.
+
+Table 1 lists the node classification accuracy of  $\mathrm{ML}^2$ -GCL and baselines with a logistic regression model. The results indicate that  $\mathrm{ML}^2$ -GCL achieves state-of-the-art results with respect to previous methods, especially when there are relatively more classes. For example,  $\mathrm{ML}^2$ -GCL achieves  $83.0\%$  and  $73.3\%$  accuracy on Amazon-Computers and Wiki-CS,
+
+Table 2. Link prediction results (%) on 6 datasets. Best results are colored: first, second, third. Models with stars adopt graph augmentation strategy. Models in italic type use at least 2 views.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Cora</td><td colspan="2">Citeseer</td><td colspan="2">Pubmed</td><td colspan="2">Amazon-Photo</td><td colspan="2">Amazon-Computers</td><td colspan="2">Wiki-CS</td></tr><tr><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td><td>AUC</td><td>AP</td></tr><tr><td>Spectral (Ng et al., 2001)</td><td>84.6</td><td>88.5</td><td>80.5</td><td>85</td><td>84.2</td><td>87.8</td><td>83.6</td><td>84.3</td><td>86.7</td><td>87.3</td><td>81.2</td><td>80.3</td></tr><tr><td>DeepWalk (Perozzi et al., 2014)</td><td>83.1</td><td>85</td><td>80.5</td><td>83.6</td><td>84.4</td><td>84.1</td><td>83.2</td><td>85.1</td><td>87.2</td><td>87.5</td><td>82.1</td><td>81.5</td></tr><tr><td>GAE (Schulman et al., 2016)</td><td>91</td><td>92</td><td>89.5</td><td>89.9</td><td>96.4</td><td>96.5</td><td>89.9</td><td>89.6</td><td>93</td><td>93.2</td><td>83.2</td><td>82.1</td></tr><tr><td>VGAE (Kipf &amp; Welling, 2016)</td><td>91.4</td><td>92.6</td><td>90.8</td><td>92</td><td>94.4</td><td>94.7</td><td>89.3</td><td>88.8</td><td>92.6</td><td>92.8</td><td>82.5</td><td>83.4</td></tr><tr><td>ARGE (Pan et al., 2018)</td><td>92.4</td><td>93.2</td><td>91.9</td><td>93</td><td>96.8</td><td>97.1</td><td>91.5</td><td>92.2</td><td>92.5</td><td>92.7</td><td>83.5</td><td>84.2</td></tr><tr><td>ARVGA (Pan et al., 2018)</td><td>92.4</td><td>92.6</td><td>92.4</td><td>93</td><td>96.5</td><td>96.8</td><td>92.1</td><td>91.9</td><td>93.1</td><td>92.8</td><td>82.6</td><td>83.9</td></tr><tr><td>GRACE* (Zhu et al., 2020)</td><td>90.9</td><td>91</td><td>92.1</td><td>92.2</td><td>97</td><td>97.1</td><td>90.8</td><td>89.3</td><td>91.6</td><td>91.2</td><td>89.5</td><td>88.5</td></tr><tr><td>GCA* (Zhu et al., 2021)</td><td>91.4</td><td>91.5</td><td>92</td><td>92.6</td><td>96.3</td><td>96.5</td><td>92.3</td><td>91.3</td><td>92.5</td><td>91.5</td><td>86.4</td><td>86.3</td></tr><tr><td>GDCL* (Zhao et al., 2021)</td><td>91.7</td><td>90.9</td><td>91.9</td><td>92</td><td>96.5</td><td>96.3</td><td>93.1</td><td>93.6</td><td>93.1</td><td>93.2</td><td>85.2</td><td>84.6</td></tr><tr><td>ProGCL* (Xia et al., 2022b)</td><td>92.9</td><td>93.5</td><td>93.1</td><td>93.3</td><td>96.1</td><td>96.7</td><td>92.6</td><td>93.5</td><td>94.5</td><td>94.2</td><td>83.6</td><td>84.1</td></tr><tr><td>AUGCL* (Niu et al., 2024)</td><td>93.3</td><td>93.2</td><td>92.5</td><td>92.8</td><td>96.3</td><td>96.5</td><td>94.2</td><td>93.9</td><td>93.7</td><td>93.9</td><td>88.9</td><td>88.5</td></tr><tr><td>ML2-GCL</td><td>96.9</td><td>95.9</td><td>96</td><td>95.1</td><td>98.7</td><td>98.5</td><td>95.1</td><td>94.2</td><td>96.9</td><td>96.6</td><td>89.8</td><td>88.9</td></tr></table>
+
+Table 3. Ablation study on node classification task for positive/negative pairs weights. The metric is the Accuracy(%)  
+
+<table><tr><td>Wp</td><td>Wn</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Amazon-Photo</td><td>Amazon-Computers</td><td>Wiki-CS</td></tr><tr><td>-</td><td>-</td><td>81.2 ± 1.3</td><td>71.0 ± 0.9</td><td>79.5 ± 2.0</td><td>89.5 ± 1.4</td><td>81.1 ± 2.4</td><td>70.0 ± 1.9</td></tr><tr><td>-</td><td>✓</td><td>81.9 ± 1.3</td><td>71.2 ± 0.9</td><td>80.1 ± 1.7</td><td>89.6 ± 2.0</td><td>82.0 ± 2.7</td><td>72.2 ± 1.9</td></tr><tr><td>✓</td><td>-</td><td>82.5 ± 1.2</td><td>71.7 ± 1.1</td><td>79.8 ± 1.8</td><td>90.4 ± 1.2</td><td>82.9 ± 2.1</td><td>72.5 ± 2.2</td></tr><tr><td>✓</td><td>✓</td><td>83.0 ± 1.3</td><td>71.7 ± 1.0</td><td>82.1 ± 1.8</td><td>90.7 ± 1.6</td><td>83.0 ± 2.1</td><td>73.3 ± 1.7</td></tr></table>
+
+a  $3.2\%$  and  $3.0\%$  relative improvement over previous state-of-the-art respectively.
+
+# 4.3. Link Prediction
+
+Link prediction aims to predict whether there is an edge between two nodes in graphs, or to predict the weights of edges. The goal of this task is to capture potential relations between nodes through learning node representations. It can be widely applied in fields such as social network analysis, protein-protein interaction prediction in biological networks, and recommendation systems.
+
+For link prediction, we consider our  $\mathrm{ML}^2$ -GCL with multiple baselines including Spectral (Ng et al., 2001), DeepWalk (Perozzi et al., 2014), GAE (Schulman et al., 2016), VGAE (Kipf & Welling, 2016), ARGE (Pan et al., 2018), ARVGA (Pan et al., 2018), GRACE (Zhu et al., 2020), GCA (Zhu et al., 2021), GDCL (Zhao et al., 2021), ProGCL (Xia et al., 2022b) and AUGCL (Niu et al., 2024). We utilize the Area Under Curve (AUC) and Average Precision (AP) to evaluate the performance. For chosen hyper-parameters see Appendix C.
+
+The results shown in Table 2 suggest that  $\mathbf{ML}^2$ -GCL significantly outperforms all the other baselines on 6 datasets under different evaluation criteria, highlighting its potential on link prediction task. For example, it achieves  $96.9\%$  AUC
+
+on Cora, i.e., a  $3.6\%$  relative improvement over previous state-of-the-art.
+
+It is noteworthy that  $\mathrm{ML}^2$ -GCL achieves the state-of-the-art results on both node classification and link prediction benchmarks with a unified approach. Unlike previous unsupervised models, we do not devise a specialized encoder for each task.
+
+# 4.4. Ablation Study
+
+In this section, we remove positive/negative pairs weights of  $\mathrm{ML}^2$ -GCL to study the impact of each component. The results are shown in Table 3. We notice that the removal of either positive pairs weights  $W^p$  or negative pairs weights  $W^n$  leads to poorer performance, which demonstrates the effectiveness of our proposed  $\mathrm{ML}^2$ -GCL. In addition, it is obvious that compared with the negative pairs weights  $W^n$ , the positive pairs weights  $W^p$  is more crucial for  $\mathrm{ML}^2$ -GCL, which is mainly due to its role on recovery of global nonlinear structure.
+
+# 4.5. Hyperparameters Analysis
+
+Figure 3 illustrates the node classification accuracy of  $\mathbf{ML}^2$ -GCL with varying values of the temperature parameter  $\tau$  and the hidden dim on 6 datasets. It is noteworthy that the optimal temperature parameter is around 0.5 on most of the
+
+![](images/afdd3d22317af46cbc67eee4e422d8dd50a77a95b9f3a73dcd7b7145b3919979.jpg)  
+Figure 3. Sensitivity analysis of the temperature parameter  $\tau$  and the hidden dim on node classification task.
+
+![](images/48ae91f6a92a9824aae83e77c8106d931c6efa52f1ade948e6be67c6eefad841.jpg)  
+(a)NCLA
+
+![](images/427a84f351af7ef9b74d99f4d0c26d6f576de1ad55eb1d9d7051196963f3d649.jpg)  
+(b) GTCA
+
+![](images/2c16520bfbcf36e262beee19d8d4cdd094d05b26dd5764234d1375a0dc77a9af.jpg)  
+(c)  $\mathrm{ML}^2$  -GCL  
+Figure 4. Visualization of NCLA, GTCA and  $\mathbf{ML}^2$  -GCL embeddings on Amazon-Computers dataset with t-SNE.
+
+datasets. In general, with the increase of the hidden dim, the classification performance shows an upward trend on most of the datasets such as Pubmed, Amazon-Photo and Amazon-Computers. We notice that  $\mathrm{ML}^2$ -GCL achieves the best performance on Wiki-CS when the hidden dim is 128. This is mainly due to the fact that the original feature dimension of Wiki-CS is only 300. Increasing the hidden dim will introduce irrelevant or redundant information.
+
+# 4.6. Embeddings Visualization
+
+To provide a more intuitive presentation of the node embeddings, we utilize t-SNE (Van der Maaten & Hinton, 2008) to visualize the node embeddings of NCLA (Shen et al., 2023), GTCA (Liang et al., 2025) and  $\mathbf{ML}^2$ -GCL on Amazon-Computers dataset. As Figure 4 shows, different colors represent different classes. It is obvious that  $\mathbf{ML}^2$ -GCL can distinguish much more different classes of nodes effectively compared with NCLA and GTCA.
+
+# 5. Conclusion
+
+In this paper, we marry manifold learning with graph contrastive learning and develop  $\mathrm{ML}^2$ -GCL. We achieve global nonlinear structure recovery from locally linear fits, and thus inherit the merits of both manifold learning and GCN. It allows us to develop a new paradigm of contrastive learning to eliminate extra views generation and pairwise distances calculation, thus avoiding the semantic disturbance and high computational complexity completely. In addition, we conduct positive sampling with the optimal closed-form solution of a typical quadratic programming problem. Theoretical analysis proves its existence. Extensive experiments verify both effectiveness and lightweight of  $\mathrm{ML}^2$ -GCL. In future, we will investigate deep integration mechanisms between manifold learning and dynamic graph structures to address nonlinear evolution patterns in temporal data or dynamic networks.
+
+# Acknowledgements
+
+This work is supported by National Natural Science Foundation of China (No.62376142, U21A20473).
+
+# 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.
+
+# References
+
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+
+# A. Theoretical Proofs
+
+# A.1. Proof of Proposition 1
+
+Proposition 1. Equation 4 without constraints has an optimal closed solution  $\pmb{w}_i^p = (\pmb{X}_i\pmb{X}_i^T + \lambda \pmb{I})^{-1}\pmb{X}_i\pmb{x}_i$ , where  $\pmb{w}_i^p \in \mathbb{R}^{|\mathcal{P}_i|}$  is the reconstruction weight vector of  $v_i$ ,  $\pmb{X}_i \in \mathbb{R}^{|\mathcal{P}_i| \times F}$  is the positive pairs embedding matrix of  $v_i$  and  $\pmb{x}_i \in \mathbb{R}^F$  is the embedding vector of  $v_i$ .
+
+Proof. First, we review Equation 4 of our proposed ML $^2$ -GCL method and rewrite the objective for each anchor point  $v_{i}, i = 1,2,\dots,N$  without constraints as:
+
+$$
+l _ {i} \left(\boldsymbol {w} _ {i} ^ {p}\right) = \left\| \boldsymbol {x} _ {i} - \sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \boldsymbol {x} _ {j} \right\| _ {2} ^ {2} + \lambda \left\| \boldsymbol {w} _ {i} ^ {p} \right\| _ {2} ^ {2} \tag {10}
+$$
+
+Then, we take the derivative of  $l_{i}$  with respect to  $\pmb{w}_i^p$
+
+$$
+\frac {\mathrm {d} l _ {i}}{\mathrm {d} \boldsymbol {w} _ {i} ^ {p}} = - 2 \boldsymbol {X} _ {i} ^ {T} \boldsymbol {x} _ {i} + 2 \boldsymbol {X} _ {i} ^ {T} \boldsymbol {X} _ {i} \boldsymbol {w} _ {i} ^ {p} + 2 \lambda \boldsymbol {w} _ {i} ^ {p} \tag {11}
+$$
+
+Let  $\frac{\mathrm{d}l_i}{\mathrm{d}\boldsymbol{w}_i^p} = 0$  we obtain
+
+$$
+\left(\boldsymbol {X} _ {i} ^ {T} \boldsymbol {X} _ {i} + \lambda \boldsymbol {I}\right) \boldsymbol {w} _ {i} ^ {p} = \boldsymbol {X} _ {i} ^ {T} \boldsymbol {x} _ {i} \tag {12}
+$$
+
+Considering  $\mathbf{X}_i^T\mathbf{X}_i\succeq \mathbf{0}$  and  $\lambda >0$ , we may conclude the second derivative of  $l_{i}$  with respect to  $w_{i}^{p}$
+
+$$
+\frac {\mathrm {d} ^ {2} l _ {i}}{\mathrm {d} \left(\boldsymbol {w} _ {i} ^ {p}\right) ^ {2}} = 2 \boldsymbol {X} _ {i} ^ {T} \boldsymbol {X} _ {i} + 2 \lambda \boldsymbol {I} \succ \mathbf {0} \tag {13}
+$$
+
+Hence,  $\pmb{w}_i^p = (\pmb{X}_i^T\pmb{X}_i + \lambda \pmb{I})^{-1}\pmb{X}_i^T\pmb{x}_i$  is the extreme point of Equation 10 and also the optimal solution of Equation 4.
+
+# A.2. Proof of Proposition 2
+
+Proposition 2. The optimal closed solution of Equation 4 can be derived with non-negativity and normalization constraints on solution of Proposition 1. Specifically, we may input the weights into the ReLU activation function, i.e.,  $\pmb{w}_i^p = \mathrm{ReLU}(\pmb{w}_i^p)$  and normalize the weights, i.e.,  $\pmb{w}_i^p = \pmb{w}_i^p / \sum_{j=1}^{N} w_{ij}^p$ .
+
+Proof. As  $\mathrm{ReLU}(\pmb{w}_i^p) = \max(\mathbf{0}, \pmb{w}_i^p)$ , we may infer that  $w_{ij}^p = \max(0, w_{ij}^p) \geq 0, i = 1, 2, \dots, N, v_j \in \mathcal{P}_i$ . In addition, we can derive conclusion as follows
+
+$$
+\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} = \sum_ {v _ {j} \in \mathcal {P} _ {i}} \frac {w _ {i j} ^ {p}}{\sum_ {j = 1} ^ {N} w _ {i j} ^ {p}} = \frac {\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p}}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} + \sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {p}} = 1, i = 1, 2, \dots , N \tag {14}
+$$
+
+where  $w_{ik}^p = 0, v_k \in \mathcal{N}_i$ .
+
+# A.3. Proof of Theorem 1
+
+Theorem 1. Suppose for a given anchor point  $v_{i}$  in a graph, other nodes can be divided into its positive pairs or negative pairs. The training objective is equivalent to graph embedding as given:
+
+$$
+\ell \Longleftrightarrow \frac {\sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}\right)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right)} \tag {15}
+$$
+
+Proof. We rewrite the loss function of anchor point  $v_{i}$  with Taylor expansion of the first order as:
+
+$$
+\begin{array}{l} \ell (\boldsymbol {z} _ {i}) = \log \frac {\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp (s (\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}) / \tau) + \sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} \exp (s (\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}) / \tau)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp (s (\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}) / \tau)} \\ = \log \left(1 + \frac {\sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} \exp \left(s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}\right) / \tau\right)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp \left(s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right) / \tau\right)}\right) \tag {16} \\ \approx \frac {\sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} \exp \left(s (\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}) / \tau\right)}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp \left(s (\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}) / \tau\right)} \\ \end{array}
+$$
+
+We extend the loss function of  $v_{i}$  to the form of the entire graph  $\mathcal{G}$  and remove the temperature hyperparameter  $\tau$ :
+
+$$
+\begin{array}{l} \mathcal {L} = \frac {1}{N} \sum_ {i = 1} ^ {N} \ell (\boldsymbol {z} _ {i}) \\ \propto \sum_ {i = 1} ^ {N} \frac {\sum_ {v _ {k} \in \mathcal {N} _ {i}} w _ {i k} ^ {n} \exp (s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {k}\right))}{\sum_ {v _ {j} \in \mathcal {P} _ {i}} w _ {i j} ^ {p} \exp (s \left(\boldsymbol {z} _ {i} , \boldsymbol {z} _ {j}\right))} \tag {17} \\ \end{array}
+$$
+
+Different from typical graph embedding, each positive pair or negative pair is associated with its own weight. The overall objective is calculated with the exponential ratios sum of the negative and positive similarities with anchor points.
+
+# B. Dataset Descriptions
+
+We evaluate the performance of  $\mathbf{ML}^2$ -GCL on node-level and edge-level tasks, i.e., node classification and link prediction. We conduct experiments on 6 widely used datasets including Cora, Citeseer, Pubmed, Amazon-Photo, Amazon-Computers and Wiki-CS. The detailed statistics are summarized in Table 4.
+
+- Cora, Citeseer and PubMed (Sen et al., 2008) are well-known citation network datasets, where nodes represent scientific papers and edges represent citation relations. Node features are bag-of-words vectors of papers, and labels represent domains of papers.  
+- Amazon-Photo and Amazon-Computers (Shchur et al., 2018) are two networks of co-purchase relationships constructed from Amazon. Nodes are product and edges exist when two products are frequently co-purchased. Each node has a bag-of-words feature encoding product reviews and is labeled with its product category.  
+- Wiki-CS (Mernyei & Cangea, 2020) is a reference network from Wikipedia references. Nodes correspond to articles about computer science and edges are hyperlinks between the articles. Articles are labeled with 10 related subfields, and their features are calculated as the average of pre-trained Glove (Pennington et al., 2014) word embeddings.
+
+Table 4. Statistics of datasets used in experiments.  
+
+<table><tr><td>Dataset</td><td># Nodes</td><td># Edges</td><td># Features</td><td># Labels</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>1,433</td><td>7</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,732</td><td>3,703</td><td>6</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>500</td><td>3</td></tr><tr><td>Amazon-Photo</td><td>7,650</td><td>119,081</td><td>745</td><td>8</td></tr><tr><td>Amazon-Computers</td><td>13,752</td><td>245,861</td><td>767</td><td>10</td></tr><tr><td>Wiki-CS</td><td>11,701</td><td>216,123</td><td>300</td><td>10</td></tr></table>
+
+# C. Hyperparameter Choices
+
+In hyperparameter search, we attempt to adjust the value of  $k$  and  $\lambda$  in  $\mathrm{ML}^2$ -GCL, as well as other deep learning hyperparameters including temperature parameter  $\tau$ , hidden dim, learning rate, dropout and weight decay. We apply the grid search strategy to choose the optimal hyperparameters. Specifically, we search  $k$  in [1, 5],  $\lambda$  in  $\{0, 0.01, 0.1, 1, 10, 100\}$ , hidden dim from  $\{128, 256, 500, 512\}$ , learning rate from  $\{0.0005, 0.001, 0.005, 0.01\}$  and weight decay in  $\{5e-5, 1e-4, 5e-4, 7e-4\}$ . Tables 5-6 give the hyperparameters specifications for  $\mathrm{ML}^2$ -GCL on node classification and link prediction tasks.
+
+Table 5. Hyperparameters specifications for  ${\mathrm{{ML}}}^{2}$  -GCL on node classification task.  
+
+<table><tr><td></td><td>Cora</td><td>CiteSeer</td><td>Pubmed</td><td>Amazon-Photo</td><td>Amazon-Computers</td><td>Wiki-CS</td></tr><tr><td>k</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>λ</td><td>0.01</td><td>1</td><td>100</td><td>10</td><td>0.01</td><td>10</td></tr><tr><td>Temperature τ</td><td>0.7</td><td>1</td><td>0.5</td><td>0.5</td><td>0.3</td><td>0.5</td></tr><tr><td># Hidden dim</td><td>500</td><td>256</td><td>512</td><td>512</td><td>512</td><td>128</td></tr><tr><td>Learning rate</td><td>0.0005</td><td>0.005</td><td>0.001</td><td>0.001</td><td>0.0005</td><td>0.01</td></tr><tr><td># Epochs</td><td>40</td><td>20</td><td>80</td><td>120</td><td>240</td><td>280</td></tr><tr><td>Dropout</td><td>0.5</td><td>0.4</td><td>0.5</td><td>0.3</td><td>0.5</td><td>0.4</td></tr><tr><td>Weight decay</td><td>5e-5</td><td>7e-4</td><td>5e-4</td><td>5e-4</td><td>5e-4</td><td>1e-4</td></tr></table>
+
+Table 6. Hyperparameters specifications for  ${\mathrm{{ML}}}^{2}$  -GCL on link prediction task.  
+
+<table><tr><td></td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Amazon-Photo</td><td>Amazon-Computers</td><td>Wiki-CS</td></tr><tr><td>k</td><td>3</td><td>3</td><td>2</td><td>2</td><td>4</td><td>1</td></tr><tr><td>λ</td><td>10</td><td>0.001</td><td>1</td><td>100</td><td>100</td><td>1</td></tr><tr><td>Temperature τ</td><td>0.4</td><td>0.5</td><td>0.3</td><td>0.4</td><td>0.3</td><td>0.3</td></tr><tr><td># Hidden dim</td><td>500</td><td>256</td><td>500</td><td>500</td><td>256</td><td>500</td></tr><tr><td>Learning rate</td><td>0.0005</td><td>0.01</td><td>0.0001</td><td>0.0001</td><td>0.01</td><td>0.001</td></tr><tr><td># Epochs</td><td>200</td><td>100</td><td>200</td><td>100</td><td>200</td><td>40</td></tr><tr><td>Dropout</td><td>0.4</td><td>0.4</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Weight decay</td><td>5e-5</td><td>5e-5</td><td>5e-5</td><td>1e-4</td><td>5e-5</td><td>5e-5</td></tr></table>
+
+# D. Memory Usage
+
+Table 7 shows the GPU memory of  $\mathrm{ML}^2$ -GCL and baselines including GRACE, GCA, AFGRL, NCLA and GTCA on node classification task. The results indicate that  $\mathrm{ML}^2$ -GCL has less memory requirement compared with existing methods due to its closed-form solution of positive pairs weights and removal of pairwise distances calculation. Cross-view contrastive methods such as GRACE require additional storage for the embedding matrices of multiple augmented views, as well as the indices of positive and negative pairs, thereby increasing memory requirement.
+
+Table 7. The GPU memory (GiBs) usage across 6 datasets on node classification task. Best results are colored: first, second, third.  
+
+<table><tr><td>Model</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Amazon-Photo</td><td>Amazon-Computers</td><td>Wiki-CS</td></tr><tr><td>GRACE</td><td>0.47</td><td>0.55</td><td>13.17</td><td>2.11</td><td>6.23</td><td>4.46</td></tr><tr><td>GCA</td><td>0.58</td><td>0.69</td><td>14.32</td><td>2.34</td><td>6.61</td><td>5.01</td></tr><tr><td>AFGRL</td><td>0.37</td><td>0.48</td><td>12.59</td><td>2.12</td><td>5.94</td><td>4.41</td></tr><tr><td>NCLA</td><td>0.67</td><td>1.02</td><td>14.56</td><td>2.25</td><td>7.16</td><td>7.71</td></tr><tr><td>GTCA</td><td>1.64</td><td>2.64</td><td>22.03</td><td>4.09</td><td>13.14</td><td>10.89</td></tr><tr><td>ML²-GCL</td><td>0.33</td><td>0.45</td><td>12.15</td><td>2.01</td><td>5.74</td><td>4.22</td></tr></table>
+
+We show part of the information from Table 7 as a scatterplot to visualize the different scaling properties of  $\mathbf{ML}^2$ -GCL and
+
+![](images/e39a7d66bbd762288957187188b14212effdec183904a53838163e894e1358f6.jpg)  
+Figure 5. Memory usage of NCLA, GTCA and  $\mathrm{ML}^2$ -GCL across 6 standard datasets on node classification task.
+
+NCLA, GTCA intuitively in Figure 5. Among the top 3 ranked methods on node classification task,  $\mathbf{ML}^2$ -GCL consumes the least GPU memory usage. As the number of nodes increases, its memory usage increases slower compared with NCLA and GTCA.

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+# MMInference: Accelerating Pre-filling for Long-Context Visual Language Models via Modality-Aware Permutation Sparse Attention
+
+Yucheng Li $^{1*}$  Huiqiang Jiang $^{2§}$  Chengruidong Zhang $^{2}$  Qianhui Wu $^{2}$  Xufang Luo $^{2}$  Surin Ahn $^{2}$  Amir H. Abdi $^{2}$  Dongsheng Li $^{2}$  Jianfeng Gao $^{2}$  Yuqing Yang $^{2}$  Lili Qiu $^{2}$
+
+# Abstract
+
+The integration of long-context capabilities with visual understanding unlocks unprecedented potential for Vision Language Models (VLMs). However, the quadratic attention complexity during the pre-filling phase remains a significant obstacle to real-world deployment. To overcome this limitation, we introduce MMInference (Multi-modality Million tokens Inference), a dynamic sparse attention method that accelerates the pre-filling stage for long-context multi-modal inputs. First, our analysis reveals that the temporal and spatial locality of video input leads to a unique sparse pattern, the Grid pattern. Simultaneously, VLMs exhibit markedly different sparse distributions across different modalities. We introduce a permutation-based method to leverage the unique Grid pattern and handle modality boundary issues. By offline search the optimal sparse patterns for each head, MMInference constructs the sparse distribution dynamically based on the input. We also provide optimized GPU kernels for efficient sparse computations. Notably, MMInference integrates seamlessly into existing VLM pipelines without any model modifications or fine-tuning. Experiments on multi-modal benchmarks—including Video QA, Captioning, Vision-NIAH, and Mixed-Modality-NIAH—with state-of-the-art long-context VLMs (LongVila, LlavaVideo, VideoChat-Flash, Qwen2.5-VL) show that MMInference accelerates the pre-filling stage by up to  $8.3 \times$  at 1M tokens while maintaining accuracy. Our code is available at https://aka.ms/MMInference.
+
+![](images/122390b437415ba8c34f9b45fc9b715a83d4b5b987062f26f3c0be10edf89315.jpg)  
+Figure 1: Dynamic sparse attention pipelines leverage sparse loading with dense computation (Zheng et al., 2023) to enable hardware-efficient acceleration. MMInference adopts a bottom-up system-algorithm co-design that accounting for both the mathematical equivalence constraints of sparse loading and the locality properties of real-world attention patterns.
+
+# 1. Introduction
+
+Scaling the context size of Vision Language Models (VLMs) allows them to handle extended temporal information from long video and text inputs, which is crucial for various applications including robotics (Black et al., 2024; Prasad et al., 2024; Cheang et al., 2024), autonomous driving (Hu et al., 2023; Wang et al., 2024c; Gao et al., 2024), and healthcare (Liu et al., 2024b). In addition, Zhang et al. (2024b) and Chen et al. (2025) show that scaling the context size of VLMs can improve the resolution in the temporal dimension and lead to better performance in video understanding tasks.
+
+However, due to the quadratic complexity of attention, processing long multi-modal inputs (i.e., the pre-fill stage) can take minutes prior to auto-regressive decoding. As shown in Fig. 2a, this leads to significant Time-to-First-Token latency, which hinders the wide adoption of long-context VLMs in real-world applications. Previous work (Child et al., 2019; Liu et al., 2022; 2024a; Yuan et al., 2025; Lu et al., 2025) re
+
+![](images/243d32e66ea5ca62b53593a8ad7744bead4fe57493b6c2bdc7345453ebe5f834.jpg)  
+(a) VLMs' attention incurs heavy cost.
+
+![](images/fbdede0620d45244706ac193d2381ed43afd3e50350b5655c55104ce6499742d.jpg)  
+(b) VLMs' attention is sparse.  
+Figure 2: (a) Latency breakdown of the pre-filling stage, with 256 tokens per frame. (b) How much element in attention needs to be computed to achieve  $95\%$  recall in a 128k context. (c) Low attention recall when reusing the top-k indices from a different request. Visualizations are based on LongVILA-7B-1M (Chen et al., 2025) with a single A100.
+
+![](images/c23c5abdd423c33d9581544219deab9a38bdd4a52b67d2e9285480af6e105261.jpg)  
+(c) Sparsity of VLMs' attention is dynamic.
+
+veals that attention matrices are typically sparse, prompting the development of sparse attention methods such as Sparse Transformer (Child et al., 2019), Swin Transformer (Liu et al., 2021), and StreamingLLM (Xiao et al., 2024). More recently, MInference (Jiang et al., 2024) proposes to use dynamic sparse attention that estimates the sparse index online, and leverages optimized GPU kernels for end-to-end acceleration. However, these methods fail to exploit the unique sparse patterns in long-context VLMs, and struggle with mixed or interleaved modalities, limiting their applicability without compromising performance.
+
+Unlike long-text contexts, video and image inputs in VLMs exhibit spatiotemporal locality, forming grid-like attention patterns with evenly spaced vertical and horizontal lines (Fig. 3a). In mixed-modality inputs, clear modality boundaries emerge: attention across modalities diverges significantly from intra-modality attention (Fig. 3b). These factors pose unique challenges for exploiting sparsity to accelerate the pre-fill stage.
+
+In this paper, we present MMInference, a permutation-based dynamic sparse attention method that significantly reduces attention FLOPs, accelerating the pre-fill stage of long-context VLMs. First, MMInference identifies the grid heads and leverages a row- and column-wise permutation to gather the sparse grid for efficient hardware computation. Next, we detect Query-boundary and 2D-boundary patterns to address inter-modality boundary issues, and apply a modality-wise permutation to isolate intra-modality regions. This results in a consecutive sparse index within each modality, permitting efficient hardware implementation of sparse computing. Finally, a Modality-Aware Sparse Attention Search Algorithm is devised to fine-tune both inter- and intra-modality patterns offline, to optimize performance with minimal overhead.
+
+We conduct extensive experiments using four state-of-the-art long-context VLMs, Llava-Video (Zhang et al., 2024b), LongVila (Chen et al., 2025), VideoChat-Flash (Li et al.,
+
+2025) and Qwen2.5-VL (Bai et al., 2025), across diverse video understanding tasks such as video captioning (Maaz et al., 2024), video question answering (Yu et al., 2019; Xiao et al., 2021; Mangalam et al., 2023; Fu et al., 2024), and video information retrieval (Zhang et al., 2024a). Additionally, we propose the Mixed-Modality Needle in a Hackathon task to assess multi-modal input performance. Our method effectively addresses modality boundaries, significantly accelerates the prefilling stage, and maintains high accuracy. With a 1M-length context, it achieves speedups of up to  $8.3 \times$  and  $1.7 \times$  over FlashAttention-2 and MInference, respectively.
+
+# 2. Attention Heads in VLMs
+
+The sparsity of the attention operation in pre-trained text-only LLMs, particularly in long-context scenarios, has been extensively studied (Wu et al., 2025; Ribar et al., 2024; Jiang et al., 2024; Li et al., 2024), showing that only  $3\%$  of attention weights are activated while achieving a recall rate of  $96.8\%$ . Similarly, VLMs also demonstrate notable dynamic sparsity in long-context scenarios. This section examines the shared and distinct properties of text-only and multi-modal LLMs in long-context scenarios, focusing on attention sparsity, sparse patterns, and modality boundaries.
+
+# 2.1. Multi-modality Attention is Dynamically Sparse
+
+As illustrated in Fig. 2a, for a  $128\mathrm{k} \times 128\mathrm{k}$  attention matrix in VLMs, retaining only the top  $5.78\%$  of attention weights on average suffices to recall  $95\%$  of total attention, indicating that each token attends only to a limited subset of tokens, even in long sequences. However, VLMs exhibit lower sparsity than text-only LLMs, where only  $1.79\%$  of weights achieve a  $95\%$  recall rate. Notably, the bottom layers in VLMs (e.g., the first four layers in LongVila) show reduced sparsity. Yet, due to variability across attention heads,  $52.3\%$  of heads in VLMs require less than  $2\%$  of
+
+![](images/7bbc3510e4411b0ea5696aaf03f5a2b0e57db44356e88cd478c0cbe8922ca979.jpg)  
+(a) Grid pattern.
+
+![](images/49e9ed0cc0e7fda74e9fe0c6c7b9c9415477d34dd37f7b2dc71dd9e81d45f069.jpg)  
+(b) Q-Boundary pattern.
+
+![](images/bcb3492e368433b3f0c39976def62e1bd70f3d4a0cff8ba18d06acea51cdbbf1.jpg)  
+(c) 2D-Boundary pattern.
+
+![](images/8d08c28fa51ece58b8410e900841ac71cca6808eaa0344fe1b767163c0f2e83e.jpg)  
+(d) Permuted Grid pattern.
+
+![](images/23d26e566a891056eb214b8b025ccd771d8c501630c71bc3d2a7eeca4579badb.jpg)  
+(e) Permuted Q-Boundary pattern.
+
+![](images/40ed4f1493efae551c23f9fcbbacc55c20c8b894fc172732d2cc8bfa975e0e0f.jpg)  
+(f) Permuted 2D-Boundary pattern.  
+Figure 3: Visualization of pre- vs. post-permutation sparsity attention patterns in VLMs.
+
+attention to be recalled. This highlights substantial computational redundancy in VLMs, especially in long-context scenarios.
+
+Similarly to LLMs, while the sparse nature of attention matrices remains consistent across inputs, the specific distributions of sparse attention are highly dynamic. As shown in Fig. 2c, reusing top-k indices for  $95\%$  attention recall (derived from Fig. 2b) across different contexts leads to a significant drop in performance.
+
+# 2.2. The Grid Head in VLMs
+
+In long-context language modeling, efficient attention mechanisms like sliding window attention (Jiang et al., 2023) and StreamingLLM (Xiao et al., 2024) exploit the locality property of text sequences. However, multi-modal inputs introduce unique geometric structures that redefine locality. As shown in Child et al. (2019), image patches exhibit locality along both vertical and horizontal directions, forming local window and slash-like patterns. Similarly, video inputs maintain locality across temporal and spatial dimensions, with frame-based sampling yielding more regular and
+
+predictable patterns.
+
+We observe that certain VLM attention heads exhibit a grid pattern. While the grid's stride and starting position vary with context, the horizontal and vertical lines are evenly spaced and often symmetrical—a distinct behavior compared to text-only LLMs (Jiang et al., 2024; Lai et al., 2025). Fig. 3a visualizes a grid head, demonstrating how local tokens in temporal and spatial dimensions are evenly distributed within the attention map, with attention focused primarily on these local tokens.
+
+# 2.3. Modality Boundaries in Multi-Modal Input
+
+The input format of VLMs differs significantly from text-only LLMs. A dedicated vision encoder generates visual representations, which are processed alongside text embeddings by the LLM. Despite pretraining on large-scale datasets, the interactions and processing patterns between modalities vary considerably, leading to distinct modality boundaries in attention (Tu et al., 2025), as illustrated in Fig. 3b and 3c.
+
+Inter-modality Attention Pattern
+
+![](images/fe503ec6a22c5ec829e449a8740c08509bcdc22505f80e66c019365ba4a7af09.jpg)  
+No-Boundary head
+
+![](images/c3b62628f96b69ea58dd28c3fa9b359d7b3857c8e34362f9e9f715146a3a4da0.jpg)  
+K-Boundary head
+
+![](images/c0c25851b0c139634c21aab3ccdd8023b0cac8a9862ad180026795524ba01214.jpg)  
+Q-Boundary head
+
+![](images/4a47f6a682409d00c0f08df1bd5df57dc3d3eb395c47a264056a9f3c182e37b9.jpg)  
+3 2D-Boundary head
+
+Intra-modality Attention Pattern
+
+![](images/010d9dca22e73832e754bdd5f481886163c9b2cc0ee8e9349bfa631f1d1606a1.jpg)  
+1  $\Lambda$  -shape head
+
+![](images/b2fb68dda3a7fa096d3aa623678e543b7574c90f221bb73cf201db49d0fb89ab.jpg)  
+2 vertical-slash head
+
+![](images/95d51dfd8ccd96b6980a82c2b0fd005434d174ef876f6f2c84fc3854e31aba24.jpg)  
+3 grid head  
+Figure 4: The framework of MMInference, encompassing both inter- and intra-modality sparse attention patterns.
+
+![](images/028eac13069b7f20e9dd9ced058d27f7455b10e8a9d6bf66b32d2779665e4e35.jpg)
+
+Specifically, we observe two key characteristics: 1) Intra-modality consistency: Attention within each modality follows a consistent pattern. For instance, the vision region in Fig. 3b exhibits a clear slash pattern, where critical elements are effectively clustered. 2) Modality-separated continuity: Patterns within a modality can be interrupted by boundaries from other modalities. As shown in Fig. 3b, vision slashes are segmented by the boundary introduced by the language region.
+
+We categorize the modality boundary patterns of VLMs into four distinct types: No-Boundary, K-Boundary, Q-Boundary, and 2D-Boundary, as illustrated in Figs. 3 and 4. 1) No Boundary and K-Boundary exhibit either no clear modality boundary or a boundary only along the key dimension, as shown in Fig. 9. Since continuity is maintained along the query dimension, these heads can be efficiently handled using intra-modality sparse patterns. 2) Q-Boundary refers to attention modality boundaries across the query dimension. For example, in Fig. 3b, sparse patterns like Text-to-Video and Video-to-Video appear interconnected, forming a trapezoidal structure, while a clear boundary separates Visual-to-Text and Text-to-Visual attention. 3) 2D-Boundary occurs when modality boundaries are present in both query and key dimensions. As shown in Fig. 3c, the 2D modality boundary segments attention weights into distinct blocks. Additionally, our analysis of Audio LMs (Chu et al., 2024) and end-to-end multimodal LMs (Xu et al., 2025a; Li et al., 2025) reveals that the cross-modality boundary phenomenon persists across these architectures. These boundaries pose unique challenges and hinder direct application of existing sparse attention methods to multi-modal inputs.
+
+# 2.4. Sparse Distributions Continuity Across Boundaries
+
+Although sparsity patterns in VLMs are often discontinuous across modalities due to modality boundaries, we find that sparsity distributions can remain continuous across these boundaries and extrapolate to other regions of the same modality. For example, in Fig. 3b, the slash lines maintain the same relative position across different areas of the vision modality. In a more complex case, Fig. 3c shows interleaved vision and text modalities forming a mixed structure. However, by spatially aggregating regions of the same modality, we observe that sparsity patterns can extend beyond local regions and often exhibit global extrapolation potential. The upper-left region in Fig. 3c exemplifies this, where the grid pattern, initially separated by textual boundaries, becomes consecutive after spatial clustering in both row and column dimensions. To validate this observation, we conducted a quantitative attention recall experiment on mixed-modality inputs, as detailed in §4.6.
+
+# 3. MMInference
+
+Following the analysis in §2, we propose MMInference to accelerate the pre-filling stage of long-context VLMs as shown in Fig. 4. The framework consists of three modules, covering both inter- and intra-modality sparse patterns: 1) the novel Grid sparse attention, together with the A-shape and Vertical-Slash patterns (Jiang et al., 2024) forms the intra-modality attention; 2) Q-Boundary and 2D-Boundary mix-modality patterns; 3) Modality-aware sparse attention search algorithm. We first perform offline pattern search to identify different patterns for each attention head. Then we
+
+use online dynamic sparse approximation to build the sparse index, and finally we perform dynamic sparse computation using optimized GPU kernels.
+
+# 3.1. Grid Head in Multi-Modality
+
+To better leverage the inductive bias in visual modalities (e.g., images, videos) and the vertical and horizontal structures in attention patterns, we propose a permutation-based dynamic sparse attention for grid head, as shown in Algo. 1.
+
+Algorithm 1 Grid Head  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , stride space  $s_g\in \phi_g$    
+# Approximate stride and phase (last_q = 64)   
+ $\widehat{\pmb{A}}\gets$  softmax  $\left(Q_{[-\mathrm{last}.q:]}K^{\top} / \sqrt{d} +m_{\mathrm{casual}}\right)$    
+# Online search grid stride and phase   
+ $b_{r},\gets 0$    
+for  $i\gets 1$  to  $|\phi_g|$  do if maxview(A,sg,i)) >br then  $s_g\gets s_{g,i},p_g\gets \mathrm{argmax}(v i e w(\widehat{A},s_{g,i}))$ $\pmb {b}_r\gets \max (\mathrm{view}(\widehat{\pmb{A}},s_{g,i}))$  end   
+end for
+
+Permute Q, K, V tensors  
+ $\overline{\mathbf{Q}}, \overline{\mathbf{K}}, \overline{\mathbf{V}} \gets$  permute  $(\mathbf{Q})$ , permute  $(\mathbf{K})$ , permute  $(\mathbf{V})$   
+# Dynamic block sparse attention w/ FlashAttention  
+(only the last and rightmost block)  
+ $\mathbf{A} \gets$  softmax  $\left(\text{sparse}(\overline{\mathbf{Q}\mathbf{K}}^{\top}, s_{g}, p_{g}) / \sqrt{d}\right)$   
+# Sparse mixed scores and values  
+ $\mathbf{y} \gets$  sparse  $(\mathbf{A}\overline{\mathbf{V}}, s_{g}, p_{g})$   
+return  $\mathbf{y}$
+
+Specifically, we first perform an online search to determine the stride and phase of grid pattern. Since only a view operation is applied to the approximate attention matrix  $\widehat{A}$ , the actual latency overhead remains minimal. Next, we use the identified grid stride and phase to permute the  $Q$ ,  $K$ , and  $V$  tensors to compute sparse attention efficiently (see Fig. 3d). In our implementation, instead of explicitly permuting  $Q$ ,  $K$ , and  $V$ , we optimize computational efficiency by dynamically loading and writing these tensors within the kernel, minimizing the overhead associated with tensor transpositions. In addition to Grid sparse attention, we also employ A-shape and Vertical-Slash attention for intra-modality operation, see Appendix C.3 for more details.
+
+# 3.2. Hybrid Modality Sparse Attention
+
+As analyzed in §2 and illustrated in Fig. 3, modality boundaries exist in multi-modal LLMs. We classify these boundaries into four patterns: No-Boundary, K-Boundary, Q-Boundary, and 2D-Boundary. As the sparse index is continuous along the query dimension for both the No-Boundary
+
+and K-Boundary heads, we can directly apply the three intra-modality attention globally. However, for Q-Boundary and 2D-Boundary, MMInference uses a permutation-based approach to efficiently handle these modality boundaries.
+
+Q-Boundary Head As shown in Fig.3b, Fig.3e, and §2.4, the Q-Boundary pattern shows a clear separation across modality, but the sparse distribution remains continuous within each modality. Building on this insight, we propose a row-wise permutation (Algorithm 2) that groups tokens of the same modality by permuting  $Q$ , and then applies offline-optimized sparse attention (A-shape, Vertical-Slash, and Grid Head) for intra-modality processing. Note that we leverage the final segment of each modality's queries to dynamically approximate the sparse indices and extrapolate to the entire modality. This method enables flexibility in handling fragmented multi-modality inputs. Additionally, instead of explicitly permuting tensors, our implementation performs dynamic loading and writing inside the kernel for optimized efficiency.
+
+Algorithm 2 Q-Boundary Head  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , modality type index  $\pmb{i}_m$  , modality type set  $m\in \phi_{m}$    
+#Permute Q tensors based on modality  $\overline{\boldsymbol{Q}}\gets$  permute  $(Q,i_m)$    
+# Looping over the modalities in query dimension   
+ $\pmb {y}\leftarrow \mathbf{0}$    
+for  $i\gets 1$  to  $|\phi_m|$  do   
+# Intra-modality sparse attention for each modality w/FlashAttention   
+ $A_{mi}\gets$  softmax(sparse  $(\overline{\mathbf{Q}}_{mi}\pmb{K}^{\top},\pmb{i}_{mi}) / \sqrt{d})$ $\pmb{y}_{mi}\gets$  sparse(AmiV)   
+# Update the modality output to the final output   
+ $\pmb {y}\leftarrow \pmb{y}_{mi}\cup \pmb{y}$    
+end for   
+return  $\pmb{y}$
+
+2D-Boundary Head Beyond Query-Boundary, there are attention heads that exhibit modality boundaries in both query and key dimensions, as shown in Fig. 3c. Given a query token, attention to key tokens from different modalities varies significantly, and queries from different modalities focus on keys in highly diverse patterns. To address 2D modality boundaries, we design a 2D permutation approach that groups  $Q$ ,  $K$ , and  $V$  according to their modalities. This allows us to leverage intra-modality continuity to handle each part of 2D boundary pattern separately and efficiently. We further illustrate this approach in Fig. 3f and it detailed in Algorithm 3. Specifically, we perform permutation on both row- and column-wise for  $Q$ ,  $K$ , and  $V$ , and then iteratively traverse each modality pair to compute dynamic
+
+Table 1: Performance (%) of different models and different methods on video understanding tasks evaluated at frames from 110 to 256.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">FLOPs</td><td>VideoDC</td><td>ActNet-QA</td><td>EgoSchema</td><td>Next-QA</td><td>PerceptionTest</td><td colspan="2">VideoMME</td><td rowspan="2">Avg.</td></tr><tr><td>test</td><td>test</td><td>test</td><td>mc</td><td>val</td><td>w/o sub.</td><td>w/ sub.</td></tr><tr><td></td><td></td><td>Llama-Video-7B</td><td colspan="6"># Frames: 110; Total # tokens: 20,240</td><td></td></tr><tr><td>Full Attention</td><td>100%</td><td>3.66</td><td>59.6</td><td>57.0</td><td>81.2</td><td>66.1</td><td>64.7</td><td>71.0</td><td>57.6</td></tr><tr><td>SF-fixed</td><td>4.8%</td><td>3.26</td><td>57.3</td><td>53.3</td><td>79.8</td><td>62.9</td><td>59.9</td><td>67.1</td><td>54.8</td></tr><tr><td>SF-strided</td><td>41.4%</td><td>3.45</td><td>58.5</td><td>56.1</td><td>80.6</td><td>64.4</td><td>61.4</td><td>68.5</td><td>56.1</td></tr><tr><td>A-shape</td><td>48.2%</td><td>3.56</td><td>56.0</td><td>51.6</td><td>79.8</td><td>65.7</td><td>54.4</td><td>65.6</td><td>53.8</td></tr><tr><td>Tri-shape</td><td>49.0%</td><td>3.58</td><td>59.3</td><td>54.5</td><td>80.3</td><td>66.1</td><td>63.6</td><td>70.1</td><td>56.7</td></tr><tr><td>VisionZip</td><td>35.2%</td><td>1.35</td><td>42.1</td><td>40.5</td><td>69.5</td><td>41.4</td><td>44.9</td><td>62.1</td><td>43.1</td></tr><tr><td>MInference</td><td>78.8%</td><td>3.64</td><td>59.6</td><td>57.0</td><td>80.6</td><td>66.1</td><td>64.6</td><td>71.0</td><td>57.5</td></tr><tr><td>Ours</td><td>47.3%</td><td>3.58</td><td>59.8</td><td>57.1</td><td>80.1</td><td>66.2</td><td>64.5</td><td>71.8</td><td>57.6</td></tr><tr><td></td><td></td><td>LongVILA-7B</td><td colspan="6"># Frames: 256; Total # tokens: 65,800</td><td></td></tr><tr><td>Full Attention</td><td>100%</td><td>2.76</td><td>59.5</td><td>61.9</td><td>80.7</td><td>58.1</td><td>60.1</td><td>65.1</td><td>55.5</td></tr><tr><td>SF-fixed</td><td>2.2%</td><td>1.99</td><td>51.3</td><td>59.6</td><td>76.5</td><td>55.5</td><td>57.1</td><td>63.0</td><td>52.1</td></tr><tr><td>SF-strided</td><td>26.6%</td><td>2.58</td><td>56.0</td><td>61.4</td><td>76.7</td><td>55.5</td><td>53.6</td><td>59.2</td><td>52.2</td></tr><tr><td>A-shape</td><td>29.1%</td><td>2.75</td><td>56.6</td><td>60.9</td><td>75.0</td><td>55.3</td><td>49.1</td><td>59.6</td><td>51.3</td></tr><tr><td>Tri-shape</td><td>29.3%</td><td>2.63</td><td>58.1</td><td>62.0</td><td>77.8</td><td>56.2</td><td>59.3</td><td>63.3</td><td>54.2</td></tr><tr><td>VisionZip</td><td></td><td></td><td></td><td>OOM</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MInference</td><td>47.0%</td><td>2.77</td><td>59.7</td><td>62.2</td><td>79.1</td><td>57.8</td><td>60.0</td><td>65.2</td><td>55.2</td></tr><tr><td>Ours</td><td>31.8%</td><td>2.84</td><td>60.2</td><td>62.2</td><td>79.4</td><td>57.8</td><td>60.0</td><td>65.5</td><td>55.4</td></tr><tr><td></td><td></td><td>Qwen2.5-VL-7B-Instruct</td><td colspan="6"># Frames: 256; Total # tokens: 33,950</td><td></td></tr><tr><td>Full Attention</td><td>100%</td><td>3.71</td><td>58.3</td><td>64.3</td><td>85.4</td><td>68.7</td><td>64.7</td><td>71.3</td><td>59.5</td></tr><tr><td>Ours</td><td>41.3%</td><td>3.75</td><td>58.0</td><td>63.9</td><td>84.9</td><td>68.9</td><td>65.1</td><td>70.9</td><td>59.4</td></tr></table>
+
+Algorithm 3 2D-Boundary Head  
+```txt
+Input:  $Q, K, V \in \mathbb{R}^{S \times d_h}$ , modality type index  $i_m$ , modality type set  $m \in \phi_m$
+```
+
+```txt
+Permute  $Q$ ,  $K$ ,  $V$  tensors based on modality
+```
+
+```latex
+$\overline{\mathbf{Q}} \gets$  permute  $(\mathbf{Q}, \mathbf{i}_m)$ ,  $\overline{\mathbf{K}} \gets$  permute  $(\mathbf{K}, \mathbf{i}_m)$
+```
+
+```txt
+$\overline{V}\gets$  permute  $(V,i_{m})$
+```
+
+```txt
+Looping over the modalities in pairs
+```
+
+```latex
+$y\gets 0$
+```
+
+```txt
+for  $i\gets 1$  to  $\left|\phi_m\right|$  do
+```
+
+```txt
+for  $j\gets 1$  to  $|\phi_m|$  do
+```
+
+```txt
+Dynamic sparse attention for each modality pair w/ FlashAttention
+```
+
+```latex
+$m_{mi,mj}\gets$  buildmask  $\left(i_{mi},i_{mj}\right)$
+```
+
+```latex
+$A_{mi,mj}\gets \mathrm{softmax}($
+```
+
+```txt
+sparse( $\overline{\mathbf{Q}}_{mi}\overline{\mathbf{K}}_{mj}^{\top}, \boldsymbol{i}_{mi}, \boldsymbol{i}_{mj}) / \sqrt{d} + \boldsymbol{m}_{mi,mj}$ )
+```
+
+```latex
+$\pmb{y}_{mi,mj} \gets \text{sparse}(\pmb{A}_{mi,mj} \overline{\pmb{V}}_{mj})$
+```
+
+```txt
+Update the modality output to the final output
+```
+
+```latex
+$\pmb {y}\leftarrow \pmb{y}_{mi,mj}\cup \pmb{y}$
+```
+
+```txt
+end for
+```
+
+```txt
+end for
+```
+
+```txt
+return  $\pmb{y}$
+```
+
+sparse attention. The 2D-Boundary requires constructing an attention mask and searching for sparse patterns in cross-modality regions. For example, in Fig. 3f, we build modality boundary indices for Vision-to-Text (bottom-left) and Text-to-Vision (upper-right) attention. This mask index construction is implemented in Triton (Tillet et al., 2019).
+
+# 3.3. Modality-Aware Sparse Attention Search Algorithm
+
+Due to modality boundaries in VLMs, we propose a modality-aware sparse attention pattern search algorithm (see Algorithm 4). The process unfolds in three steps: 1) intra-modality search within each modality following (Jiang et al., 2024), 2) cross-modality search across all modality pairs, and 3) inter-modality search informed by the results of the first two steps.
+
+# 4. Experiments
+
+In this section, we address two key questions: (i) How effective MMInference is? We evaluate our method on three general long-video tasks: long-video understanding, Video Needle in a Haystack, and Video-Text Needle in a Haystack. These benchmarks cover long-video captioning, open-ended QA, multiple-choice QA, mixed-modality tasks, and retrieval tasks, providing a comprehensive assessment of MMInference's effectiveness across diverse long-video scenarios. (ii) How efficient MMInference is? We analyze end-to-end latency and its breakdown to thoroughly evaluate the efficiency of MMInference.
+
+# 4.1. Dataset and Baselines
+
+Implementation Details Our experiments are conducted on two state-of-the-art long-video VLMs: Llava-
+
+![](images/3c29002ea385dbd477c7ad4b1d6e043b37e9018d338268a22742ecec3a62d6ef.jpg)  
+(a) MMInference in V-NIAH
+
+![](images/7b4c257f17f166d4294450b98e2adf629a02ef7b4812e5a8e33019f98e6d3ddc.jpg)  
+(b) FullAttention in V-NIAH
+
+![](images/9ba26a9c6c8690ef9fa465db9f561566586654ecad2ad659e56d46573763081d.jpg)  
+(c) MMInference in MM-NIAH
+
+![](images/13907a2de5eaf38c085ac9f36d3692ce54c8e4caf64118002be1f9c246c3da2f.jpg)  
+(d) FullAttention in MM-NIAH  
+Figure 5: V-NIAH (Zhang et al., 2024a) and MM-NIAH results using LongVila-Qwen2-7B-1M (Chen et al., 2025).
+
+Video (Zhang et al., 2024b) and LongVILA (Chen et al., 2025). We follow the MInference experimental setup, configuring the corresponding search space while adopting optimal configurations from prior work for other methods. We adjust the local window sizes of A-shape and tri-shape patterns to align FLOPs with our method. For MInference, we adopt its optimal configuration, which results with FLOPs approximately twice as high as our method's in VLMs. Our implementation leverages Triton (Tillet et al., 2019), FlashAttention (Dao, 2024), and dynamic sparse compiler PIT (Zheng et al., 2023). For the Vertical-Slash and Grid Head patterns, we set  $last_{q} = 64$ . Latency experiments are performed on a single NVIDIA A100 using bfloat16, with greedy decoding to ensure stable results. Additional implementation details are provided in Appendix C.
+
+Dataset Our evaluation uses the official metrics and scripts provided by these tasks. Additionally, we introduce a Mixed-Modality Needle in a Haystack (MM-NIAH) task to assess VLMs' retrieval capabilities on mixed-modality inputs. Dataset details are provided in Appendix D.
+
+(i) Video Understanding Tasks: These include ActNetQA (Yu et al., 2019), EgoSchema (Mangalam et al., 2023), Next-QA (Xiao et al., 2021), PerceptionTest (Patraucean et al., 2024), VideoDC (Lab, 2024), and VideoMME (Fu et al., 2024). These benchmarks span five categories, covering tasks such as captioning and video question answering. Input lengths range from 110 frames (e.g., 20k) to 256
+
+frames (e.g., 66k) in Llava-Video (Zhang et al., 2024b) and LongVILA (Chen et al., 2025).
+
+(ii) Video Needle in a Haystack (V-NIAH) (Zhang et al., 2024a): A long-video retrieval task testing VLMs' performance with tokens of up to 6k frames (e.g., 1.1M tokens), where inserted images are placed at various positions.  
+(iii) Mixed-Modality Needle in a Haystack (MM-NIAH): To evaluate VLMs in mixed-modality scenarios, we construct a mix-modality version of NIAH. Specifically,  $25\%$  of the input consists of text segments inserted at the document level across different frames in long-video inputs, forming a mix-modality haystack. All other settings align with V-NIAH, including the multi-choice VQA task with randomly inserted images. This benchmark tests input lengths of up to 4.5k frames (e.g., 1.1M tokens).
+
+Baselines We include five training-free sparse attention approaches, one visual token compression method, and also incorporate FlashAttention-2 (Dao, 2024) as a baseline. 1) SparseTransformer (Fixed) (Child et al., 2019): Retains attention within each segment and allows all tokens to attend to the segment's initial tokens. 2) SparseTransformer (Strided) (Child et al., 2019): Employs local windows with dilated attention. 3) A-Shape (Xiao et al., 2024): Preserves only the sink token with local attention. 4) Tri-Shape (LI et al., 2025; Acharya et al., 2024): Extends A-Shape by enabling full attention for all tokens to the last window's
+
+![](images/a2c2c480c6b4c1b2f62df1d5d5b2218c8efc80629bbc70f36da80e05dfb2a0a6.jpg)  
+(a) All Textual Context
+
+![](images/38b53c34986ae39e25938a258555358ff92f67dcd1956e804e2f2c38d9ec8067.jpg)  
+(b) Visual Context Inserted  
+Figure 6: Transition of sparse patterns from textual context to visual context. (a) The vertical-slash pattern for all textual context. (b) Grid pattern appears when visual modality is appended. (c) Grid pattern dominates.
+
+![](images/70751045a3af30065c1c1c7c47cdd90cf0ae9db40366d4bcdb1bbf7727895874.jpg)  
+(c) More Visual Context
+
+![](images/97b16c9ecca9b52172b7128b24fe0f11de070ed00fb5e3982741817ac8e6750c.jpg)  
+(d) All Visual Context
+
+queries. 5) Vertical-Slash Pattern (Jiang et al., 2024): Focuses on specific tokens (vertical lines) and tokens at fixed intervals (slash lines). 6) VisionZip (Yang et al., 2024): A visual token compression method that reduces the number of visual tokens per frame by evaluating tokens based on their attention scores and discarding less important ones. Full details on implementation, hyperparameters, and illustrations for our baselines can be found in Appendix C.
+
+# 4.2. Long Video Understanding
+
+Table 1 presents the performance of different methods on video understanding tasks. The results show that: 1) Our method and MInference closely approximate full attention across all tasks while requiring only half the FLOPs of MInference. 2) Static sparse patterns, such as A-shape and Tri-shape, perform reasonably well on most tasks but experience a notable performance drop in multi-choice VQA tasks like EgoSchema. Additionally, the slight increase in query full attention in Tri-shape effectively improves performance. 3) Among SF patterns, the slash pattern better preserves performance. Even when using SF-fixed with only  $2\% -5\%$  of FLOPs, it still maintains strong performance on most tasks.
+
+# 4.3. Video Needle In A Haystack
+
+Fig. 5a, 5b, and 13 show the performance of different models on V-NIAH, revealing notable differences in handling long-context video retrieval as the number of processed frames increases: 1) Our method achieves results nearly identical to full attention. 2) A-shape struggles with mid-context information even at 300 frames, while Tri-shape maintains full performance until 3.9k frames (i.g. 700K tokens) before a sharp decline. 3) SF-fixed degrades at 2.1k frames (i.g. 350K tokens), while SF-strided surpasses Tri-shape, holding performance until 4.5k frames (i.g. 825K tokens). 4) MInference preserves VLM retrieval well, with only slight degradation beyond 4.8K frames.
+
+# 4.4. Mixed-Modality Needle In A Haystack
+
+Beyond V-NIAH, we introduce a mixed-modality NIAH test to evaluate the performance of different sparse methods on video-text inputs, in Fig. 5c, 5d, and 14. Mixed-modality inputs lead to more pronounced performance degradation across all methods. However, by incorporating inter-modality sparse patterns, our method maintains performance close to full attention, especially when compared to MInference and ours w/o inter-modality. Notably, Tri-shape and MInference show significant drops at 1.8k frames (i.g. 440K tokens) and 2.7k frames (i.g. 660K tokens).
+
+# 4.5. Latency
+
+![](images/0faf643fcf0bd90279b3b1dc8231ddf7124f1a1fa9d341da892552866dee06e4.jpg)  
+Figure 7: End-to-End Latency.
+
+Fig. 7 and 16 present end-to-end and kernel-level latency across different context sizes. The grid pattern significantly outperforms the vertical-slash pattern in sparsity, achieving a  $2 - 3 \times$  speedup even at 1M tokens. Additionally, the grid pattern achieves an end-to-end speedup of  $8.3 \times$  and a kernel-level speedup of  $12 \times$ .
+
+# 4.6. Analysis
+
+Transition of Sparse Patterns Across Modalities Since LLMs and VLMs exhibit different sparse patterns, we examine the interplay between the Grid and Vertical-Slash pattern. As shown in Fig. 6, Llava-Video-7B primarily uses Vertical-Slash pattern for purely textual inputs. However, once a visual input is appended, it transitions to a Grid pattern to
+
+Table 2: Performance (%) on video understanding tasks based on VideoChat-Flash (Li et al., 2025) at frames 512 with 8k tokens.  
+
+<table><tr><td rowspan="2">Model</td><td>VideoDC</td><td>ActNet-QA</td><td>EgoSchema</td><td>Next-QA</td><td>PerceptionTest</td><td colspan="2">VideoMME</td><td rowspan="2">Avg.</td></tr><tr><td>test</td><td>test</td><td>test</td><td>mc</td><td>val</td><td>w/o sub.</td><td>w/ sub.</td></tr><tr><td>VideoChat-Flash</td><td>3.21</td><td>53.6</td><td>57.0</td><td>81.2</td><td>69.1</td><td>63.2</td><td>70.5</td><td>56.8</td></tr><tr><td>w/ MMInference</td><td>3.19</td><td>54.3</td><td>57.3</td><td>79.8</td><td>69.1</td><td>63.0</td><td>70.2</td><td>56.7</td></tr></table>
+
+capture the geometric structure of the visual content. This shift occurs at the modality boundary, creating a more structured arrangement of vertical and horizontal intervals. Such behavior highlights the need for distinct sparsity strategies in visual and mixed-modality contexts, rather than simply reusing sparse patterns from LLMs for VLMs.
+
+![](images/7167d43a225feb1eca1210bf052c12fcc565314af510bdffbf8ce8e1e31b9f70.jpg)  
+Figure 8: The sparse index does not effectively extrapolate from text to the visual modality. However, an index built within the same modality can generalize across modality boundaries.
+
+Sparse Index Across Modalities In Fig. 8, the sparse index achieves high recall for textual regions but fails to generalize to visual ones. To address this, we construct a sparse index from the visual modality and evaluate it on separate visual segments, each separated by modality boundaries. Remarkably, this approach extrapolates effectively across all visual segments, even when interspersed with textual boundaries. As shown in Fig. 8, the sparse index achieves high recall in the textual but fails to generalize to the visual. To address this, we construct a sparse index using the visual modality and evaluate it across distinct regions of the visual modality, separated by modality boundaries. Remarkably, this approach successfully extrapolates to all visual regions even when interrupted by text-induced boundaries.
+
+Integrate with token compression methods As shown in Table 2, our method integrates seamlessly with token compression techniques, enabling near-lossless performance while supporting longer or higher-resolution video inputs. Specifically, VideoChat-Flash reduces tokens per frame from 196 to 16 at the ViT stage, while our method further applies sparse attention in the LLM decoder. Results demonstrate strong performance retention across benchmarks.
+
+# 5. Related Work
+
+Long-Context Vision Language Models Recent VLMs have extended their context length to support long multi-modal inputs (Zhang et al., 2024a; Chen et al., 2025; Wang et al., 2024b; Team et al., 2024), enabling applications such as long-video understanding (Fu et al., 2024; Xiao et al., 2021; Wang et al., 2024a; Bai et al., 2025), multi-modal retrieval (Zhang et al., 2024a), and multi-modal chain-of-thought reasoning (Qwen, 2024). For instance, Zhang et al. (2024a) transfer long-context capabilities from base LLMs to vision tasks, Chen et al. (2025) introduce multi-modal sequence parallelism to accelerate video fine-tuning, and Zhang et al. (2024b) emphasize the role of data calibration and synthetic data in boosting VLM performance.
+
+Efficiency Optimization for VLMs While long-context VLMs achieve high accuracy, their high inference cost limits practical use in long-video scenarios. A common strategy is vision token compression—reducing video feature resolution by dropping or merging less important visual tokens (Bolya et al., 2023; Chen et al., 2024; Shen et al., 2024; He et al., 2024; Tu et al., 2025; Weng et al., 2024; Wen et al., 2024). RNN-Transformer hybrids are also used (Wang et al., 2024b) to balance efficiency and context length. However, these methods often assume inputs are long videos paired with short text, focusing solely on visual token optimization, while overlooking mixed-modality inputs critical for multi-turn interactions (Huang et al., 2024). Recently, Xu et al. (2025b) applied dynamic sparse attention to long-context VLMs, but their approach ignores modality-specific inductive biases and is limited to single-modality video tasks.
+
+# 6. Conclusion
+
+We propose MMInference, a modality-aware permutation sparse attention method that accelerates long-context VLMs. It features permutation-based grid sparse attention, Q-boundary/2D-boundary patterns for mixed-modality boundaries, and a Modality-Aware Sparse Attention Search Algorithm. Our optimized GPU kernels enable end-to-end acceleration. Experiments on video understanding tasks, V-NIAH and MM-NIAH using Llava-Video and LongVila demonstrate that MMInference preserves full-attention performance while achieving up to  $8.3 \times$  speedup at 1M tokens.
+
+# 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.
+
+# References
+
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+
+# A. Modality-Aware Sparse Attention Search Algorithm
+
+In Algorithm 4, we detail the procedure for selecting the optimal sparse attention pattern for each attention head under a constrained FLOPs budget. The algorithm jointly determines the best pattern and its configuration (e.g., stride size in grid attention, number of vertical/slash lines in VS pattern) to maximize accuracy. We first construct a kernel-aware search space, where all candidate patterns have comparable real-world FLOPs based on GPU kernel measurements—rather than theoretical estimates—to ensure practical efficiency.
+
+We then evaluate each candidate using a reference example and select the configuration that maximizes attention recall, using the actual attention output as the objective. This recall-based scoring incorporates the V matrix and builds on FlashAttention (Dao, 2024), enabling end-to-end pattern selection with minimal memory overhead and improved performance.
+
+Algorithm 4 Modality-aware Sparse Attention Pattern Search  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , inter-modality search space  $\rho_{\mathrm{inter}}$  , intramodality search space  $\rho_{\mathrm{intra}}$  , modality type set  $m\in \phi_m$  , optimized sparse pattern P   
+# Intra-modality sparse attention pattern search   
+for  $i\gets 1$  to  $|\phi_m|$  do  $p_{mi}\gets$  KernelAwareSearch  $(Q,K,V,m_i)$ $\mathrm{P}\leftarrow \mathrm{P}\cup p_{mi}$    
+end for   
+# Cross-modality sparse attention pattern search   
+for  $i\gets 1$  to  $|\phi_m|$  do for  $j\gets 1$  to  $|\phi_m|$  do  $p_{mi,mj}\gets$  KernelAwareSearch  $(Q,K,V,m_i,mj)$ $\mathrm{P}\leftarrow \mathrm{P}\cup p_{mi,mj}$    
+end for   
+end for   
+# Inter-modality sparse attention pattern search   
+for  $i\gets 1$  to  $|\rho_{\mathrm{inter}}|$  do  $p_i\gets$  argmin (|sparse  $(Q,K,V,i)-$  attention  $(Q,K,V)$  |  $\mathrm{P}\leftarrow \mathrm{P}\cup p_i$    
+end for   
+return P
+
+![](images/f3f3f78a1983c7ea7d5ba9811805c9d9b4ad219e35ba31e5e2cb63e34c72d979.jpg)  
+(a) K-Boundary pattern.  
+Figure 9: Additional inter-modality sparse pattern.
+
+![](images/12b367160c21bf02b55d927ae3c320b83d3dc8cfd8cb69a0a00b941017f2e5a6.jpg)  
+(b) No-Boundary pattern.
+
+![](images/9b1b7b3c69052f5dca9b4c7ff797c9bb01876c86dc310ee376117bbc6c822721.jpg)  
+(a) A-shape
+
+![](images/edc8469a943b70b3197ce3afe2bd7222c485849a4879747b29347c5bd46b591f.jpg)  
+(b) SF-fixed
+
+![](images/230fc826fb65daa1863cd7f2435d12ee3532488ccfd11220268d73ace604d55d.jpg)  
+(c) SF-strided
+
+![](images/11255ec73d6dfb4bfe307d1806fbf9688080686a4fb912ea42b98a258a2b778c.jpg)  
+(d) Tri-shape  
+Figure 10: The baselines of sparse attention in our experiments.
+
+![](images/f6f4055c77fd1dc2d54aadeaa8b543609ecea831d3287483ee2038c2cfbf6d95.jpg)  
+(e) Vertical-Slash (MInference)
+
+# B. Pattern Analysis
+
+# B.1. Additional Mix-modality Pattern
+
+In §2, we explain how the grid pattern naturally arises from the geometric structure of vision inputs. Fig. 9 further illustrates two additional patterns in the mixed-modality search space: the K-Boundary and No-Boundary patterns. Notably, both patterns incur no additional cost compared to pure intra-modality attention, as their sparse indices can be computed across all rows without extra computation.
+
+# B.2. Additional Sparse Attention Pattern Visualization
+
+We further analyze the sparse patterns in Qwen2.5-VL (Wang et al., 2024a) with dynamic resolution inputs and in VideoChat-Flash (Li et al., 2025) under visual token compression, across both video benchmark and mixed-modality inputs, as shown in Fig.17 and Fig.18.
+
+# C. Experiment Details
+
+# C.1. Vision Language Models
+
+We use two state-of-the-art VLMs in our experiments: LongVILA (Chen et al., 2025) and Llava-Video (Zhang et al., 2024b). Llava-Video supports varying numbers of frames (32, 64, 110) for video understanding, and as reported, performance improves with more frames. Thus, we adopt the 110-frame variant for benchmarking. For LongVILA, we use the 256-frame version (LongVILA-256Frame) with a 128K context length for video understanding benchmarks, and the 1M-token version (LongVILA-1M), designed for retrieval tasks, for the V-NIAH evaluation.
+
+Table 3: Hyperparameters detail of baselines.  
+
+<table><tr><td>Method</td><td>Hyperparameters</td></tr><tr><td>A-shape</td><td>Sink = 128, Local = 4096</td></tr><tr><td>SF-fixed</td><td>Local = token_per_frame, vline_stride = token_per_frame</td></tr><tr><td>SF-strided</td><td>Local = token_per_frame, vline_stride = token_per_frame</td></tr><tr><td>Tri-shape</td><td>Sink = 128, Local = 4096, Bottom = 128</td></tr><tr><td>MInference</td><td>Vertical_size ∈ {1000, 2000, 4000}, SLash_size ∈ {1024, 2048, 4096, 6144}</td></tr><tr><td>VisionZip</td><td>dominant = 54, contextual = 10</td></tr></table>
+
+# C.2. Baselines
+
+We include five sparse attention baselines in our experiments: A-shape (Xiao et al., 2024), SF-fixed (Child et al., 2019), SF-strided (Child et al., 2019), Tri-shape (LI et al., 2025), MInference (Jiang et al., 2024), and VisionZip (Yang et al., 2024). Fig. 10 illustrates the attention patterns of these baselines.
+
+While VisionZip (Yang et al., 2024) is primarily a visual token compression method—compressing vision tokens using attention scores from the vision encoder before passing them to the LLM—it is included for comparison as it reduces FLOPs in the pre-filling stage and offers insight into token compression approaches.
+
+# C.3. A-shape and Vertical-Slash
+
+A-shape and Vertical-Slash are used for intra-modality attention, alongside our newly proposed Grid pattern.
+
+At inference time, we estimate the attention matrix online to dynamically determine the spatial layout of sparse indices, conditioned on the assigned pattern and actual input. Sparse attention is then computed using our optimized GPU kernels. Note that while the masks for Vertical-Slash and Grid patterns are dynamically generated, A-shape uses a static mask, incurring no additional overhead beyond sparse computation.
+
+A-shape head. A-shape is a static sparse pattern that includes the first seven initial tokens along with a local attention window.
+
+Vertical-Slash head. Due to the continuity of vertical and slash lines, we matmul the last query vector  $\mathbf{Q}_{[-\mathrm{last}.q]}$  and key vector  $\pmb{K}$  to produce the estimated attention matrix  $\widehat{\pmb{A}}$ , which, in turn, is used to determine the indices for the vertical  $i_v$  and slash  $i_s$  lines. After obtaining the sparse indices for the vertical and slash lines, we convert them into a sparse format  $i_{vs}$ . Using these sparse indices, we perform block-sparse calculations of the attention weights and attention output.
+
+# C.4. Permutation for the Grid Pattern and Across Modality
+
+We illustrate how the permutation is applied to the Grid pattern and the Q-boundary and 2D-boundary patterns in Fig. 11 and Fig. 11.
+
+![](images/1d6f04b0adcb893be374021d5af8efaa7263e2764bb7876503f9cec0435dd8bd.jpg)  
+(a) Before Permutation
+
+![](images/8f29435bf900f2b9ac57feb3f2a6c192f72d1161ce994b5bd694ad2fd1cab7df.jpg)  
+(b) Row-wise Permutation  
+Figure 11: Permutation for the Grid Pattern. (a) Before permutation. (b) Row-wise permutation. (c) Column-wise permutation.
+
+![](images/1ed2c9d6de5d010b86f0bae9cc23811130e17f6d269d72d6370d85ab5bfdef11.jpg)  
+(c) Column-wise Permutation
+
+![](images/b1ff34bbae5407002336be48adb21d0bcb3cd90478171cbbe7cc925a1b8f51b7.jpg)  
+(a) Mix-modality
+
+![](images/9cf91e2d88b190dd7d48f484ec58da3ffbf96889eedbf7f77ba5c825dc366f50.jpg)  
+(b) Q-wise Permutation  
+Figure 12: Permutation for mix-modality context. (a) Mix-modality. (b) Q-wise permutation. (c) K-wise permutation.
+
+![](images/2c0819cc40d50f826c75a012b145224aa3523760f1086f069c2847ba8d712fa7.jpg)  
+(c) K-wise Permutation
+
+# C.5. Search Space
+
+Following (Jiang et al., 2024), we set the target FLOPs  $t$  to be the same as 1k global tokens and 4k local window tokens in the  $A$ -shape pattern. Additionally, we use only one sample as our calibration set from the egoschema task with no more than 25K tokens, which exhibits strong generalization and stability across different lengths and domains. The search time is approximately 15 minutes on a single A100. This pattern search is individually conducted for each model: Llava-Video-7B, LongVila-256Frame, and LongVila-1M. The search space is shown in Table 4.
+
+# D. Benchmark Details
+
+We evaluate our method on several video understanding benchmarks that test different aspects of video comprehension:
+
+EgoSchema EgoSchema (Mangalam et al., 2023) is a diagnostic benchmark for very long-form video language understanding, structured as a multiple-choice question answering task. The benchmark requires models to answer questions about egocentric videos by selecting from given options (labeled A through E). The evaluation can be performed either on the full set via submission to an evaluation server, or on a released subset of 500 questions for direct scoring.
+
+Video-MME Video-MME (Fu et al., 2024) is a comprehensive multi-modal evaluation benchmark that tests MLLMs across diverse video types and temporal dimensions. It spans 6 primary visual domains with 30 subfields and includes videos ranging from 11 seconds to 1 hour in duration. The benchmark comprises 900 videos totaling 254 hours, with 2,700 manually annotated question-answer pairs. It evaluates models' ability to process not just video frames but also integrated multi-modal inputs like subtitles and audio.
+
+NExT-QA NExT-QA (Xiao et al., 2021) focuses on advancing video understanding from basic description to explaining temporal actions. It features both multiple-choice and open-ended QA tasks that target three key aspects: causal action reasoning, temporal action reasoning, and common scene comprehension. The benchmark is specifically designed to evaluate models' ability to reason about actions beyond superficial scene descriptions.
+
+Perception Test The Perception Test (Patraucean et al., 2023) perceves perception and reasoning skills across video, audio, and text modalities. It contains 11.6k real-world videos with an average length of 23 seconds, featuring perceptually interesting situations. The benchmark tests four key skills (Memory, Abstraction, Physics, Semantics) and various types of reasoning (descriptive, explanatory, predictive, counterfactual). Videos are densely annotated with six types of labels: multiple-choice QA, grounded video QA, object tracks, point tracks, temporal action segments, and sound segments.
+
+ActivityNet-QA ActivityNet-QA (Yu et al., 2019) is a large-scale VideoQA dataset consisting of 58,000 QA pairs on 5,800 complex web videos derived from the ActivityNet dataset. The benchmark is fully annotated and designed to test models' understanding of complex web videos through question answering. Unlike automatically generated datasets, ActivityNet-QA features human-annotated questions and answers, making it particularly valuable for evaluating real-world video understanding capabilities.
+
+<table><tr><td>Attention Type</td><td>Parameters</td></tr><tr><td>Grid Attention</td><td>(frame_stride, True, False, False, 1024) 
+(frame_stride, False, True, False, 1024) 
+(frame_stride, False, False, True, 1024) 
+(frame_stride, True, True, False, 1024) 
+(frame_stride, False, True, True, 1024) 
+(frame_stride, True, True, True, 1024) 
+(stride, True, False, False, 1024) 
+(stride, False, True, False, 1024) 
+(stride, False, False, True, 1024) 
+(stride, True, True, False, 1024) 
+(stride, False, True, True, 1024) 
+(stride, True, True, True, 1024)</td></tr><tr><td>A-shape</td><td>(128, 1024) 
+(128, 2048) 
+(128, 4096)</td></tr><tr><td>Vertical-Slash</td><td>(1000, 1024) 
+(1000, 2048) 
+(2000, 2048) 
+(1000, 3096) 
+(2000, 3096) 
+(1000, 4096) 
+(2000, 4096) 
+(3500, 200) 
+(1000, 2500)</td></tr></table>
+
+Table 4: The search space for each attention pattern: 1) Grid Attention: (stride, use hline, use vline, use slash, max stride); 2) A-shape: (sink, local); 3) Vertical-Slash: (vertical size, slash size)
+
+Video Detail Description (VideoDC) VideoDC (Lab, 2024) focuses on comprehensive video understanding through detailed descriptions. The benchmark consists of question-answer pairs generated with GPT-3.5, where questions prompt for detailed descriptions focusing on main subjects, their actions, and background scenes. The evaluation assesses the quality and completeness of video descriptions generated by models.
+
+# E. Additional Experiments Results
+
+# E.1. Additional Video Needle In A Haystack Results
+
+we further present the results of the Video Needle In A Haystack task with our baselines. The results of our method and full attention is shown in Fig. 5.
+
+# E.2. Additional Mixed-Modality Needle In A Haystack Results
+
+We further present the results of the Mixed-Modality Needle In A Haystack task with our baselines and the inter-modality variant of our method. The results of full attention and MMInference are shown in Fig. 5.
+
+# E.3. Latency Breakdown
+
+As shown in Fig. 16, we present the micro-benchmark results of various sparse attention methods across different context lengths.
+
+![](images/68abe7127626802a457b94f0cc3e23a1adddcce5168d55e698da6397a99c35de.jpg)  
+(a) A-shape
+
+![](images/216654f6c96503cbbb71b17f492f7d0e464eca43fb230429fe0c869cb59398fe.jpg)  
+(b) Tri-shape
+
+![](images/03559178afd6bf53dd2611810cea9f1d48cd55d7222a48dcca28ef9d83b315c1.jpg)  
+(c) SF-fixed
+
+![](images/fe0d7f9adcbdf0b9588ff7c609bf4a8e2f90958fdc6cc6a1da1fc0c5fbb89413.jpg)  
+(d) SF-strided
+
+![](images/117640b602ae5de66ef372b5d67456738e5079dc80bab665a764a95497fd5b90.jpg)  
+(e) MInference  
+Figure 13: Video Needle In A Haystack (Zhang et al., 2024a) results using LongVila-Qwen2-7B-1M (Chen et al., 2025).
+
+# E.4. VS Pattern vs. Grid Pattern
+
+Both VS pattern and Grid pattern achieve strong performance on video understanding and V-NIAH tasks. However, due to the grid attention pattern observed in VLMs, the overlap between blocks covered by diagonal lines in the VS pattern is minimal, reducing sparsity within the kernel. This explains why VS pattern exhibits significantly higher latency compared to Grid pattern. Additionally, leveraging permutation-based optimization effectively reduces the number of blocks involved in kernel computation, thereby lowering latency while maintaining comparable performance.
+
+# F. Sparse Attention in DiT
+
+Recently, many efficient DiT methods (Hassani et al., 2023; Xi et al., 2025; Zhang et al., 2025; Xu et al., 2025b; Ding et al., 2025) have adopted sparse attention to accelerate long video generation. We note that these methods can also benefit from permutation-based transformations to achieve kernel-efficient implementations. For example, the 2D/3D sliding window attention in NATTEN can be converted into dense tensor core computation via permutation, as illustrated in Fig. 15. Similarly, the temporal head in Sparse VideoGen (Xi et al., 2025) and the anti-diagonal structure in xAttention (Xu et al.,
+
+![](images/2f742c31f02c1bcaa64782379ba7ec356cf8b040c1d97dc5218a6f96d6a736be.jpg)  
+(a) A-shape
+
+![](images/d20824d9f660a34b5b9e5b28bddcd9a0fd9d656843231a9427dc6bd41281ff36.jpg)  
+(b) Tri-shape
+
+![](images/310abefec8808959954ef00a39673c99aa23979b0cda7543f0b9c7708e084bd5.jpg)  
+(c) MInference
+
+![](images/a7b4fb92ce235393adf4c05e8fbb717b5cbbbfe80f753ebb16d853bdfc10847f.jpg)  
+(d) MMInference w/o Inter-modality  
+Figure 14: Mixed-Modality Needle In A Haystack results using LongVila-Qwen2-7B-1M (Chen et al., 2025).
+
+2025b) can be restructured through permutation to enable sparse loading with dense computation, significantly speeding up DiT inference, especially in long-context scenarios.
+
+# G. Kernel Implementation
+
+As shown in Algorithms 5, 6, and 7, we provide implementation details of the FlashAttention-based kernels. The Grid-shape kernel in Algorithm 5 integrates block-sparse FlashDecoding (Qwen, 2023), which sparsifies the query loading, with block-sparse FlashAttention-2, which sparsifies the key loading. The Q-Boundary kernel in Algorithm 6 introduces sparsity
+
+![](images/5b0c60119d2e1ef70c3a54f0daa649ac32f3901c79faf1f1a1bf54532abdecd1.jpg)  
+(a) Natten  
+Figure 15: Permutation-based implementation of 2D/3D sliding window attention (Hassani et al., 2023) enables efficient sparse attention optimization for DiT architectures.
+
+![](images/ff4b7a9e54275f511ea14c67a84f4587ffc23f9cf6c35cfb15857d69c662846c.jpg)  
+(b) Permutated Natten
+
+![](images/2fe6a201f1e5d4844b0aa437727990c3eec1e7e4c5a60f57ee29b09ec5301c2d.jpg)  
+Figure 16: The latency breakdown of a single attention kernel for four sparse attention patterns and FlashAttention (Dao, 2024) across different context windows in a single A100, including the index time for dynamic sparse approximation and building dynamic sparsity. At 1M tokens, the latency for Grid is  $358\mathrm{ms}$ .
+
+along the query dimension using FlashAttention-2 (Dao, 2024), while the 2D-Boundary kernel in Algorithm 7 applies sparsity along both the query and key dimensions.
+
+Algorithm 5 Grid-Shape Flash Attention  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , block size  $B$  , stride size  $\sigma$  query start index  $s_q$  , key start index  $s_k$  # Sparse load in  $\pmb{K}$  using FlashAttention for  $i\gets 1$  to  $N$  do Scale  $\tau \leftarrow \sqrt{\frac{1}{d_h}}$  Load  $Q_{\mathrm{chip}}\gets Q^{[i\times B:(i + 1)\times B]}\in \mathbb{R}^{B\times d_h}$  Initialize  $O_{\mathrm{chip}}\gets (0)^{B\times d_h}\in \mathbb{R}^{B\times d_h}$  Initialize  $m\gets (-\inf)^{B}\in \mathbb{R}^{B}$  Initialize  $l\gets (0)^{B}\in \mathbb{R}^{B}$  # Sparse load in  $\pmb{Q}$  using FlashDecoding for  $i\gets 1$  to  $N_{\sigma}$  do Load  $Q_{\mathrm{chip}}\gets Q^{[i\times B:(i + 1)\times B]\times \sigma +s_q}\in \mathbb{R}^{B\times d_h}$  Initialize  $O_{\mathrm{chip}}\gets (0)^{B\times d_h}\in \mathbb{R}^{B\times d_h}$  Initialize  $m\gets (-\inf)^{B}\in \mathbb{R}^{B}$  Initialize  $l\gets (0)^{B}\in \mathbb{R}^{B}$  # Loop in K for  $j\gets 1$  to  $M_{\sigma}$  do Load  $K_{\mathrm{chip}}\gets K^{[j\times B:(j + 1)\times B]\times \sigma +s_k}\in \mathbb{R}^{B\times d_h}$  Load  $K_{\mathrm{chip}}\gets K^{[j\times B:(j + 1)\times B]\times \sigma +s_k}\in \mathbb{R}^{B\times d_h}$  Load  $V_{\mathrm{chip}}\gets V^{[j\times B:(j + 1)\times B]\times \sigma +s_k}\in \mathbb{R}^{B\times d_h}$ $S\gets \tau Q_{\mathrm{chip}}K_{\mathrm{chip}}^T$ $S\gets \mathrm{mask}(S)$ $m_{new}^{i}\gets \max (m^{i},\mathrm{rowmax}(S))\in \mathbb{R}^{B}$ $S\gets S - m_{new}^{i}$ $P\gets \exp (S)$ $l_{new}^{i}\gets \mathrm{rowsum}(S)$ $\alpha \leftarrow \exp (m_i^i -m_i^i)$ $l^i\gets \alpha l^i +l_i^i$ $O_{\mathrm{chip}}\gets \alpha O_{\mathrm{chip}} + PV_{\mathrm{chip}}$  end for #Write outputs  $O_{\mathrm{chip}}\gets \mathrm{diag}(l^i)^{-1}O_{\mathrm{chip}}$  Save  $O_i\gets O_{\mathrm{chip}}$  end for #Write outputs  $O_{\mathrm{chip}}\gets \mathrm{diag}(l^i)^{-1}O_{\mathrm{chip}}$  Save  $O_i\gets O_{\mathrm{chip}}$  end for
+
+Algorithm 6 Q-Boundary Flash Attention  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , block size  $B$  modality index   
+ $I_{m}$  , sparse attention kernel  $\mathbf{Op}_m$    
+Scale  $\tau \leftarrow \sqrt{\frac{1}{d_h}}$    
+Initialize  $O\gets (0)^{S\times d_h}\in \mathbb{R}^{S\times d_h}$    
+# Loop modality and parallelized in GPU   
+for  $m\in \{\mathrm{text},\mathrm{vision},\dots,\}$  do   
+for  $i\gets 1$  to  $N_{m}$  do Load index  $I_{chip}\gets I_m^{[i\times B:(i + 1)\times B]}\in \mathbb{R}^B$  Load  $Q_{chip}\gets Q^{I_{chip}}\in \mathbb{R}^{B\times d_h}$  Initialize  $O_{chip}\gets (0)^{B\times d_h}\in \mathbb{R}^{B\times d_h}$  Initialize  $m\gets (-\inf)^B\in \mathbb{R}^B$  Initialize  $l\gets (0)^B\in \mathbb{R}^B$  # Loop in K using modality sparse attention   
+ $O_{chip},m,l\gets O_{p_m}(Q_{chip},K,V,O_{chip},m,l)$  # Write outputs w/ modality index   
+ $O_{chip}\gets \mathrm{diag}(l^i)^{-1}O_{chip}$  Save  $O_i^{I_{chip}}\gets O_{chip}$    
+end for   
+end for
+
+Algorithm 7 2D-Boundary Flash Attention  
+Input:  $Q,K,V\in \mathbb{R}^{S\times d_h}$  , block size  $B$  modality index   
+ $I_{m}$  , sparse attention kernel  $\mathbf{Op}_m$    
+Scale  $\tau \leftarrow \sqrt{\frac{1}{d_h}}$    
+Initialize  $O\gets (0)^{S\times d_h}\in \mathbb{R}^{S\times d_h}$    
+# Loop modality and parallelized in GPU   
+for  $m_q\in \{\mathrm{text},\mathrm{vision},\dots,\}$  do   
+for  $i\gets 1$  to  $N_{m,q}$  do   
+Load index  $I_{chip,q}\gets I_{m,q}^{[i\times B:(i + 1)\times B]}\in \mathbb{R}^B$    
+Load  $Q_{chip}\gets Q^{I_{chip,q}}\in \mathbb{R}^{B\times d_h}$    
+Initialize  $O_{chip}\gets (0)^{B\times d_h}\in \mathbb{R}^{B\times d_h}$    
+Initialize  $m\gets (-\inf)^B\in \mathbb{R}^B$    
+Initialize  $l\gets (0)^B\in \mathbb{R}^B$    
+# Loop in K and modality   
+for  $m_k\in \{\mathrm{text},\mathrm{vision},\dots,\}$  do   
+for  $j\gets 1$  to  $M_{m,k}$  do   
+Load index  $I_{chip,k}\gets I_{m,k}^{[j\times B:(j + 1)\times B]}\in \mathbb{R}^B$    
+Load  $K_{chip}\gets K^{I_{chip,k}}\in \mathbb{R}^{B\times d_h}$    
+Load  $V_{chip}\gets V^{I_{chip,k}}\in \mathbb{R}^{B\times d_h}$ $O_{chip},m,l\gets O_p(Q_{chip},K_{chip},V_{chip},O_{chip},m,l)$  end for   
+end for   
+# Write outputs w/ modality index   
+ $O_{chip}\gets \mathrm{diag}(l^i)^{-1}O_{chip}$    
+Save  $O_i^{I_{chip,q}}\gets O_{chip}$    
+end for   
+end for
+
+![](images/728ef99d8ef08cb1d9dceeba68ba21b45502da270e7d3f572aa75fafe05c33a6.jpg)
+
+![](images/3ac3dd6a891e657e7b9e8c614344e464f07de5869f76559e383dd50f7fa123ae.jpg)  
+(a) Qwen2.5-VL on EgoSchema
+
+![](images/3d6f43d7043e4a43165c1d5689ef736b2938866a6281839827e275d59f1561d5.jpg)
+
+![](images/4004ac12ab6bb7637c665a267b6f3bb73273cfb48aaeeddab67d874984fb10b7.jpg)
+
+![](images/ebd0f44dba6244c9aa6a86ca5bda723dd4f8a2498e2dddee31633f9a4ca1d516.jpg)
+
+![](images/627288db4f1c67bebe8b9f4985d9e11e383edc2126d829c71b723757b4f3aef8.jpg)  
+(b) VideoChat on EgoSchema
+
+![](images/620ae105c9c262d26de261af8f66f45c5fb1a2505047a1e76306873b94112018.jpg)
+
+![](images/5bf13dd9aaea2f6100387af7a2581999f422a365e0123c8afb97fa402d653dc0.jpg)
+
+![](images/fd99a0e833ed85b8df18648f4694dab0949778d6ce408d0faed92da223c58954.jpg)
+
+![](images/98b054258ed07e05f3c5514bd16057c805f38ffa23ec95cccfeead992f21882f.jpg)  
+(c) Qwen2.5-VL on VideoMME
+
+![](images/cf02b571ef514bd7b4f1a82413b4c74a90698e28111074417f0750f8cf716f58.jpg)
+
+![](images/653d04b292ffd96d81e4e1c8294b00a5f1061413b5ccd75a8730101bf73aa767.jpg)
+
+![](images/0055c567e40b09854cae0d7cf4a85819b435014f3c61f81a3eee08046204b982.jpg)  
+Figure 17: Visualization of sparse attention patterns in Qwen2.5-VL with dynamic resolution input and VideoChat-Flash with visual token compression across different benchmarks.
+
+![](images/5ffa2416289c2f2e55df9508dad5f85e58450af706a8d9a43856a7b7854e89f5.jpg)  
+(d) VideoChat on VideoMME
+
+![](images/340a52cb80efb272a1d4ca4b887a46a3ecd4badecd6b85a50d70cf4b850c68d4.jpg)
+
+![](images/293b9b856403aa8ccf75a3818608a640694f518e3a8044fbf7609c4dc68b0ab0.jpg)
+
+![](images/2459281dcceb5feef3e3698646543c0d051a787c51d5f17083e1eb10304fb440.jpg)
+
+![](images/30b81a47dcf9142aa656dda96e7292d455c3044f0e45eb494aa9ac26c570b1e5.jpg)  
+(a) Qwen2.5-VL on Mix-modality
+
+![](images/5fa5b4e18a81104ed14d3845d78d346535187425ab1f6938a1263dd69dcfff58.jpg)
+
+![](images/e7a11ee930325f44a4057f210365d6377de7a0b70dd6dcbfe1e018b1375ee02c.jpg)
+
+![](images/7e1a40fe5fbb4574bb48263551bf0ae7c8e9e59dc461a89129f5ce7ec931b5a1.jpg)  
+Figure 18: Visualization of sparse attention patterns in Qwen2.5-VL with dynamic resolution input and VideoChat-Flash with visual token compression with mix-modality inputs.
+
+![](images/e301121b74fb42ce484aa84e822f6e67f166be43efa91b04a8ab7efea95f8925.jpg)  
+(b) VideoChat on Mix-modality
+
+![](images/713b5e6ae51c5b3b18aba08af6b1e9bbb4900565d92162354a0dabccdc1cebb4.jpg)
+
+![](images/8650758b53eed7272e66ce0608edcb27b598e42ebdb8252e51c58da4defd123f.jpg)

2025/MMInference_ Accelerating Pre-filling for Long-Context Visual Language Models via Modality-Aware Permutation Sparse Attention/images.zip

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