ICML 2026 Papers

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices through ad-hoc choices. We introduce _Collapsed Effective Operators_, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a strict spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing and enables the inclusion of topological features in neural network architectures via positional encoding.

This paper presents a theoretical investigation into the generalization capabilities of cross-domain knowledge distillation. Utilizing a high-dimensional asymptotic analysis of a linear teacher–student model, we characterize the excess risk while accounting for both model and covariate shifts. Our results provide a formal guarantee for the efficacy of distillation: even when the source and target domains differ substantially, there still may exist a regime where the student model achieves superior generalization ability over the student-only baseline. Moreover, we identify a \textit{crossed double descent} phenomenon: the excess risk can vary non-monotonically with the teacher’s and student’s dimension-to-sample-size ratios. These results provide rigorous insight into when and why distillation helps across domains.

The rapid growth of modern training datasets has significantly increased computational cost, motivating dataset pruning~(DP) methods which retain only a subset of informative samples to reduce training cost. Existing pruning criteria typically rely on either intrinsic signals that assess samples independently or extrinsic signals that promote diversity via pairwise relations. While effective in their own specific regimes, each captures only one aspect of sample utility and lacks robustness across different pruning ratios or data distribution. In this work, we present a unified graph-based DP framework. By modeling the dataset as a weighted graph, where node weights encode intrinsic value and edge weights encode extrinsic value, DP can be cast as a Maximum Weight Clique Problem (MWCP). Although MWCP is NP-hard, its structure admits a principled greedy solution based on sample-wise marginal gains. Under a few mild conditions, we further prove that this unified objective enjoys a formal approximation guarantee, which applies to a broad family of importance metrics and provides practical design guidelines. Extensive experiments show that our method outperforms existing DP methods while substantially reducing training cost, reducing training time by over 40\% without sacrificing accuracy on ImageNet-1k with ResNet-50.

Knowledge distillation is crucial for compressing large Vision–Language Models (VLMs) into efficient architectures. While prior VLM research has primarily focused on reasoning tasks like visual question answering, multimodal embedding learning, a key component for large-scale retrieval, has received comparatively less attention. Existing distillation methods typically align static global representations, overlooking hierarchical feature structure and fine-grained cross-modal interactions. This leads to a structural gap where student models fail to inherit object-level semantics and spatial relationships from teachers. To address this limitation, we propose **HieRD**, a Hierarchical Representation Distillation framework that preserves hierarchical structure within and across modalities throughout the distillation process by leveraging clustered visual tokens and multi-granular alignment with phrase-level text. Experimental results on multimodal embedding and downstream tasks show that HieRD consistently outperforms strong baselines, reflecting the effectiveness of its fine-grained semantic and spatial modeling, while enabling compact and efficient embedding models.

Chain-of-Thought (CoT) prompting has shown promise in enhancing the reasoning capabilities of large language models (LLMs) on text-attributed graphs (TAGs). This work reframes CoT-based graph learning through the principle of clustering as reasoning, offering a $k$-means interpretation of how iterative reasoning operates over graph-structured data. We observe that existing graph CoT methods rely on disjoint architectures and fixed graph representations, limiting step-by-step semantic-topological interaction and interpretability. To overcome this limitation, we propose a unified framework named KCoT that integrates CoT reasoning with graph representation learning. Our key theoretical result reveals a formal mathematical correspondence between a Transformer block and the $k$-means algorithm, allowing reasoning to be interpreted as iterative assignment and update steps. Based on this insight, we introduce a Semantic Discriminating Prompt that explicitly formulates these steps as structured CoT reasoning, together with a structure-grounded alignment strategy to fuse topological priors with evolving thought-conditioned representations. Experiments on standard benchmarks demonstrate consistent improvements over state-of-the-art methods, validating clustering as a principled mechanism for CoT-based graph learning.

Graph foundation models (GFMs), pretrained on massive graph data, have transformed graph machine learning by supporting general-purpose reasoning across diverse graph tasks and domains. Existing GFMs pretrained with fixed-hop subgraph sampling impose a fixed receptive field, causing scale mismatch on diverse tasks, which often require heterogeneous and unknown structural contexts beyond a fixed sampling scale. We propose **R-GFM**, a Riemannian Graph-of-Graphs (GoG) based foundation model, that treats *structural scale* as a first-class citizen in modeling. R-GFM constructs a multi-scale GoG over-sampled subgraphs at different hop distances and learns geometry-adaptive representations from Riemannian manifolds. Theoretical analysis shows that R-GFM reduces structural domain generalization error compared to fixed-scale GFMs. Experiments on various datasets demonstrate that R-GFM achieves state-of-the-art performance, with up to a **49\%** relative improvement on downstream tasks.

Graph Contrastive Learning (GCL), which trains graph encoders by maximizing similarity between positive samples and minimizing it between negative ones, has emerged as a mainstream graph pre-training paradigm. It is widely recognized that positive samples are essential in GCLs. Ideally, maximizing the similarity of positive samples enables graph encoders to capture intrinsic semantic and patterns of graph data. However, we discover an interesting phenomenon: GCLs can achieve competitive performance even without positive samples. This motivates us to revisit the fundamental mechanism of positive samples in GCLs. From the perspective of Dirichlet energy, we theoretically finds that message passing, a key mechanism in graph encoders, trivializes the maximization of positive samples, preventing GCLs from effectively learning from positive samples. To address this, we propose SPGCL to mitigate the trivialization caused by message passing and restore the learning efficacy of positive samples. Specifically, we find that high Dirichlet energy features help positive samples provide effective learning signals while low Dirichlet energy features contribute little to positive learning signal but is useful for positive sampling. Based on this, SPGCL propagates only high Dirichlet energy features and uses low energy features to construct a probability matrix for reliable positive sampling. Extensive experiments demonstrate the effectiveness of SPGCL.

Federated graph learning (FGL) enables collaborative training on graph data across multiple clients. With the rise of large language models (LLMs), textual attributes in FGL graphs are gaining attention. Text-attributed graph federated learning (TAG-FGL) improves FGL by explicitly leveraging LLMs to process and integrate these textual features. However, current TAG-FGL methods face three main challenges: \textbf{(1) Overhead.} LLMs for processing long texts incur high token and computation costs. To make TAG-FGL practical, we introduce graph condensation (GC) to reduce computation load, but this choice also brings new issues. \textbf{(2) Suboptimal.} To reduce LLM overhead, we introduce GC into TAG-FGL by compressing multi-hop texts/neighborhoods into a condensed core with fixed LLM surrogates (summaries/embeddings). However, this one-shot condensation is often not client-adaptive, leading to suboptimal performance. \textbf{(3) Interpretability.} LLM-based condensation further introduces a black-box bottleneck: summaries lack faithful attribution and clear grounding to specific source spans, making local inspection and auditing difficult. To address the above issues, we propose \textbf{DANCE}, a new TAG-FGL paradigm with GC. To improve \textbf{suboptimal} performance, DANCE performs round-wise, model-in-the-loop condensation refresh using the latest global model. To enhance \textbf{interpretability}, DANCE preserves provenance by storing locally inspectable evidence packs that trace predictions to selected neighbors and source text spans. Across 8 TAG datasets, DANCE improves accuracy by \textbf{2.33\%} at an \textbf{8\%} condensation ratio, with \textbf{33.42\%} fewer tokens per condensed node than TAG-FGL baselines.

Deep neural networks learn feature representations via complex geometric transformations of the input data manifold. Despite the models' empirical success across domains, our understanding of neural feature representations is still incomplete. In this work we investigate neural feature geometry through the lens of discrete geometry. Since the input data manifold is typically unobserved, we approximate it using geometric graphs that encode local similarity structure. We provide theoretical results on the evolution of these graphs during training, showing that nonlinear activations play a crucial role in shaping feature geometry in feedforward neural networks. Moreover, we discover that the geometric transformations resemble a discrete Ricci flow on these graphs, suggesting that neural feature geometry evolves analogous to Ricci flow. This connection is supported by experiments on over 20,000 feedforward neural networks trained on binary classification tasks across both synthetic and real-world datasets. We observe that the emergence of class separability corresponds to the emergence of community structure in the associated graph representations, which is known to relate to discrete Ricci flow dynamics. Building on these insights, we introduce a novel framework for locally evaluating geometric transformations through comparison with discrete Ricci flow dynamics. Our experimental results further suggest connections between the evolution of feature geometry, and training time and network depth.

We present a fully dynamic data structure that supports edge and node updates and cluster membership queries for the Normalised Cut problem with strong theoretical guarantees. Furthermore, our data structure outperforms the state of the art significantly on real world datasets. At the heart of our data structure is the novel notion of *Just-in-Time Sampling Trees*. The worst-case edge update time of our data structure is $O(\log n)$ where $n$ is the number of nodes in the current graph. Let $d_{\max}$ be the maximum degree of the current graph, let $T_{NC}(n',k')$ be the running time of an $\alpha$-approximation algorithm for the Normalised Cut problem on $n'$ vertices and $k'$ clusters, and let $\text{vol}(Y)$ be the sum of the unweighted degrees of all nodes in a set $Y$. The worst-case query time of our data structure to label all nodes in $Y$ is $O\left(kd_{max}^2 \log(n) + \gamma(n,\epsilon,k,Y)\right)$, with approximation ratio $\alpha \frac{1+\epsilon}{1-\epsilon}$, where $\gamma(n,\epsilon,k,Y)$ is defined to be $\log(n)\log^\star(n)\epsilon^{-4}k^2 + T_{NC}(\epsilon^{-4}k^2,k) + \epsilon^{-8}k^4 +vol(Y)$. Assuming $d_{\max}$ is polylogarithmic, as is the case with many sparse real-world graphs, our method achieves the best known trade-off between query time and update time.

As generative AI achieves hyper-realism, superficial artifact detection has become obsolete. While prevailing methods rely on resource-intensive fine-tuning of black-box backbones, we propose that forgery detection capability is already encoded within pre-trained models rather than requiring end-to-end retraining. To elicit this intrinsic capability, we propose the discriminative neural anchors (DNA) framework, which employs a coarse-to-fine excavation mechanism. First, by analyzing feature decoupling and attention distribution shifts, we pinpoint critical intermediate layers where the focus of the model logically transitions from global semantics to local anomalies. Subsequently, we introduce a triadic fusion scoring metric paired with a curvature-truncation strategy to strip away semantic redundancy, precisely isolating the forgery-discriminative units (FDUs) inherently imprinted with sensitivity to forgery traces. Moreover, we introduce HIFI-Gen, a high-fidelity synthetic benchmark built upon the very latest models, to address the lag in existing datasets. Experiments demonstrate that by solely relying on these anchors, DNA achieves superior detection performance even under few-shot conditions. Furthermore, it exhibits remarkable robustness across diverse architectures and against unseen generative models, validating that waking up latent neurons is more effective than extensive fine-tuning.

Time-series forecasting is critical in various scenarios, such as energy, transportation, and public health. However, most existing forecasters rely primarily on one-way inference, \textit{i.e.}, mapping \textbf{history} to \textbf{target}, and overlook the structural information provided by a revised natural chain (``\textbf{history} (model input) -- \textbf{target} (ground-truth output) -- \textbf{post-target continuation}''). The post-target continuation records how trajectories evolve after the target, which can help stabilize forecasting, but it is not observable at inference time. In this work, we aim to obtain an approximate proxy of the post-target continuation for the current input, providing structural knowledge for bidirectional forecasting. This idea is instantiated as KUP-BI (Knowledge Utilization Paradigm with Bidirectional Inspiration), a new time-series modeling paradigm that distills continuation-style knowledge (as an approximate post-target continuation proxy) from a \emph{train-only} historical library and integrates it into standard forecasting backbones. The input stream and the continuation-proxy stream are fused via a lightweight feature-level gating module. This design does not introduce information beyond what is already contained in the training trajectories; instead, it provides a structured inductive bias that helps backbones exploit typical continuation patterns rather than relying solely on parametric extrapolation. Experimental results on six public datasets show that KUP-BI consistently improves the forecasting performance of state-of-the-art models, with small additional overhead.

Graph coarsening reduces the size of a graph while preserving certain properties. Most existing methods preserve either spectral or spatial characteristics. Recent research shows that topology-preserving coarsening methods maintain GNN performance on coarsened graphs but suffer from exponential time complexity. To address these problems, we propose Scalable Topology-Preserving Graph Coarsening (STPGC) by introducing the concepts of graph strong collapse and graph edge collapse extended from algebraic topology. STPGC comprises three new algorithms, GStrongCollapse, GEdgeCollapse, and NeighborhoodConing, based on these two concepts, which eliminate dominated nodes and edges while rigorously preserving topological features. We further prove that STPGC preserves the GNN receptive field and develop approximate algorithms to accelerate GNN training. Experiments on node classification with GNNs demonstrate the efficiency and effectiveness of STPGC.

Graph coarsening is a graph dimensionality reduction technique that aims to construct a smaller and more tractable graph while preserving the essential structural and semantic properties of the original graph. However, most existing methods rely on pair-wise similarity matching, where each node independently searches for its best partner based on global information. This selfishness matching paradigm incurs substantial computational and memory overhead. To address this problem, we shift to a non-selfishness principle that prioritizes the collective interference of neighborhood in coarsening, and propose an efficient method named NOPE, which achieves linear memory consumption and near-linear computational complexity in the number of nodes. Furthermore, we derive a faster variant NOPE*, which reduces O(\Delta\cdot d) interference evaluation to O(d) based on the local isotropy assumption, and consequently alleviates the computational bottleneck for high-degree nodes. Experimental results show that NOPE* achieves $1.8–10\times$ speedup over NOPE and surpass almost all baselines with 1-3 orders of magnitude acceleration. Meanwhile, learning on coarsened graphs yields comparable performance to original graphs, and can even show superior performance over LLM-based graph reasoning owing to compact graph information. The code can be available at https://github.com/dazonglian/NOPE-main.

Model merging combines multiple fine-tuned models into a single model by $\textit{adding}$ their weight updates, providing a lightweight alternative to retraining. Existing methods primarily target resolving conflicts between task updates, leaving the failure mode of over-counting shared knowledge unaddressed. We show that when tasks share aligned spectral directions (i.e., overlapping singular vectors), a simple linear combination repeatedly accumulates these directions, inflating the singular values and biasing the merged model toward shared subspaces. To mitigate this issue, we propose Singular Value Calibration (SVC), a training-free and data-free post-processing method that quantifies subspace overlap and rescales inflated singular values to restore a balanced spectrum. Across vision and language benchmarks, SVC consistently improves strong merging baselines and achieves state-of-the-art performance. Furthermore, by modifying only the singular values, SVC improves the performance of Task Arithmetic by 13.0\%. Code is available at https://github.com/lyymuwu/SVC.

Editing complex, long-form knowledge in Large Language Models remains a significant challenge due to the difficulty of maintaining generation coherence. Existing autoregressive methods like AnyEdit alleviate length constraints but rely on Fixed-window Chunking, which disregards logical structure and compromises consistency. To address this, we present AnyEdit++, a structure-aware framework incorporating Bayes-Chunk, an adaptive segmentation mechanism that dynamically identifies semantic boundaries based on Bayesian Surprise. We underpin this approach with a theoretical framework establishing two key principles: (1) Structural Independence: we prove that cross-segment interference is minimized when anchor keys are geometrically orthogonal (a condition naturally satisfied by our surprisal-based boundaries but violated by fixed windows), and (2) Causal Locality: we demonstrate that updates injected at these semantic peaks yield strictly superior control compared to arbitrary split points. Extensive experiments across mathematical reasoning, code generation, and narrative tasks demonstrate that AnyEdit++ achieves superior performance and robustness compared to state-of-the-art baselines, validating that structural awareness is critical for effective long-form knowledge editing.

Large language models (LLMs) are increasingly used in scientific discovery, system modeling, and decision-making, prompting interest in their ability to reason over complex structured data. Existing benchmarks primarily focus on static or local graph reasoning, overlooking the high-order structures in real-world systems whose global properties evolve across multiple scales. We introduce LLM4PH, a benchmark that evaluates multi-scale structural reasoning through the lens of persistent homology (PH), a topological framework for tracking structural evolution. LLM4PH decomposes the PH pipeline into interpretable reasoning tasks spanning synthetic and real-world graphs, revealing that most models struggle with reasoning over structural transitions and persistence. Beyond task-level evaluation, we perform cross-task ablations on prompt encoding and transfer, explore post-training effects, and construct a compositional PH pipeline to assess end-to-end performance. Our results provide the first in-depth view of how well LLMs bridge discrete graph structures with continuous topological abstraction, and offer insights into their potential for structure-aware scientific reasoning.

Size transfer scales Graph Convolutional Networks (GCNs) by applying models trained on sampled subgraphs to larger target graphs. However, existing theoretical guarantees are typically confined to dense graphs or restricted sparsity regimes, failing to cover the arbitrary sparsity of real-world networks. To bridge this gap, we introduce the Generalized Graphon Convolutional Network (GWCN) based on the generalized graphon theory. Unlike the classical graphon limit which vanishes in sparse settings, GWCN employs stretching to construct a non-trivial limit that preserves topological structure. We derive an explicit transfer error bound that decomposes into size-dependent and density-dependent components, providing a unified guarantee across arbitrary sparsity levels. Empirical results on real-world networks corroborate our findings, demonstrating that transfer error vanishes as graph size increases and edge density decreases.

Graph neural networks (GNNs) excel at aggregating neighbor information for classification, yet their performance is hindered by graph structural entanglement, where spurious correlations from semantically irrelevant neighbors contaminate node embeddings. This challenge is most acute for nodes near class boundaries in the embedding space, where amplified structural noise blurs decision boundaries and destabilizes predictions. Existing robust GNN methods largely treat all nodes uniformly, ignoring boundary vulnerabilities. In this paper, to improve classification performance, we tackle graph structural disentanglement by identifying boundary-region entanglement as the primary bottleneck and propose Boundary Embedding Shaping (BES), an adaptive contrastive learning GNN plug-in module that selectively suppresses spurious structural noise at decision boundaries with minimal model parameter perturbation. Extensive experiments demonstrate that BES consistently improves boundary discrimination and outperforms existing leading methods. Notably, BES boosts GCN performance by an average of 3.3% in node classification (up to 5.0% on WikiCS) and achieves superior accuracy in link prediction.