Daily Papers

Actuator degradation turns quadruped locomotion into a coordination problem requiring joints to compensate for lost actuation. Prior work suggests that morphology-aware graph policies improve learning and generalization under body perturbations. We ask whether these benefits can be strengthened by explicitly modeling higher-order mechanical structure. We represent the Unitree Go1 as a cell complex with limb- and body-level rank-2 cells and apply Hodge-based message passing. Under degradation training, the node-edge-face Hodge actor achieves the highest return on unseen actuator degradations, with higher survival and lower velocity-tracking error. These results support higher-order morphology as a useful inductive bias for whole-body compensation under actuator degradation.

Message passing neural networks have shown a lot of success on graph-structured data. However, there are many instances where message passing can lead to over-smoothing or fail when neighboring nodes belong to different classes. In this work, we introduce a simple yet general framework for improving learning in message passing neural networks. Our approach essentially upsamples edges in the original graph by adding "slow nodes" at each edge that can mediate communication between a source and a target node. Our method only modifies the input graph, making it plug-and-play and easy to use with existing models. To understand the benefits of slowing down message passing, we provide theoretical and empirical analyses. We report results on several supervised and self-supervised benchmarks, and show improvements across the board, notably in heterophilic conditions where adjacent nodes are more likely to have different labels. Finally, we show how our approach can be used to generate augmentations for self-supervised learning, where slow nodes are randomly introduced into different edges in the graph to generate multi-scale views with variable path lengths.

Message passing neural networks (MPNNs) have emerged as go-to models for learning on graph-structured data in the past decade. Despite their effectiveness, most of such models still incur severe issues such as over-smoothing and -correlation, due to their underlying objective of minimizing the Dirichlet energy and the derived neighborhood aggregation operations. In this paper, we propose the DDSM, a new MPNN model built on an optimization framework that includes the stress majorization and orthogonal regularization for overcoming the above issues. Further, we introduce the diffusion distances for nodes into the framework to guide the new message passing operations and develop efficient algorithms for distance approximations, both backed by rigorous theoretical analyses. Our comprehensive experiments showcase that DDSM consistently and considerably outperforms 15 strong baselines on both homophilic and heterophilic graphs.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

Sparse Mixture-of-Experts (MoE) layers route tokens through a handful of experts, and learning-free compression of these layers reduces inference cost without retraining. A subtle obstruction blocks every existing compressor in this family: three experts can each be pairwise compatible yet form an irreducible cycle when merged together, so any score that ranks experts on pairwise signals is structurally blind to which triples are jointly mergeable. We show the obstruction is a precise mathematical object, the harmonic kernel of the simplicial Laplacian on a 2-complex whose vertices are experts, whose edges carry KL merge barriers, and whose faces carry triplet barriers; Hodge-decomposing the edge-barrier signal isolates the kernel exactly. We turn the diagnostic into a selection objective: HodgeCover greedily covers the harmonic-critical edges and triplet-critical triangles, and a hybrid variant of HodgeCover pairs it with off-the-shelf weight pruning on survivors. On three open-weight Sparse MoE backbones under aggressive expert reduction, HodgeCover matches state-of-the-art learning-free baselines on the expert-reduction axis, leads on the aggressive-compression frontier of the hybrid axis, and uniquely balances retained mass across all four Hodge components. These results show that exposing the harmonic kernel of a learned MoE structure changes which compressor wins at the regime that matters most.

Message passing neural networks (MPNNs) have been shown to suffer from the phenomenon of over-squashing that causes poor performance for tasks relying on long-range interactions. This can be largely attributed to message passing only occurring locally, over a node's immediate neighbours. Rewiring approaches attempting to make graphs 'more connected', and supposedly better suited to long-range tasks, often lose the inductive bias provided by distance on the graph since they make distant nodes communicate instantly at every layer. In this paper we propose a framework, applicable to any MPNN architecture, that performs a layer-dependent rewiring to ensure gradual densification of the graph. We also propose a delay mechanism that permits skip connections between nodes depending on the layer and their mutual distance. We validate our approach on several long-range tasks and show that it outperforms graph Transformers and multi-hop MPNNs.

A prominent paradigm for graph neural networks is based on the message-passing framework. In this framework, information communication is realized only between neighboring nodes. The challenge of approaches that use this paradigm is to ensure efficient and accurate long-distance communication between nodes, as deep convolutional networks are prone to oversmoothing. In this paper, we present a novel method based on time derivative graph diffusion (TIDE) to overcome these structural limitations of the message-passing framework. Our approach allows for optimizing the spatial extent of diffusion across various tasks and network channels, thus enabling medium and long-distance communication efficiently. Furthermore, we show that our architecture design also enables local message-passing and thus inherits from the capabilities of local message-passing approaches. We show that on both widely used graph benchmarks and synthetic mesh and graph datasets, the proposed framework outperforms state-of-the-art methods by a significant margin

Message-passing graph neural networks (MPNNs) emerged as powerful tools for processing graph-structured input. However, they operate on a fixed input graph structure, ignoring potential noise and missing information. Furthermore, their local aggregation mechanism can lead to problems such as over-squashing and limited expressive power in capturing relevant graph structures. Existing solutions to these challenges have primarily relied on heuristic methods, often disregarding the underlying data distribution. Hence, devising principled approaches for learning to infer graph structures relevant to the given prediction task remains an open challenge. In this work, leveraging recent progress in exact and differentiable k-subset sampling, we devise probabilistically rewired MPNNs (PR-MPNNs), which learn to add relevant edges while omitting less beneficial ones. For the first time, our theoretical analysis explores how PR-MPNNs enhance expressive power, and we identify precise conditions under which they outperform purely randomized approaches. Empirically, we demonstrate that our approach effectively mitigates issues like over-squashing and under-reaching. In addition, on established real-world datasets, our method exhibits competitive or superior predictive performance compared to traditional MPNN models and recent graph transformer architectures.

Message passing mechanism contributes to the success of GNNs in various applications, but also brings the oversquashing problem. Recent works combat oversquashing by improving the graph spectrums with rewiring techniques, disrupting the structural bias in graphs, and having limited improvement on oversquashing in terms of oversquashing measure. Motivated by unitary RNN, we propose Graph Unitary Message Passing (GUMP) to alleviate oversquashing in GNNs by applying unitary adjacency matrix for message passing. To design GUMP, a transformation is first proposed to make general graphs have unitary adjacency matrix and keep its structural bias. Then, unitary adjacency matrix is obtained with a unitary projection algorithm, which is implemented by utilizing the intrinsic structure of unitary adjacency matrix and allows GUMP to be permutation-equivariant. Experimental results show the effectiveness of GUMP in improving the performance on various graph learning tasks.

In this work we develop a new method, named Sub-graph Permutation Equivariant Networks (SPEN), which provides a framework for building graph neural networks that operate on sub-graphs, while using a base update function that is permutation equivariant, that are equivariant to a novel choice of automorphism group. Message passing neural networks have been shown to be limited in their expressive power and recent approaches to over come this either lack scalability or require structural information to be encoded into the feature space. The general framework presented here overcomes the scalability issues associated with global permutation equivariance by operating more locally on sub-graphs. In addition, through operating on sub-graphs the expressive power of higher-dimensional global permutation equivariant networks is improved; this is due to fact that two non-distinguishable graphs often contain distinguishable sub-graphs. Furthermore, the proposed framework only requires a choice of k-hops for creating ego-network sub-graphs and a choice of representation space to be used for each layer, which makes the method easily applicable across a range of graph based domains. We experimentally validate the method on a range of graph benchmark classification tasks, demonstrating statistically indistinguishable results from the state-of-the-art on six out of seven benchmarks. Further, we demonstrate that the use of local update functions offers a significant improvement in GPU memory over global methods.

Graph neural networks are popular architectures for graph machine learning, based on iterative computation of node representations of an input graph through a series of invariant transformations. A large class of graph neural networks follow a standard message-passing paradigm: at every layer, each node state is updated based on an aggregate of messages from its neighborhood. In this work, we propose a novel framework for training graph neural networks, where every node is viewed as a player that can choose to either 'listen', 'broadcast', 'listen and broadcast', or to 'isolate'. The standard message propagation scheme can then be viewed as a special case of this framework where every node 'listens and broadcasts' to all neighbors. Our approach offers a more flexible and dynamic message-passing paradigm, where each node can determine its own strategy based on their state, effectively exploring the graph topology while learning. We provide a theoretical analysis of the new message-passing scheme which is further supported by an extensive empirical analysis on a synthetic dataset and on real-world datasets.

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

Message Passing Neural Networks (MPNNs) are instances of Graph Neural Networks that leverage the graph to send messages over the edges. This inductive bias leads to a phenomenon known as over-squashing, where a node feature is insensitive to information contained at distant nodes. Despite recent methods introduced to mitigate this issue, an understanding of the causes for over-squashing and of possible solutions are lacking. In this theoretical work, we prove that: (i) Neural network width can mitigate over-squashing, but at the cost of making the whole network more sensitive; (ii) Conversely, depth cannot help mitigate over-squashing: increasing the number of layers leads to over-squashing being dominated by vanishing gradients; (iii) The graph topology plays the greatest role, since over-squashing occurs between nodes at high commute (access) time. Our analysis provides a unified framework to study different recent methods introduced to cope with over-squashing and serves as a justification for a class of methods that fall under graph rewiring.

Block coordinate descent (BCD) methods are widely used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can significantly improve the progress made by each BCD iteration. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with sparse dependencies between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active-set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization.

Graph neural networks (GNNs) have become an indispensable tool for analyzing relational data. In the literature, classical GNNs may be classified into three variants: convolutional, attentional, and message-passing. While the standard message-passing variant is highly expressive, its typical pair-wise messages nevertheless only consider the features of the center node and each neighboring node individually. This design fails to incorporate the rich contextual information contained within the broader local neighborhood, potentially hindering its ability to learn complex relationships within the entire set of neighboring nodes. To address this limitation, this work first formalizes the concept of neighborhood-contextualization, rooted in a key property of the attentional variant. This then serves as the foundation for generalizing the message-passing variant to the proposed neighborhood-contextualized message-passing (NCMP) framework. To demonstrate its utility, a simple, practical, and efficient method to parametrize and operationalize NCMP is presented, leading to the development of the proposed Soft-Isomorphic Neighborhood-Contextualized Graph Convolution Network (SINC-GCN). A preliminary analysis on a synthetic binary node classification problem then underscores both the expressivity and efficiency of the proposed GNN architecture. Overall, the paper lays the foundation for the novel NCMP framework as a practical path toward further enhancing the graph representational power of classical GNNs.

Neural architectures that learn potential energy surfaces from molecular data have undergone fast improvement in recent years. A key driver of this success is the Message Passing Neural Network (MPNN) paradigm. Its favorable scaling with system size partly relies upon a spatial distance limit on messages. While this focus on locality is a useful inductive bias, it also impedes the learning of long-range interactions such as electrostatics and van der Waals forces. To address this drawback, we propose Ewald message passing: a nonlocal Fourier space scheme which limits interactions via a cutoff on frequency instead of distance, and is theoretically well-founded in the Ewald summation method. It can serve as an augmentation on top of existing MPNN architectures as it is computationally inexpensive and agnostic to architectural details. We test the approach with four baseline models and two datasets containing diverse periodic (OC20) and aperiodic structures (OE62). We observe robust improvements in energy mean absolute errors across all models and datasets, averaging 10% on OC20 and 16% on OE62. Our analysis shows an outsize impact of these improvements on structures with high long-range contributions to the ground truth energy.

Loopy Belief Propagation (LBP) is a widely used approximate inference algorithm in probabilistic graphical models, with applications in computer vision, error correction codes, protein folding, program analysis, etc. However, LBP faces significant computational challenges when applied to large-scale program analysis. While GPU (Graphics Processing Unit) parallel computing provides a promising solution, existing approaches lack support for flexible update strategies and have yet to integrate logical constraints with GPU acceleration, leading to suboptimal practical performance. This paper presents a GPU-accelerated LBP algorithm for program analysis. To support the diverse update strategies required by users, we propose a unified representation for specifying arbitrary user-defined update strategies, along with a dependency analysis algorithm. Furthermore, building on previous work that leverages the local structure of Horn clauses to simplify message passing, we group messages to minimize warp divergence and better utilize GPU resources. Experimental results on datarace analysis over eight real-world Java programs show that our approach achieves an average speedup of 2.14times over the state-of-the-art sequential approach and 5.56times over the state-of-the-art GPU-based approach, while maintaining high accuracy.

Fairness in graph neural networks has been actively studied recently. However, existing works often do not explicitly consider the role of message passing in introducing or amplifying the bias. In this paper, we first investigate the problem of bias amplification in message passing. We empirically and theoretically demonstrate that message passing could amplify the bias when the 1-hop neighbors from different demographic groups are unbalanced. Guided by such analyses, we propose BeMap, a fair message passing method, that leverages a balance-aware sampling strategy to balance the number of the 1-hop neighbors of each node among different demographic groups. Extensive experiments on node classification demonstrate the efficacy of BeMap in mitigating bias while maintaining classification accuracy. The code is available at https://github.com/xiaolin-cs/BeMap.

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.

Creating fast and accurate force fields is a long-standing challenge in computational chemistry and materials science. Recently, several equivariant message passing neural networks (MPNNs) have been shown to outperform models built using other approaches in terms of accuracy. However, most MPNNs suffer from high computational cost and poor scalability. We propose that these limitations arise because MPNNs only pass two-body messages leading to a direct relationship between the number of layers and the expressivity of the network. In this work, we introduce MACE, a new equivariant MPNN model that uses higher body order messages. In particular, we show that using four-body messages reduces the required number of message passing iterations to just two, resulting in a fast and highly parallelizable model, reaching or exceeding state-of-the-art accuracy on the rMD17, 3BPA, and AcAc benchmark tasks. We also demonstrate that using higher order messages leads to an improved steepness of the learning curves.

We present a novel framework for learning on CW-complex structured data points. Recent advances have discussed CW-complexes as ideal learning representations for problems in cheminformatics. However, there is a lack of available machine learning methods suitable for learning on CW-complexes. In this paper we develop notions of convolution and attention that are well defined for CW-complexes. These notions enable us to create the first Hodge informed neural network that can receive a CW-complex as input. We illustrate and interpret this framework in the context of supervised prediction.

Neural message passing algorithms for semi-supervised classification on graphs have recently achieved great success. However, for classifying a node these methods only consider nodes that are a few propagation steps away and the size of this utilized neighborhood is hard to extend. In this paper, we use the relationship between graph convolutional networks (GCN) and PageRank to derive an improved propagation scheme based on personalized PageRank. We utilize this propagation procedure to construct a simple model, personalized propagation of neural predictions (PPNP), and its fast approximation, APPNP. Our model's training time is on par or faster and its number of parameters on par or lower than previous models. It leverages a large, adjustable neighborhood for classification and can be easily combined with any neural network. We show that this model outperforms several recently proposed methods for semi-supervised classification in the most thorough study done so far for GCN-like models. Our implementation is available online.

Efficient computation of graph diffusion equations (GDEs), such as Personalized PageRank, Katz centrality, and the Heat kernel, is crucial for clustering, training neural networks, and many other graph-related problems. Standard iterative methods require accessing the whole graph per iteration, making them time-consuming for large-scale graphs. While existing local solvers approximate diffusion vectors through heuristic local updates, they often operate sequentially and are typically designed for specific diffusion types, limiting their applicability. Given that diffusion vectors are highly localizable, as measured by the participation ratio, this paper introduces a novel framework for approximately solving GDEs using a local diffusion process. This framework reveals the suboptimality of existing local solvers. Furthermore, our approach effectively localizes standard iterative solvers by designing simple and provably sublinear time algorithms. These new local solvers are highly parallelizable, making them well-suited for implementation on GPUs. We demonstrate the effectiveness of our framework in quickly obtaining approximate diffusion vectors, achieving up to a hundred-fold speed improvement, and its applicability to large-scale dynamic graphs. Our framework could also facilitate more efficient local message-passing mechanisms for GNNs.

This paper proposes a message-passing mechanism to address language modelling. A new layer type is introduced that aims to substitute self-attention for unidirectional sequence generation tasks. The system is shown to be competitive with existing methods: Given N tokens, the computational complexity is O(N logN) and the memory complexity is O(N) under reasonable assumptions. In the end, the Dispatcher layer is seen to achieve comparable perplexity to prior results while being more efficient.

In this paper we propose a neural message passing approach to augment an input 3D indoor scene with new objects matching their surroundings. Given an input, potentially incomplete, 3D scene and a query location, our method predicts a probability distribution over object types that fit well in that location. Our distribution is predicted though passing learned messages in a dense graph whose nodes represent objects in the input scene and edges represent spatial and structural relationships. By weighting messages through an attention mechanism, our method learns to focus on the most relevant surrounding scene context to predict new scene objects. We found that our method significantly outperforms state-of-the-art approaches in terms of correctly predicting objects missing in a scene based on our experiments in the SUNCG dataset. We also demonstrate other applications of our method, including context-based 3D object recognition and iterative scene generation.

Supervised learning on molecules has incredible potential to be useful in chemistry, drug discovery, and materials science. Luckily, several promising and closely related neural network models invariant to molecular symmetries have already been described in the literature. These models learn a message passing algorithm and aggregation procedure to compute a function of their entire input graph. At this point, the next step is to find a particularly effective variant of this general approach and apply it to chemical prediction benchmarks until we either solve them or reach the limits of the approach. In this paper, we reformulate existing models into a single common framework we call Message Passing Neural Networks (MPNNs) and explore additional novel variations within this framework. Using MPNNs we demonstrate state of the art results on an important molecular property prediction benchmark; these results are strong enough that we believe future work should focus on datasets with larger molecules or more accurate ground truth labels.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Topological deep learning has emerged for modeling higher-order relational structures beyond pairwise interactions that standard graph neural networks fail to capture. Although combinatorial complexes offer a unified topological framework, most existing topological deep learning methods rely on local message passing via attention mechanisms, which incur quadratic complexity and remain low-dimensional, limiting scalability and rank-aware information aggregation in higher-order complexes.We propose Combinatorial Complex Mamba (CCMamba), the first unified mamba-based neural framework for learning on combinatorial complexes. CCMamba reformulates message passing as a selective state-space modeling problem by organizing multi-rank incidence relations into structured sequences processed by rank-aware state-space models. This enables adaptive, directional, and long range information propagation in linear time without self attention. We further establish the theoretical analysis that the expressive power upper-bound of CCMamba message passing is the 1-Weisfeiler-Lehman test. Experiments on graph, hypergraph, and simplicial benchmarks demonstrate that CCMamba consistently outperforms existing methods while exhibiting improved scalability and robustness to depth.

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Recent advances have discussed cell complexes as ideal learning representations. However, there is a lack of available machine learning methods suitable for learning on CW complexes. In this paper, we derive an explicit expression for the Wasserstein distance between cell complex signal distributions in terms of a Hodge-Laplacian matrix. This leads to a structurally meaningful measure to compare CW complexes and define the optimal transportation map. In order to simultaneously include both feature and structure information, we extend the Fused Gromov-Wasserstein distance to CW complexes. Finally, we introduce novel kernels over the space of probability measures on CW complexes based on the dual formulation of optimal transport.

Complex Query Answering (CQA) over incomplete Knowledge Graphs (KGs) is a challenging task. Recently, a line of message-passing-based research has been proposed to solve CQA. However, they perform unsatisfactorily on negative queries and fail to address the noisy messages between variable nodes in the query graph. Moreover, they offer little interpretability and require complex query data and resource-intensive training. In this paper, we propose a Neural-Symbolic Message Passing (NSMP) framework based on pre-trained neural link predictors. By introducing symbolic reasoning and fuzzy logic, NSMP can generalize to arbitrary existential first order logic queries without requiring training while providing interpretable answers. Furthermore, we introduce a dynamic pruning strategy to filter out noisy messages between variable nodes. Experimental results show that NSMP achieves a strong performance. Additionally, through complexity analysis and empirical verification, we demonstrate the superiority of NSMP in inference time over the current state-of-the-art neural-symbolic method. Compared to this approach, NSMP demonstrates faster inference times across all query types on benchmark datasets, with speedup ranging from 2times to over 150times.

Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes). They do not, however, consider the spatial direction from one atom to another, despite directional information playing a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a direction in coordinate space. These directional message embeddings are rotationally equivariant since the associated directions rotate with the molecule. We propose a message passing scheme analogous to belief propagation, which uses the directional information by transforming messages based on the angle between them. Additionally, we use spherical Bessel functions and spherical harmonics to construct theoretically well-founded, orthogonal representations that achieve better performance than the currently prevalent Gaussian radial basis representations while using fewer than 1/4 of the parameters. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet outperforms previous GNNs on average by 76% on MD17 and by 31% on QM9. Our implementation is available online.

Graph-based recommender systems have achieved remarkable effectiveness by modeling high-order interactions between users and items. However, such approaches are significantly undermined by popularity bias, which distorts the interaction graph's structure, referred to as topology bias. This leads to overrepresentation of popular items, thereby reinforcing biases and fairness issues through the user-system feedback loop. Despite attempts to study this effect, most prior work focuses on the embedding or gradient level bias, overlooking how topology bias fundamentally distorts the message passing process itself. We bridge this gap by providing an empirical and theoretical analysis from a Dirichlet energy perspective, revealing that graph message passing inherently amplifies topology bias and consistently benefits highly connected nodes. To address these limitations, we propose Test-time Simplicial Propagation (TSP), which extends message passing to higher-order simplicial complexes. By incorporating richer structures beyond pairwise connections, TSP mitigates harmful topology bias and substantially improves the representation and recommendation of long-tail items during inference. Extensive experiments across five real-world datasets demonstrate the superiority of our approach in mitigating topology bias and enhancing recommendation quality.

Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes. Understanding and mitigating oversquashing has recently received significant attention from the research community. In this paper, we continue this line of work by analyzing oversquashing through the lens of the effective resistance between nodes in the input graph. Effective resistance intuitively captures the ``strength'' of connection between two nodes by paths in the graph, and has a rich literature spanning many areas of graph theory. We propose to use total effective resistance as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use. We further develop an algorithm to identify edges to be added to an input graph to minimize the total effective resistance, thereby alleviating oversquashing. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies for improving the performance of GNNs.

Graph neural networks have moved from a niche representation-learning technique to the default model class wherever data carry relational structure. The interesting question is no longer whether message passing helps on a given dataset, but where graph structure earns its computational cost and where it does not. This survey organises the field around a single design space, derives the spectral and spatial formulations from shared first principles, and connects expressive power to the Weisfeiler-Leman hierarchy with explicit statements of what current architectures can and cannot separate. Against that methodological backbone we examine twelve application domains, among them recommendation and social networks, knowledge graphs and language-model integration, drug discovery and molecular property learning, healthcare and neuroscience, computer vision, traffic and urban computing, power and renewable-energy systems, wireless and sixth-generation networks, fraud and cybersecurity, industrial prognostics, materials science, and climate modelling. For each domain we specify the graph-construction choices and their costs, identify which architecture families dominate and why, and separate reported gains from artefacts of weak baselines or favourable splits. A cross-domain comparison exposes recurring patterns: heterophily and scale undercut the same models almost everywhere, temporal graphs remain harder than their static counterparts, and the architectures that top public leaderboards are seldom the ones that reach deployment. We treat over-smoothing, over-squashing, robustness, distribution shift, fairness, and explainability not as a closing checklist but as the constraints that decide adoption.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

Vision-Language models (VLMs) pre-trained on large corpora have demonstrated notable success across a range of downstream tasks. In light of the rapidly increasing size of pre-trained VLMs, parameter-efficient transfer learning (PETL) has garnered attention as a viable alternative to full fine-tuning. One such approach is the adapter, which introduces a few trainable parameters into the pre-trained models while preserving the original parameters during adaptation. In this paper, we present a novel modeling framework that recasts adapter tuning after attention as a graph message passing process on attention graphs, where the projected query and value features and attention matrix constitute the node features and the graph adjacency matrix, respectively. Within this framework, tuning adapters in VLMs necessitates handling heterophilic graphs, owing to the disparity between the projected query and value space. To address this challenge, we propose a new adapter architecture, p-adapter, which employs p-Laplacian message passing in Graph Neural Networks (GNNs). Specifically, the attention weights are re-normalized based on the features, and the features are then aggregated using the calibrated attention matrix, enabling the dynamic exploitation of information with varying frequencies in the heterophilic attention graphs. We conduct extensive experiments on different pre-trained VLMs and multi-modal tasks, including visual question answering, visual entailment, and image captioning. The experimental results validate our method's significant superiority over other PETL methods.

Currently, prominent Transformer architectures applied on graphs and meshes for shape analysis tasks employ traditional attention layers that heavily utilize spectral features requiring costly eigenvalue decomposition-based methods. To encode the mesh structure, these methods derive positional embeddings, that heavily rely on eigenvalue decomposition based operations, e.g. on the Laplacian matrix, or on heat-kernel signatures, which are then concatenated to the input features. This paper proposes a novel approach inspired by the explicit construction of the Hodge Laplacian operator in Discrete Exterior Calculus as a product of discrete Hodge operators and exterior derivatives, i.e. (L := star_0^{-1} d_0^T star_1 d_0). We adjust the Transformer architecture in a novel deep learning layer that utilizes the multi-head attention mechanism to approximate Hodge matrices star_0, star_1 and star_2 and learn families of discrete operators L that act on mesh vertices, edges and faces. Our approach results in a computationally-efficient architecture that achieves comparable performance in mesh segmentation and classification tasks, through a direct learning framework, while eliminating the need for costly eigenvalue decomposition operations or complex preprocessing operations.

Graph Neural Networks (GNNs) have shown promising potential in graph representation learning. The majority of GNNs define a local message-passing mechanism, propagating information over the graph by stacking multiple layers. These methods, however, are known to suffer from two major limitations: over-squashing and poor capturing of long-range dependencies. Recently, Graph Transformers (GTs) emerged as a powerful alternative to Message-Passing Neural Networks (MPNNs). GTs, however, have quadratic computational cost, lack inductive biases on graph structures, and rely on complex Positional/Structural Encodings (SE/PE). In this paper, we show that while Transformers, complex message-passing, and SE/PE are sufficient for good performance in practice, neither is necessary. Motivated by the recent success of State Space Models (SSMs), such as Mamba, we present Graph Mamba Networks (GMNs), a general framework for a new class of GNNs based on selective SSMs. We discuss and categorize the new challenges when adopting SSMs to graph-structured data, and present four required and one optional steps to design GMNs, where we choose (1) Neighborhood Tokenization, (2) Token Ordering, (3) Architecture of Bidirectional Selective SSM Encoder, (4) Local Encoding, and dispensable (5) PE and SE. We further provide theoretical justification for the power of GMNs. Experiments demonstrate that despite much less computational cost, GMNs attain an outstanding performance in long-range, small-scale, large-scale, and heterophilic benchmark datasets.

It has been shown that a message passing neural networks (MPNNs), a popular family of neural networks for graph-structured data, are at most as expressive as the first-order Weisfeiler-Leman (1-WL) graph isomorphism test, which has motivated the development of more expressive architectures. In this work, we analyze if the limited expressiveness is actually a limiting factor for MPNNs and other WL-based models in standard graph datasets. Interestingly, we find that the expressiveness of WL is sufficient to identify almost all graphs in most datasets. Moreover, we find that the classification accuracy upper bounds are often close to 100\%. Furthermore, we find that simple WL-based neural networks and several MPNNs can be fitted to several datasets. In sum, we conclude that the performance of WL/MPNNs is not limited by their expressiveness in practice.

Rotary position encodings make attention sensitive to relative position, but extending them to graphs requires choosing how graph structure enters the rotation. Previous works assign each node a rotation from spectral coordinates, so the rotary factor between two nodes depends only on their endpoints and cannot distinguish the routes connecting them. We introduce Analytic-Walk Rotary Positional Encodings (AW-RoPE), which place the rotations on edges and sum the transported features over all walks, so contributions along different routes can reinforce or cancel. An exact variant evaluates the complete sum by a differentiable linear solve, and a sparse variant truncates it at a finite depth. We prove forward and parameter-derivative truncation bounds at fixed inputs and parameters. Both variants act on projected queries and keys, and the sparse recurrence also augments message-passing networks. Across five synthetic tasks both variants reduce nRMSE by 15--58% relative to the strongest baseline, and on real superpixel, peptide and OGB benchmarks the sparse recurrence attains the best mean on every dataset with Performer kernels and on twelve of thirteen datasets with GIN. Analysis shows that AW-RoPE can distinguish routes whose only cue is how two endpoints are connected, while node-wise rotary encodings cannot.

This work introduces DADAO: the first decentralized, accelerated, asynchronous, primal, first-order algorithm to minimize a sum of L-smooth and mu-strongly convex functions distributed over a given network of size n. Our key insight is based on modeling the local gradient updates and gossip communication procedures with separate independent Poisson Point Processes. This allows us to decouple the computation and communication steps, which can be run in parallel, while making the whole approach completely asynchronous, leading to communication acceleration compared to synchronous approaches. Our new method employs primal gradients and does not use a multi-consensus inner loop nor other ad-hoc mechanisms such as Error Feedback, Gradient Tracking, or a Proximal operator. By relating the inverse of the smallest positive eigenvalue of the Laplacian matrix chi_1 and the maximal resistance chi_2leq chi_1 of the graph to a sufficient minimal communication rate between the nodes of the network, we show that our algorithm requires O(nfrac{L{mu}}log(1{epsilon})) local gradients and only O(nchi_1chi_2frac{L{mu}}log(1{epsilon})) communications to reach a precision epsilon, up to logarithmic terms. Thus, we simultaneously obtain an accelerated rate for both computations and communications, leading to an improvement over state-of-the-art works, our simulations further validating the strength of our relatively unconstrained method. We also propose a SDP relaxation to find the optimal gossip rate of each edge minimizing the total number of communications for a given graph, resulting in faster convergence compared to standard approaches relying on uniform communication weights. Our source code is released on a public repository.

Message passing neural networks (MPNNs) have emerged as the most popular framework of graph neural networks (GNNs) in recent years. However, their expressive power is limited by the 1-dimensional Weisfeiler-Lehman (1-WL) test. Some works are inspired by k-WL/FWL (Folklore WL) and design the corresponding neural versions. Despite the high expressive power, there are serious limitations in this line of research. In particular, (1) k-WL/FWL requires at least O(n^k) space complexity, which is impractical for large graphs even when k=3; (2) The design space of k-WL/FWL is rigid, with the only adjustable hyper-parameter being k. To tackle the first limitation, we propose an extension, (k,t)-FWL. We theoretically prove that even if we fix the space complexity to O(n^k) (for any kgeq 2) in (k,t)-FWL, we can construct an expressiveness hierarchy up to solving the graph isomorphism problem. To tackle the second problem, we propose k-FWL+, which considers any equivariant set as neighbors instead of all nodes, thereby greatly expanding the design space of k-FWL. Combining these two modifications results in a flexible and powerful framework (k,t)-FWL+. We demonstrate (k,t)-FWL+ can implement most existing models with matching expressiveness. We then introduce an instance of (k,t)-FWL+ called Neighborhood^2-FWL (N^2-FWL), which is practically and theoretically sound. We prove that N^2-FWL is no less powerful than 3-WL, and can encode many substructures while only requiring O(n^2) space. Finally, we design its neural version named N^2-GNN and evaluate its performance on various tasks. N^2-GNN achieves record-breaking results on ZINC-Subset (0.059), outperforming previous SOTA results by 10.6%. Moreover, N^2-GNN achieves new SOTA results on the BREC dataset (71.8%) among all existing high-expressive GNN methods.

Building on our earlier program-evolution workflow guided by large language models (LLMs), we study weight-five bivariate bicycle (BB) and perturbed bivariate bicycle (PBB) codes. The resulting catalogue contains 1,142 distinct code proposals, including 1,081 nonbaseline proposals attributable to LLM-generated programs. Across the catalogue, we certify connected Calderbank--Shor--Steane (CSS) realizations [[96,4,10]], [[140,6,10]], and [[180,4,14]]. A post-search comparison certifies seven imported Lin--Pryadko archive constructions. For leading parameter triples also represented in that archive, we provide exact distance evidence, explicit bivariate presentations, and verified component reductions. A basis-independent connectivity analysis identifies 409 of the 1,142 catalogue entries as disconnected and shows that 73.1\% of the classes with exact distance certificates contain repeated connected components. Algebraic analysis organizes the connected CSS classes into order-3, order-7, and order-15 cyclotomic-kernel strata. The strongest exact connected PBB parameter point is [[216,4,10]], attained by two distinct component classes. Among the 936 distinct CSS proposals from the LLM-guided campaign with a recorded positive distance, 816 (87.18\%) are certified at dgeq5. For comparison, three random-search controls each sample 6,444 CSS code proposals uniformly without replacement, using the same per-lattice and encoded-dimension sample counts as the LLM-guided campaign. In these controls, 4,672--4,785 proposals (72.50--74.26\%) meet the same criterion. The LLM-guided campaign has the higher certified yield, while the random controls cover more connected classes. Together, these results extend LLM-guided discovery to a more constrained code family and provide a reproducible structural and exact-distance account of its strongest candidates.

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

Graph Neural Networks (GNNs) have shown great potential in the field of graph representation learning. Standard GNNs define a local message-passing mechanism which propagates information over the whole graph domain by stacking multiple layers. This paradigm suffers from two major limitations, over-squashing and poor long-range dependencies, that can be solved using global attention but significantly increases the computational cost to quadratic complexity. In this work, we propose an alternative approach to overcome these structural limitations by leveraging the ViT/MLP-Mixer architectures introduced in computer vision. We introduce a new class of GNNs, called Graph ViT/MLP-Mixer, that holds three key properties. First, they capture long-range dependency and mitigate the issue of over-squashing as demonstrated on Long Range Graph Benchmark and TreeNeighbourMatch datasets. Second, they offer better speed and memory efficiency with a complexity linear to the number of nodes and edges, surpassing the related Graph Transformer and expressive GNN models. Third, they show high expressivity in terms of graph isomorphism as they can distinguish at least 3-WL non-isomorphic graphs. We test our architecture on 4 simulated datasets and 7 real-world benchmarks, and show highly competitive results on all of them. The source code is available for reproducibility at: https://github.com/XiaoxinHe/Graph-ViT-MLPMixer.

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.

Optimizing multiple conversion objectives is a core challenge in industrial recommendation, often limited by signal erosion in rigid architectures. Existing Multi-Task Learning (MTL) methods typically enforce uniform dependency strengths across a static conversion funnel, overlooking how task correlations naturally vary based on item characteristics. Hierarchical message passing along these fixed chains leads to cumulative signal attenuation, which degrades performance on sparse, deep-funnel objectives. To address this, we propose the Personalized Task Dependency Graphs (PTDG). While respecting necessary physical causal constraints (e.g., Click -> Pay), PTDG dynamically "rewires" the intensity of dependency pathways for each item via low-rank approximation to ensure structural robustness. We implement a GCN-based propagation with hard causal masking to establish adaptive information shortcuts. Additionally, we introduce an Adaptive Progressive Masking (APM) strategy that decouples shared parameters according to task sparsity, helping to stabilize optimization. Experiments on KuaiRand1K and an industrial dataset show that PTDG significantly improves AUC on sparse conversion tasks by up to 1.45%, while maintaining comparable performance on dense objectives. Online A/B testing shows PTDG improves Conversion Rate (CVR) by 1.2% and effective Cost Per Mille (eCPM) by 1.9% relative to the baseline.

We present ModularSubsetSelection (MSS), a new algorithm for locally differentially private (LDP) frequency estimation. Given a universe of size k and n users, our varepsilon-LDP mechanism encodes each input via a Residue Number System (RNS) over ell pairwise-coprime moduli m_0, ldots, m_{ell-1}, and reports a randomly chosen index j in [ell] along with the perturbed residue using the statistically optimal SubsetSelection (SS) (Wang et al. 2016). This design reduces the user communication cost from Θbigl(ωlog_2(k/ω)bigr) bits required by standard SS (with ωapprox k/(e^varepsilon+1)) down to lceil log_2 ell rceil + lceil log_2 m_j rceil bits, where m_j < k. Server-side decoding runs in Θ(n + r k ell) time, where r is the number of LSMR (Fong and Saunders 2011) iterations. In practice, with well-conditioned moduli (i.e., constant r and ell = Θ(log k)), this becomes Θ(n + k log k). We prove that MSS achieves worst-case MSE within a constant factor of state-of-the-art protocols such as SS and ProjectiveGeometryResponse (PGR) (Feldman et al. 2022) while avoiding the algebraic prerequisites and dynamic-programming decoder required by PGR. Empirically, MSS matches the estimation accuracy of SS, PGR, and RAPPOR (Erlingsson, Pihur, and Korolova 2014) across realistic (k, varepsilon) settings, while offering faster decoding than PGR and shorter user messages than SS. Lastly, by sampling from multiple moduli and reporting only a single perturbed residue, MSS achieves the lowest reconstruction-attack success rate among all evaluated LDP protocols.

Routing has emerged as a promising strategy for balancing performance and cost in large language model (LLM) systems that combine lightweight models with powerful but expensive large models. Recent studies show that probe routing, which predicts the correctness of a small model using its hidden states, provides an effective solution in text-only LLMs. However, we observe that these probes degrade substantially when applied to multimodal LLMs (MLLMs). Through empirical analysis, we find that the presence of visual inputs weakens the separability of correctness signals in hidden states, making them harder to extract using standard probe designs. To address this challenge, we introduce two complementary approaches for improving probe routing in MLLMs. First, we propose the Attention Probe, which aggregates hidden states from the preceding layer based on attention scores to recover distributed correctness signals. Second, we present the KL-Regularized LoRA Probe (ReLope), which inserts a lightweight LoRA adapter and applies a KL regularizer to learn routing-aware representations. Comprehensive experiments show that our methods consistently outperform baselines, suggesting that improving the quality of hidden states is key to effective routing in MLLMs. Our code is available at https://github.com/Spinozaaa/ReLope.

Structured variational autoencoders (SVAEs) combine probabilistic graphical model priors on latent variables, deep neural networks to link latent variables to observed data, and structure-exploiting algorithms for approximate posterior inference. These models are particularly appealing for sequential data, where the prior can capture temporal dependencies. However, despite their conceptual elegance, SVAEs have proven difficult to implement, and more general approaches have been favored in practice. Here, we revisit SVAEs using modern machine learning tools and demonstrate their advantages over more general alternatives in terms of both accuracy and efficiency. First, we develop a modern implementation for hardware acceleration, parallelization, and automatic differentiation of the message passing algorithms at the core of the SVAE. Second, we show that by exploiting structure in the prior, the SVAE learns more accurate models and posterior distributions, which translate into improved performance on prediction tasks. Third, we show how the SVAE can naturally handle missing data, and we leverage this ability to develop a novel, self-supervised training approach. Altogether, these results show that the time is ripe to revisit structured variational autoencoders.

Graph neural networks (GNNs), in general, are built on the assumption of a static set of features characterizing each node in a graph. This assumption is often violated in practice. Existing methods partly address this issue through feature imputation. However, these techniques (i) assume uniformity of feature set across nodes, (ii) are transductive by nature, and (iii) fail to work when features are added or removed over time. In this work, we address these limitations through a novel GNN framework called GRAFENNE. GRAFENNE performs a novel allotropic transformation on the original graph, wherein the nodes and features are decoupled through a bipartite encoding. Through a carefully chosen message passing framework on the allotropic transformation, we make the model parameter size independent of the number of features and thereby inductive to both unseen nodes and features. We prove that GRAFENNE is at least as expressive as any of the existing message-passing GNNs in terms of Weisfeiler-Leman tests, and therefore, the additional inductivity to unseen features does not come at the cost of expressivity. In addition, as demonstrated over four real-world graphs, GRAFENNE empowers the underlying GNN with high empirical efficacy and the ability to learn in continual fashion over streaming feature sets.

This study utilizes community structures to address node degree biases in message-passing (MP) via learnable graph augmentations and novel graph transformers. Recent augmentation-based methods showed that MP neural networks often perform poorly on low-degree nodes, leading to degree biases due to a lack of messages reaching low-degree nodes. Despite their success, most methods use heuristic or uniform random augmentations, which are non-differentiable and may not always generate valuable edges for learning representations. In this paper, we propose Community-aware Graph Transformers, namely CGT, to learn degree-unbiased representations based on learnable augmentations and graph transformers by extracting within community structures. We first design a learnable graph augmentation to generate more within-community edges connecting low-degree nodes through edge perturbation. Second, we propose an improved self-attention to learn underlying proximity and the roles of nodes within the community. Third, we propose a self-supervised learning task that could learn the representations to preserve the global graph structure and regularize the graph augmentations. Extensive experiments on various benchmark datasets showed CGT outperforms state-of-the-art baselines and significantly improves the node degree biases. The source code is available at https://github.com/NSLab-CUK/Community-aware-Graph-Transformer.

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from oriented matroid theory which we show are both necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope and invariant under the polytope's orientation-preserving symmetries (SL(d,Z) ltimes Z^d). When implemented with a certain masking procedure, one can also guarantee that every rollout produces a fine triangulation (in 2D). On unseen polygons with N_pts leq 40, dualGNN is the most uniform FRT sampler we tested, and even a model trained on a single polygon generalizes well to other polygons. The model is small (sim92k parameters), trains in sim7.5 hours on a single consumer GPU, and runs without modification on an M1 MacBook Pro. We apply dualGNN to string theory, uniformly sampling Calabi-Yau threefolds at h^{1,1}=86 and consistent with uniformity at h^{1,1}=128. This is an order of magnitude beyond previous learned methods with a model sim1000times smaller. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN .

Graph Neural Networks (GNNs) have seen significant advances in recent years, yet their application to multigraphs, where parallel edges exist between the same pair of nodes, remains under-explored. Standard GNNs, designed for simple graphs, compute node representations by combining all connected edges at once, without distinguishing between edges from different neighbors. There are some GNN architectures proposed specifically for multigraphs, yet these architectures perform only node-level aggregation in their message passing layers, which limits their expressive power. Furthermore, these approaches either lack permutation equivariance when a strict total edge ordering is absent, or fail to preserve the topological structure of the multigraph. To address all these shortcomings, we propose MEGA-GNN, a unified framework for message passing on multigraphs that can effectively perform diverse graph learning tasks. Our approach introduces a two-stage aggregation process in the message passing layers: first, parallel edges are aggregated, followed by a node-level aggregation of messages from distinct neighbors. We show that MEGA-GNN is not only permutation equivariant but also universal given a strict total ordering on the edges. Experiments show that MEGA-GNN significantly outperforms state-of-the-art solutions by up to 13\% on Anti-Money Laundering datasets and is on par with their accuracy on real-world phishing classification datasets in terms of minority class F1 score.

Diffusion models have exhibited exciting capabilities in generating images and are also very promising for video creation. However, the inference speed of diffusion models is limited by the slow sampling process, restricting its use cases. The sequential denoising steps required for generating a single sample could take tens or hundreds of iterations and thus have become a significant bottleneck. This limitation is more salient for applications that are interactive in nature or require small latency. To address this challenge, we propose Partially Conditioned Patch Parallelism (PCPP) to accelerate the inference of high-resolution diffusion models. Using the fact that the difference between the images in adjacent diffusion steps is nearly zero, Patch Parallelism (PP) leverages multiple GPUs communicating asynchronously to compute patches of an image in multiple computing devices based on the entire image (all patches) in the previous diffusion step. PCPP develops PP to reduce computation in inference by conditioning only on parts of the neighboring patches in each diffusion step, which also decreases communication among computing devices. As a result, PCPP decreases the communication cost by around 70% compared to DistriFusion (the state of the art implementation of PP) and achieves 2.36sim 8.02times inference speed-up using 4sim 8 GPUs compared to 2.32sim 6.71times achieved by DistriFusion depending on the computing device configuration and resolution of generation at the cost of a possible decrease in image quality. PCPP demonstrates the potential to strike a favorable trade-off, enabling high-quality image generation with substantially reduced latency.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

With the increasing prevalence of graph-structured data, multi-view graph clustering has been widely used in various downstream applications. Existing approaches primarily rely on a unified message passing mechanism, which significantly enhances clustering performance. Nevertheless, this mechanism limits its applicability to heterophilous situations, as it is fundamentally predicated on the assumption of homophily, i.e., the connected nodes often belong to the same class. In reality, this assumption does not always hold; a moderately or even mildly homophilous graph is more common than a fully homophilous one due to inevitable heterophilous information in the graph. To address this issue, in this paper, we propose a novel SiMilarity-enhanced Homophily for Multi-view Heterophilous Graph Clustering (SMHGC) approach. By analyzing the relationship between similarity and graph homophily, we propose to enhance the homophily by introducing three similarity terms, i.e., neighbor pattern similarity, node feature similarity, and multi-view global similarity, in a label-free manner. Then, a consensus-based inter- and intra-view fusion paradigm is proposed to fuse the improved homophilous graph from different views and utilize them for clustering. The state-of-the-art experimental results on both multi-view heterophilous and homophilous datasets collectively demonstrate the strong capacity of similarity for unsupervised multi-view heterophilous graph learning. Additionally, the consistent performance across semi-synthetic datasets with varying levels of homophily serves as further evidence of SMHGC's resilience to heterophily.

Machine learning interatomic potentials trained on first-principles reference data are becoming valuable tools for computational physics, biology, and chemistry. Equivariant message-passing neural networks, including transformers, achieve state-of-the-art accuracy but rely on cutoff-based graphs, limiting their ability to capture long-range effects such as electrostatics or dispersion, as well as electron delocalization. While long-range correction schemes based on inverse power laws of interatomic distances have been proposed, they are unable to communicate higher-order geometric information and are thus limited in applicability. To address this shortcoming, we propose the use of equivariant, rather than scalar, charges for long-range interactions, and design a graph neural network architecture, LOREM, around this long-range message passing mechanism. We consider several datasets specifically designed to highlight non-local physical effects, and compare short-range message passing with different receptive fields to invariant and equivariant long-range message passing. Even though most approaches work for careful dataset-specific choices of their model hyperparameters, LOREM works consistently without such changes, with excellent benchmark performance.

Factorisation-based Models (FMs), such as DistMult, have enjoyed enduring success for Knowledge Graph Completion (KGC) tasks, often outperforming Graph Neural Networks (GNNs). However, unlike GNNs, FMs struggle to incorporate node features and generalise to unseen nodes in inductive settings. Our work bridges the gap between FMs and GNNs by proposing ReFactor GNNs. This new architecture draws upon both modelling paradigms, which previously were largely thought of as disjoint. Concretely, using a message-passing formalism, we show how FMs can be cast as GNNs by reformulating the gradient descent procedure as message-passing operations, which forms the basis of our ReFactor GNNs. Across a multitude of well-established KGC benchmarks, our ReFactor GNNs achieve comparable transductive performance to FMs, and state-of-the-art inductive performance while using an order of magnitude fewer parameters.

Mixed-Integer Linear Programming (MILP) is a cornerstone of combinatorial optimization, yet solving large-scale instances remains a significant computational challenge. Recently, Graph Neural Networks (GNNs) have shown promise in accelerating MILP solvers by predicting high-quality solutions. However, we identify that existing methods misalign with the intrinsic structure of MILP problems at two levels. At the leaning objective level, the Binary Cross-Entropy (BCE) loss treats variables independently, neglecting their relative priority and yielding plausible logits. At the model architecture level, standard GNN message passing inherently smooths the representations across variables, missing the natural competitive relationships within constraints. To address these challenges, we propose CoCo-MILP, which explicitly models inter-variable Contrast and intra-constraint Competition for advanced MILP solution prediction. At the objective level, CoCo-MILP introduces the Inter-Variable Contrastive Loss (VCL), which explicitly maximizes the embedding margin between variables assigned one versus zero. At the architectural level, we design an Intra-Constraint Competitive GNN layer that, instead of homogenizing features, learns to differentiate representations of competing variables within a constraint, capturing their exclusionary nature. Experimental results on standard benchmarks demonstrate that CoCo-MILP significantly outperforms existing learning-based approaches, reducing the solution gap by up to 68.12% compared to traditional solvers. Our code is available at https://github.com/happypu326/CoCo-MILP.

While diffusion language models (DLMs) offer a promising alternative to autoregressive models (ARs), existing open-source DLMs suffer from high inference latency. This bottleneck is mainly due to the attention's quadratic complexity with respect to context length in computing all query-key pairs. Intuitively, to reduce this complexity, a natural strategy is to restrict attention to sparse patterns that retain only the most relevant connections. Such approaches are well-established in ARs, where attention follows fixed and clearly defined sparse patterns. However, in DLMs, we observe distinct sparsity behaviors: (1) attention patterns vary across heads, (2) attention patterns in each head remain highly similar across denoising steps, and (3) early denoising steps are critical for generation. These findings render sparse attention methods designed for ARs largely incompatible with DLMs, as they fail to capture head-specific structures and risk degrading generation when applied in early denoising steps. To address these challenges, we propose SparseD, a novel sparse attention method for DLMs. Leveraging the observations, SparseD only requires pre-computing head-specific sparse patterns one time, and reuses them across all steps. This prevents recomputing sparse patterns at each denoising step. Meanwhile, SparseD uses full attention in the early steps, then switches to sparse attention later to maintain generation quality. Together, these establish SparseD as a practical and efficient solution for deploying DLMs in long-context applications. Experimental results demonstrate that SparseD achieves lossless acceleration, delivering up to 1.50times speedup over FlashAttention at a 64k context length with 1,024 denoising steps.

A key requirement for graph neural networks is that they must process a graph in a way that does not depend on how the graph is described. Traditionally this has been taken to mean that a graph network must be equivariant to node permutations. Here we show that instead of equivariance, the more general concept of naturality is sufficient for a graph network to be well-defined, opening up a larger class of graph networks. We define global and local natural graph networks, the latter of which are as scalable as conventional message passing graph neural networks while being more flexible. We give one practical instantiation of a natural network on graphs which uses an equivariant message network parameterization, yielding good performance on several benchmarks.

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step followed by a projection-based measurement consistency step, often produce suboptimal results. By studying the generative sampling path, here we show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint, which can be used synergistically with the previous solvers to make the iterations close to the manifold. The proposed manifold constraint is straightforward to implement within a few lines of code, yet boosts the performance by a surprisingly large margin. With extensive experiments, we show that our method is superior to the previous methods both theoretically and empirically, producing promising results in many applications such as image inpainting, colorization, and sparse-view computed tomography. Code available https://github.com/HJ-harry/MCG_diffusion

Diffusion models based on permutation-equivariant networks can learn permutation-invariant distributions for graph data. However, in comparison to their non-invariant counterparts, we have found that these invariant models encounter greater learning challenges since 1) their effective target distributions exhibit more modes; 2) their optimal one-step denoising scores are the score functions of Gaussian mixtures with more components. Motivated by this analysis, we propose a non-invariant diffusion model, called SwinGNN, which employs an efficient edge-to-edge 2-WL message passing network and utilizes shifted window based self-attention inspired by SwinTransformers. Further, through systematic ablations, we identify several critical training and sampling techniques that significantly improve the sample quality of graph generation. At last, we introduce a simple post-processing trick, i.e., randomly permuting the generated graphs, which provably converts any graph generative model to a permutation-invariant one. Extensive experiments on synthetic and real-world protein and molecule datasets show that our SwinGNN achieves state-of-the-art performances. Our code is released at https://github.com/qiyan98/SwinGNN.

Given the damping factor α and precision tolerance ε, andersen2006local introduced Approximate Personalized PageRank (APPR), the de facto local method for approximating the PPR vector, with runtime bounded by Θ(1/(αε)) independent of the graph size. Recently, fountoulakis2022open asked whether faster local algorithms could be developed using O(1/(sqrtαε)) operations. By noticing that APPR is a local variant of Gauss-Seidel, this paper explores the question of whether standard iterative solvers can be effectively localized. We propose to use the locally evolving set process, a novel framework to characterize the algorithm locality, and demonstrate that many standard solvers can be effectively localized. Let operatorname{vol}{ (S_t)} and overlineγ_{t} be the running average of volume and the residual ratio of active nodes textstyle S_{t} during the process. We show operatorname{vol}{ (S_t)}/overlineγ_{t} leq 1/ε and prove APPR admits a new runtime bound O(operatorname{vol}(S_t)/(αoverlineγ_{t})) mirroring the actual performance. Furthermore, when the geometric mean of residual reduction is Θ(sqrtα), then there exists c in (0,2) such that the local Chebyshev method has runtime O(operatorname{vol}(S_{t})/(sqrtα(2-c))) without the monotonicity assumption. Numerical results confirm the efficiency of this novel framework and show up to a hundredfold speedup over corresponding standard solvers on real-world graphs.

We present a novel edge-level ego-network encoding for learning on graphs that can boost Message Passing Graph Neural Networks (MP-GNNs) by providing additional node and edge features or extending message-passing formats. The proposed encoding is sufficient to distinguish Strongly Regular Graphs, a family of challenging 3-WL equivalent graphs. We show theoretically that such encoding is more expressive than node-based sub-graph MP-GNNs. In an empirical evaluation on four benchmarks with 10 graph datasets, our results match or improve previous baselines on expressivity, graph classification, graph regression, and proximity tasks -- while reducing memory usage by 18.1x in certain real-world settings.

We present DYMAG, a graph neural network based on a novel form of message aggregation. Standard message-passing neural networks, which often aggregate local neighbors via mean-aggregation, can be regarded as convolving with a simple rectangular waveform which is non-zero only on 1-hop neighbors of every vertex. Here, we go beyond such local averaging. We will convolve the node features with more sophisticated waveforms generated using dynamics such as the heat equation, wave equation, and the Sprott model (an example of chaotic dynamics). Furthermore, we use snapshots of these dynamics at different time points to create waveforms at many effective scales. Theoretically, we show that these dynamic waveforms can capture salient information about the graph including connected components, connectivity, and cycle structures even with no features. Empirically, we test DYMAG on both real and synthetic benchmarks to establish that DYMAG outperforms baseline models on recovery of graph persistence, generating parameters of random graphs, as well as property prediction for proteins, molecules and materials. Our code is available at https://github.com/KrishnaswamyLab/DYMAG.

In this paper, we present an algorithm for the sparse signal recovery problem that incorporates damped Gaussian generalized approximate message passing (GGAMP) into Expectation-Maximization (EM)-based sparse Bayesian learning (SBL). In particular, GGAMP is used to implement the E-step in SBL in place of matrix inversion, leveraging the fact that GGAMP is guaranteed to converge with appropriate damping. The resulting GGAMP-SBL algorithm is much more robust to arbitrary measurement matrix A than the standard damped GAMP algorithm while being much lower complexity than the standard SBL algorithm. We then extend the approach from the single measurement vector (SMV) case to the temporally correlated multiple measurement vector (MMV) case, leading to the GGAMP-TSBL algorithm. We verify the robustness and computational advantages of the proposed algorithms through numerical experiments.

Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying subspaces of a PDE operator. We use Hodge heat flow as a controlled lens on this distinction and introduce TopoBox-3D, where tunnels and cavities vary Betti support while the exact Hodge decomposition separates the harmonic kernel from the positive spectrum. Across six architectures, models that infer topology implicitly suffer excess matched degradation in 37 of 45 model--task topology-OOD cells, yet cases that change harmonic dimension are not more strongly penalized on average. The dominant difficulty is instead spectral: the initial Rayleigh quotient is the most stable predictor of error, and spectral broadening adds information for edge and face cochains. Most strikingly, controlled probes show that explicit incidence and harmonic coordinates do not yield the best kernel-identity accuracy; nevertheless, TNO ranks first in mixed-input nonharmonic accuracy on all six tasks with nontrivial harmonic support. Together, these results establish topology as a distinct generalization axis beyond arbitrary-geometry compatibility and show that its influence extends across the Hodge spectrum rather than remaining confined to the harmonic kernel. More broadly, they suggest that global, low-frequency structural priors may help organize predictions in the faster-decaying complementary component, offering a new perspective on how neural operators may generalize across topology as well as geometry.

We propose a Semantic Ordered Statistics Decoder (sem-OSD), a soft decoder for short linear block codes carrying byte-streamed sources such as natural-language text. Sem-OSD injects a byte-level language-model (LM) prior into ordered statistics decoding (OSD) through a fused bit-level score that combines channel reliability with the LM prior, and uses it for the most-reliable basis (MRB) selection and the codeword candidate scoring. Sem-OSD enumerates two complementary test-error-pattern (TEP) families: a bit-flip family that flips up to m bits, and an LM-driven family of up to ω byte substitutions that reaches error patterns the bit-flip family cannot. The LM prior is computed by a byte-level Transformer fine-tuned for byte-level denoising. Simulation results show that, on AWGN, sem-OSD achieves block error rates (BLERs) below the finite-blocklength normal-approximation bound for uniform sources on both binary BCH(127,64) and shortened RS(16,8) over GF(256), exceeding Fossorier OSD by a 1.5 dB coding gain. On a Gilbert--Elliott burst-error channel, sem-OSD provides 4 dB and 1 dB of more coding gain than Berlekamp--Massey and OSD, respectively.

Graph Neural Networks (GNNs) that are based on the message passing (MP) paradigm generally exchange information between 1-hop neighbors to build node representations at each layer. In principle, such networks are not able to capture long-range interactions (LRI) that may be desired or necessary for learning a given task on graphs. Recently, there has been an increasing interest in development of Transformer-based methods for graphs that can consider full node connectivity beyond the original sparse structure, thus enabling the modeling of LRI. However, MP-GNNs that simply rely on 1-hop message passing often fare better in several existing graph benchmarks when combined with positional feature representations, among other innovations, hence limiting the perceived utility and ranking of Transformer-like architectures. Here, we present the Long Range Graph Benchmark (LRGB) with 5 graph learning datasets: PascalVOC-SP, COCO-SP, PCQM-Contact, Peptides-func and Peptides-struct that arguably require LRI reasoning to achieve strong performance in a given task. We benchmark both baseline GNNs and Graph Transformer networks to verify that the models which capture long-range dependencies perform significantly better on these tasks. Therefore, these datasets are suitable for benchmarking and exploration of MP-GNNs and Graph Transformer architectures that are intended to capture LRI.

Despite its outstanding performance in various graph tasks, vanilla Message Passing Neural Network (MPNN) usually fails in link prediction tasks, as it only uses representations of two individual target nodes and ignores the pairwise relation between them. To capture the pairwise relations, some models add manual features to the input graph and use the output of MPNN to produce pairwise representations. In contrast, others directly use manual features as pairwise representations. Though this simplification avoids applying a GNN to each link individually and thus improves scalability, these models still have much room for performance improvement due to the hand-crafted and unlearnable pairwise features. To upgrade performance while maintaining scalability, we propose Neural Common Neighbor (NCN), which uses learnable pairwise representations. To further boost NCN, we study the unobserved link problem. The incompleteness of the graph is ubiquitous and leads to distribution shifts between the training and test set, loss of common neighbor information, and performance degradation of models. Therefore, we propose two intervention methods: common neighbor completion and target link removal. Combining the two methods with NCN, we propose Neural Common Neighbor with Completion (NCNC). NCN and NCNC outperform recent strong baselines by large margins. NCNC achieves state-of-the-art performance in link prediction tasks. Our code is available at https://github.com/GraphPKU/NeuralCommonNeighbor.

Diffusion Language Models (DLMs) have recently emerged as a strong alternative to autoregressive language models (LMs). DLMs offer comparable accuracy with faster inference speed via parallel decoding. However, standard DLM decoding strategies relying on high-confidence tokens encounter an inherent information-theoretic bottleneck that restricts decoding progress and ultimately slows generation. We demonstrate both theoretically and empirically that prioritizing high-confidence tokens is inherently inefficient. High-probability tokens carry negligible information and strictly relying on them limits the effective progress made in each decoding round. We prove that the number of decoding rounds must grow linearly with the sample's total information (negative log-likelihood) and inversely with the per-round information budget, establishing a bits-to-rounds principle. We also propose Explore-Then-Exploit (ETE), a training-free decoding strategy that maximizes information throughput and decoding efficiency. ETE combines cross-block decoding with targeted exploration of high-uncertainty tokens to reshape the conditional distribution and trigger cascades of confident predictions. Experiments verify our theoretical bounds and demonstrate that ETE consistently reduces the required number of decoding rounds compared to confidence-only baselines without compromising generation quality.

Graph neural networks (GNNs) emerged recently as a standard toolkit for learning from data on graphs. Current GNN designing works depend on immense human expertise to explore different message-passing mechanisms, and require manual enumeration to determine the proper message-passing depth. Inspired by the strong searching capability of neural architecture search (NAS) in CNN, this paper proposes Graph Neural Architecture Search (GNAS) with novel-designed search space. The GNAS can automatically learn better architecture with the optimal depth of message passing on the graph. Specifically, we design Graph Neural Architecture Paradigm (GAP) with tree-topology computation procedure and two types of fine-grained atomic operations (feature filtering and neighbor aggregation) from message-passing mechanism to construct powerful graph network search space. Feature filtering performs adaptive feature selection, and neighbor aggregation captures structural information and calculates neighbors' statistics. Experiments show that our GNAS can search for better GNNs with multiple message-passing mechanisms and optimal message-passing depth. The searched network achieves remarkable improvement over state-of-the-art manual designed and search-based GNNs on five large-scale datasets at three classical graph tasks. Codes can be found at https://github.com/phython96/GNAS-MP.

While most autoregressive LLMs are constrained to one-by-one decoding, diffusion LLMs (dLLMs) have attracted growing interest for their potential to dramatically accelerate inference through parallel decoding. Despite this promise, the conditional independence assumption in dLLMs causes parallel decoding to ignore token dependencies, inevitably degrading generation quality when these dependencies are strong. However, existing works largely overlook these inherent challenges, and evaluations on standard benchmarks (e.g., math and coding) are not sufficient to capture the quality degradation caused by parallel decoding. To address this gap, we first provide an information-theoretic analysis of parallel decoding. We then conduct case studies on analytically tractable synthetic list operations from both data distribution and decoding strategy perspectives, offering quantitative insights that highlight the fundamental limitations of parallel decoding. Building on these insights, we propose ParallelBench, the first benchmark specifically designed for dLLMs, featuring realistic tasks that are trivial for humans and autoregressive LLMs yet exceptionally challenging for dLLMs under parallel decoding. Using ParallelBench, we systematically analyze both dLLMs and autoregressive LLMs, revealing that: (i) dLLMs under parallel decoding can suffer dramatic quality degradation in real-world scenarios, and (ii) current parallel decoding strategies struggle to adapt their degree of parallelism based on task difficulty, thus failing to achieve meaningful speedup without compromising quality. Our findings underscore the pressing need for innovative decoding methods that can overcome the current speed-quality trade-off. We release our benchmark to help accelerate the development of truly efficient dLLMs.

Byte-level hierarchical language models (LMs) have recently emerged as a robust alternative to their popular counterparts that use subword tokenization. However, generating one byte at a time remains a bottleneck for inference speed. To address this, we introduce multi-byte prediction (MBP), which generates multiple bytes in parallel, speeding up inference with minimal performance impact and no additional parameters. MBP builds on the popular multi-token prediction (MTP) paradigm with two crucial innovations. First, we introduce a variable-length prediction window that aligns with the latent tokens, or segments, of a hierarchical LM. Second, we implement a novel attention-masking scheme that enables parallel byte prediction without violating causality. We show that multi-byte prediction strikes a Pareto-optimal trade-off across multiple generative tasks, instruction following, question answering, summarization, and machine translation, achieving the best trade-off between performance and inference throughput.

Diverse outputs in text generation are necessary for effective exploration in complex reasoning tasks, such as code generation and mathematical problem solving. Such Pass@k problems benefit from distinct candidates covering the solution space. However, traditional sampling approaches often waste computational resources on repetitive failure modes. While Diffusion Language Models have emerged as a competitive alternative to the prevailing Autoregressive paradigm, they remain susceptible to this redundancy, with independent samples frequently collapsing into similar modes. To address this, we propose a training free, low cost intervention to enhance generative diversity in Diffusion Language Models. Our approach modifies intermediate samples in a batch sequentially, where each sample is repelled from the feature space of previous samples, actively penalising redundancy. Unlike prior methods that require retraining or beam search, our strategy incurs negligible computational overhead, while ensuring that each sample contributes a unique perspective to the batch. We evaluate our method on the HumanEval and GSM8K benchmarks using the LLaDA-8B-Instruct model. Our results demonstrate significantly improved diversity and Pass@k performance across various temperature settings. As a simple modification to the sampling process, our method offers an immediate, low-cost improvement for current and future Diffusion Language Models in tasks that benefit from diverse solution search. We make our code available at https://github.com/sean-lamont/odd.

Diffusion language models are a promising alternative to autoregressive models due to their potential for faster generation. Among discrete diffusion approaches, Masked diffusion currently dominates, largely driven by strong perplexity on language modeling benchmarks. In this work, we present the first scaling law study of uniform-state and interpolating discrete diffusion methods. We also show that Masked diffusion models can be made approximately 12% more FLOPs-efficient when trained with a simple cross-entropy objective. We find that perplexity is informative within a diffusion family but can be misleading across families, where models with worse likelihood scaling may be preferable due to faster and more practical sampling, as reflected by the speed-quality Pareto frontier. These results challenge the view that Masked diffusion is categorically the future of diffusion language modeling and that perplexity alone suffices for cross-algorithm comparison. Scaling all methods to 1.7B parameters, we show that uniform-state diffusion remains competitive on likelihood-based benchmarks and outperforms autoregressive and Masked diffusion models on GSM8K, despite worse validation perplexity. We provide the code, model checkpoints, and video tutorials on the project page: http://s-sahoo.github.io/scaling-dllms

Byte Pair Encoding (BPE) is a widely used tokenization algorithm, whose tokens cannot extend across pre-tokenization boundaries, functionally limiting it to representing at most full words. The BoundlessBPE and SuperBPE algorithms extend and improve BPE by relaxing this limitation and allowing the formation of superwords, which are combinations of pretokens that form phrases. However, previous implementations were impractical to train: for example, BoundlessBPE took 4.7 CPU days to train on 1GB of data. We show that supermerge candidates, two or more consecutive pretokens eligible to form a supermerge, can be aggregated by frequency much like regular pretokens. This avoids keeping full documents in memory, as the original implementations of BoundlessBPE and SuperBPE required, leading to a significant training speedup. We present a two-phase formulation of BoundlessBPE that separates first-phase learning of regular merges from second-phase learning of supermerges, producing identical results to the original implementation. We also show a near-equivalence between two-phase BoundlessBPE and SuperBPE, with the difference being that a manually selected hyperparameter used in SuperBPE can be automatically determined in the second phase of BoundlessBPE. These changes enable a much faster implementation, allowing training on that same 1GB of data in 603 and 593 seconds for BoundlessBPE and SuperBPE, respectively, a more than 600x increase in speed. For each of BoundlessBPE, SuperBPE, and BPE, we open-source both a reference Python implementation and a fast Rust implementation.

Collaborative perception empowers each agent to improve its perceptual ability through the exchange of perceptual messages with other agents. It inherently results in a fundamental trade-off between perception ability and communication cost. To address this bottleneck issue, our core idea is to optimize the collaborative messages from two key aspects: representation and selection. The proposed codebook-based message representation enables the transmission of integer codes, rather than high-dimensional feature maps. The proposed information-filling-driven message selection optimizes local messages to collectively fill each agent's information demand, preventing information overflow among multiple agents. By integrating these two designs, we propose CodeFilling, a novel communication-efficient collaborative perception system, which significantly advances the perception-communication trade-off and is inclusive to both homogeneous and heterogeneous collaboration settings. We evaluate CodeFilling in both a real-world dataset, DAIR-V2X, and a new simulation dataset, OPV2VH+. Results show that CodeFilling outperforms previous SOTA Where2comm on DAIR-V2X/OPV2VH+ with 1,333/1,206 times lower communication volume. Our code is available at https://github.com/PhyllisH/CodeFilling.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

The explosion of large-scale data in fields such as finance, e-commerce, and social media has outstripped the processing capabilities of single-machine systems, driving the need for distributed statistical inference methods. Traditional approaches to distributed inference often struggle with achieving true sparsity in high-dimensional datasets and involve high computational costs. We propose a novel, two-stage, distributed best subset selection algorithm to address these issues. Our approach starts by efficiently estimating the active set while adhering to the ell_0 norm-constrained surrogate likelihood function, effectively reducing dimensionality and isolating key variables. A refined estimation within the active set follows, ensuring sparse estimates and matching the minimax ell_2 error bound. We introduce a new splicing technique for adaptive parameter selection to tackle subproblems under ell_0 constraints and a Generalized Information Criterion (GIC). Our theoretical and numerical studies show that the proposed algorithm correctly finds the true sparsity pattern, has the oracle property, and greatly lowers communication costs. This is a big step forward in distributed sparse estimation.

Transformers for graph data are increasingly widely studied and successful in numerous learning tasks. Graph inductive biases are crucial for Graph Transformers, and previous works incorporate them using message-passing modules and/or positional encodings. However, Graph Transformers that use message-passing inherit known issues of message-passing, and differ significantly from Transformers used in other domains, thus making transfer of research advances more difficult. On the other hand, Graph Transformers without message-passing often perform poorly on smaller datasets, where inductive biases are more crucial. To bridge this gap, we propose the Graph Inductive bias Transformer (GRIT) -- a new Graph Transformer that incorporates graph inductive biases without using message passing. GRIT is based on several architectural changes that are each theoretically and empirically justified, including: learned relative positional encodings initialized with random walk probabilities, a flexible attention mechanism that updates node and node-pair representations, and injection of degree information in each layer. We prove that GRIT is expressive -- it can express shortest path distances and various graph propagation matrices. GRIT achieves state-of-the-art empirical performance across a variety of graph datasets, thus showing the power that Graph Transformers without message-passing can deliver.

We propose SLoPe, a Double-Pruned Sparse Plus Lazy Low-rank Adapter Pretraining method for LLMs that improves the accuracy of sparse LLMs while accelerating their pretraining and inference and reducing their memory footprint. Sparse pretraining of LLMs reduces the accuracy of the model, to overcome this, prior work uses dense models during fine-tuning. SLoPe improves the accuracy of sparsely pretrained models by adding low-rank adapters in the final 1% iterations of pretraining without adding significant overheads to the model pretraining and inference. In addition, SLoPe uses a double-pruned backward pass formulation that prunes the transposed weight matrix using N:M sparsity structures to enable an accelerated sparse backward pass. SLoPe accelerates the training and inference of models with billions of parameters up to 1.14times and 1.34times respectively (OPT-33B and OPT-66B) while reducing their memory usage by up to 0.77times and 0.51times for training and inference respectively.

Developing models capable of complex, multi-step reasoning is a central goal in artificial intelligence. While representing problems as graphs is a powerful approach, Graph Neural Networks (GNNs) are fundamentally constrained by their message-passing mechanism, which imposes a local bottleneck that limits global, holistic reasoning. We argue that dynamic programming (DP), which solves problems by iteratively refining a global state, offers a more powerful and suitable learning paradigm. We introduce FloydNet, a new architecture that embodies this principle. In contrast to local message passing, FloydNet maintains a global, all-pairs relationship tensor and learns a generalized DP operator to progressively refine it. This enables the model to develop a task-specific relational calculus, providing a principled framework for capturing long-range dependencies. Theoretically, we prove that FloydNet achieves 3-WL (2-FWL) expressive power, and its generalized form aligns with the k-FWL hierarchy. FloydNet demonstrates state-of-the-art performance across challenging domains: it achieves near-perfect scores (often >99\%) on the CLRS-30 algorithmic benchmark, finds exact optimal solutions for the general Traveling Salesman Problem (TSP) at rates significantly exceeding strong heuristics, and empirically matches the 3-WL test on the BREC benchmark. Our results establish this learned, DP-style refinement as a powerful and practical alternative to message passing for high-level graph reasoning.

As the adoption of large language models increases and the need for per-user or per-task model customization grows, the parameter-efficient fine-tuning (PEFT) methods, such as low-rank adaptation (LoRA) and its variants, incur substantial storage and transmission costs. To further reduce stored parameters, we introduce a "divide-and-share" paradigm that breaks the barriers of low-rank decomposition across matrix dimensions, modules and layers by sharing parameters globally via a vector bank. As an instantiation of the paradigm to LoRA, our proposed VB-LoRA composites all the low-rank matrices of LoRA from a shared vector bank with a differentiable top-k admixture module. VB-LoRA achieves extreme parameter efficiency while maintaining comparable or better performance compared to state-of-the-art PEFT methods. Extensive experiments demonstrate the effectiveness of VB-LoRA on natural language understanding, natural language generation, and instruction tuning tasks. When fine-tuning the Llama2-13B model, VB-LoRA only uses 0.4% of LoRA's stored parameters, yet achieves superior results. Our source code is available at https://github.com/leo-yangli/VB-LoRA.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

In this paper, we tackle the problem of online semi-supervised learning (SSL). When data arrive in a stream, the dual problems of computation and data storage arise for any SSL method. We propose a fast approximate online SSL algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local "representative points" that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We apply our algorithm to face recognition and optical character recognition applications to show that we can take advantage of the manifold structure to outperform the previous methods. Unlike previous heuristic approaches, we show that our method yields provable performance bounds.

Diffusion Language Models (DLMs) have been seen as a promising competitor for autoregressive language models. However, diffusion language models have long been constrained by slow inference. A core challenge is that their non-autoregressive architecture and bidirectional attention preclude the key-value cache that accelerates decoding. We address this bottleneck by proposing a KV-cache-like mechanism, delayed KV-Cache, for the denoising process of DLMs. Our approach is motivated by the observation that different tokens have distinct representation dynamics throughout the diffusion process. Accordingly, we propose a delayed and conditioned caching strategy for key and value states. We design two complementary variants to cache key and value step-by-step: (1) dKV-Cache-Decode, which provides almost lossless acceleration, and even improves performance on long sequences, suggesting that existing DLMs may under-utilise contextual information during inference. (2) dKV-Cache-Greedy, which has aggressive caching with reduced lifespan, achieving higher speed-ups with quadratic time complexity at the cost of some performance degradation. dKV-Cache, in final, achieves from 2-10x speedup in inference, largely narrowing the gap between ARs and DLMs. We evaluate our dKV-Cache on several benchmarks, delivering acceleration across general language understanding, mathematical, and code-generation benchmarks. Experiments demonstrate that cache can also be used in DLMs, even in a training-free manner from current DLMs.

Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first principles by certifying nonnegative attention weights through a self-dual homogeneous cone. Building on this observation, we show that rank-one positive semi-definite (PSD) features offer a favorable capacity--interference tradeoff. KATA recovers a parameter-free convex output gate and characterizes associative capacity through the Welch interference floor. For tolerances above this floor, KATA enlarges the state without adding parameters and admits spherical codes with exponentially many keys in the projection dimension. We implement KATA as fused Triton kernels at two operating points: a flash-attention-style forward up to {sim}1.6times FlashAttention-2 throughput, and an exact O(T) chunked-state form that reaches {sim}11times FlashAttention-2 forward throughput at 131k tokens. An associative scan of the first-order feature lowers the inter-chunk recurrence depth to O(log(T/C)) for chunk size C and averages {sim}2.4times the throughput of a matched sequential linear-attention baseline. On long-range MQAR and repeated-key overwrite, several KATA variants outperform Gated DeltaNet, with parameter counts and state sizes reported alongside accuracy. Induction preserves near-perfect recall, while kernel benchmarks show that the maps can be implemented efficiently. KATA retains 0.985 MQAR at a 16times out-of-distribution length, approaching the softmax with roughly one quarter of the KV-cache entries. Experiments on 340M-parameter LLMs reveal a feature-dependent fluency trade-off and clarify how positional embeddings, delta rules, and decay gates interact with feature geometry.

Discrete diffusion models are promising alternatives to autoregressive approaches for text generation, yet their decoding methods remain under-studied. Standard autoregressive search procedures, such as beam search, do not directly apply to iterative denoising, where hypotheses are complete intermediate sequences rather than left-to-right prefixes. Furthermore, existing diffusion decoding procedures only provide limited control over the diversity and coverage of retained hypotheses. In this work, we introduce D5P4, a beam-style decoding method tailored to discrete diffusion models, which casts intermediate beam selection as MAP inference under a partitioned Determinantal Point Process. This yields a model-internal batch objective that balances quality and diversity without external verifiers. Experiments on open-ended generation, question answering, and mathematical reasoning show that D5P4 improves diversity and pass@k coverage while matching or surpassing baseline quality and fidelity

Graph neural networks have recently achieved remarkable success in representing graph-structured data, with rapid progress in both the node embedding and graph pooling methods. Yet, they mostly focus on capturing information from the nodes considering their connectivity, and not much work has been done in representing the edges, which are essential components of a graph. However, for tasks such as graph reconstruction and generation, as well as graph classification tasks for which the edges are important for discrimination, accurately representing edges of a given graph is crucial to the success of the graph representation learning. To this end, we propose a novel edge representation learning framework based on Dual Hypergraph Transformation (DHT), which transforms the edges of a graph into the nodes of a hypergraph. This dual hypergraph construction allows us to apply message-passing techniques for node representations to edges. After obtaining edge representations from the hypergraphs, we then cluster or drop edges to obtain holistic graph-level edge representations. We validate our edge representation learning method with hypergraphs on diverse graph datasets for graph representation and generation performance, on which our method largely outperforms existing graph representation learning methods. Moreover, our edge representation learning and pooling method also largely outperforms state-of-the-art graph pooling methods on graph classification, not only because of its accurate edge representation learning, but also due to its lossless compression of the nodes and removal of irrelevant edges for effective message-passing.

Minwise hashing is a fundamental and one of the most successful hashing algorithm in the literature. Recent advances based on the idea of densification~Proc:OneHashLSH_ICML14,Proc:Shrivastava_UAI14 have shown that it is possible to compute k minwise hashes, of a vector with d nonzeros, in mere (d + k) computations, a significant improvement over the classical O(dk). These advances have led to an algorithmic improvement in the query complexity of traditional indexing algorithms based on minwise hashing. Unfortunately, the variance of the current densification techniques is unnecessarily high, which leads to significantly poor accuracy compared to vanilla minwise hashing, especially when the data is sparse. In this paper, we provide a novel densification scheme which relies on carefully tailored 2-universal hashes. We show that the proposed scheme is variance-optimal, and without losing the runtime efficiency, it is significantly more accurate than existing densification techniques. As a result, we obtain a significantly efficient hashing scheme which has the same variance and collision probability as minwise hashing. Experimental evaluations on real sparse and high-dimensional datasets validate our claims. We believe that given the significant advantages, our method will replace minwise hashing implementations in practice.

Mixture-of-Experts (MoE) and looped architectures scale models along two orthogonal axes, namely parameter capacity and effective depth. However, mainstream looped architectures rely on dense backbones that couple parameter count with per-token FLOPs, which makes it impossible to isolate the effect of iterative computation under matched budgets. To this end, we present LoopMoE, a looped MoE language model that integrates sparse routing with iterative weight-shared computation through two designs. The first is IterAdaLN, which resolves weight-sharing symmetry via a modulation signal jointly conditioned on the iteration index and the per-token hidden state. The second is a capacity-balancing strategy that recovers the attention-to-FFN active parameter ratio of well-tuned non-looped references. Together, these designs enable the first strictly controlled, head-to-head evaluation of a looped MoE against a Vanilla MoE under identical total parameters, per-token FLOPs, and active sublayer ratios. At the 3B scale, LoopMoE outperforms the Vanilla MoE on 8 of 9 downstream benchmarks with an average improvement exceeding 1 point. At the 9B scale, LoopMoE continues to outperform the matched Vanilla MoE, indicating that the architectural gain persists at larger scale. Our work establishes a controlled synthesis of sparsity and recurrence, and suggests a promising direction for looped language models.

We present the first framework to solve linear inverse problems leveraging pre-trained latent diffusion models. Previously proposed algorithms (such as DPS and DDRM) only apply to pixel-space diffusion models. We theoretically analyze our algorithm showing provable sample recovery in a linear model setting. The algorithmic insight obtained from our analysis extends to more general settings often considered in practice. Experimentally, we outperform previously proposed posterior sampling algorithms in a wide variety of problems including random inpainting, block inpainting, denoising, deblurring, destriping, and super-resolution.

Graph-based techniques and spectral graph theory have enriched the field of machine learning with a variety of critical advances. A central object in the analysis is the graph Laplacian L, which encodes the structure of the graph. We consider the problem of learning over this Laplacian in a distributed streaming setting, where new edges of the graph are observed in real time by a network of workers. In this setting, it is hard to learn quickly or approximately while keeping a distributed representation of L. To address this challenge, we present a novel algorithm, GSQUEAK, which efficiently sparsifies the Laplacian by maintaining a small subset of effective resistances. We show that our algorithm produces sparsifiers with strong spectral approximation guarantees, all while processing edges in a single pass and in a distributed fashion.

Modern ML applications increasingly rely on complex deep learning models and large datasets. There has been an exponential growth in the amount of computation needed to train the largest models. Therefore, to scale computation and data, these models are inevitably trained in a distributed manner in clusters of nodes, and their updates are aggregated before being applied to the model. However, a distributed setup is prone to Byzantine failures of individual nodes, components, and software. With data augmentation added to these settings, there is a critical need for robust and efficient aggregation systems. We define the quality of workers as reconstruction ratios in (0,1], and formulate aggregation as a Maximum Likelihood Estimation procedure using Beta densities. We show that the Regularized form of log-likelihood wrt subspace can be approximately solved using iterative least squares solver, and provide convergence guarantees using recent Convex Optimization landscape results. Our empirical findings demonstrate that our approach significantly enhances the robustness of state-of-the-art Byzantine resilient aggregators. We evaluate our method in a distributed setup with a parameter server, and show simultaneous improvements in communication efficiency and accuracy across various tasks. The code is publicly available at https://github.com/hamidralmasi/FlagAggregator

Despite the strong performance of large language models (LLMs) across a wide range of tasks, they still have reliability issues. Previous studies indicate that strong LLMs like GPT-4-turbo excel in evaluating the reliability of responses from LLMs, but face efficiency and local deployment issues. Thus, to enable weak LLMs to effectively assess the reliability of LLM responses, we propose a novel cross-query-comparison-based method called Meta Ranking (MR). Unlike previous few-shot methods that solely based on in-context learning capabilities in LLMs, MR assesses reliability by pairwisely ranking the target query-response pair with multiple reference query-response pairs. We found that MR is highly effective in error detection for LLM responses, where weak LLMs, such as Phi-2, could surpass strong baselines like GPT-3.5-turbo, requiring only five reference samples and significantly improving efficiency. We further demonstrate that MR can enhance strong LLMs' performance in two practical applications: model cascading and instruction tuning. In model cascading, we combine open- and closed-source LLMs to achieve performance comparable to GPT-4-turbo with lower costs. In instruction tuning, we use MR for iterative training data filtering, significantly reducing data processing time and enabling LLaMA-7B and Phi-2 to surpass Alpaca-13B with fewer training tokens. These results underscore the high potential of MR in both efficiency and effectiveness.

Pass@k is widely used to report the reasoning performance of LLMs, but it often produces unstable and potentially misleading rankings, especially when the number of trials (samples) is limited and computational resources are constrained. We present a principled Bayesian evaluation framework that replaces Pass@k and average accuracy over N trials (avg@N) with posterior estimates of a model's underlying success probability and credible intervals, yielding stable rankings and a transparent decision rule for differences. Evaluation outcomes are modeled as categorical (not just 0/1) with a Dirichlet prior, giving closed-form expressions for the posterior mean and uncertainty of any weighted rubric and enabling the use of prior evidence when appropriate. Theoretically, under a uniform prior, the Bayesian posterior mean is order-equivalent to average accuracy (Pass@1), explaining its empirical robustness while adding principled uncertainty. Empirically, in simulations with known ground-truth success rates and on AIME'24/'25, HMMT'25, and BrUMO'25, the posterior-based procedure achieves faster convergence and greater rank stability than Pass@k and recent variants, enabling reliable comparisons at far smaller sample counts. The framework clarifies when observed gaps are statistically meaningful (non-overlapping credible intervals) versus noise, and it naturally extends to graded, rubric-based evaluations. Together, these results recommend replacing Pass@k for LLM evaluation and ranking with a posterior-based, compute-efficient protocol that unifies binary and non-binary evaluation while making uncertainty explicit. Source code is available at https://github.com/mohsenhariri/scorio

In this work, we provide a systematic survey of Discrete Diffusion Language Models (dLLMs) and Discrete Diffusion Multimodal Language Models (dMLLMs). Unlike autoregressive (AR) models, dLLMs and dMLLMs adopt a multi-token, parallel decoding paradigm using full attention and a denoising-based generation strategy. This paradigm naturally enables parallel generation, fine-grained output controllability, and dynamic, response-aware perception. These capabilities are previously difficult to achieve with AR models. Recently, a growing number of industrial-scale proprietary d(M)LLMs, as well as a large number of open-source academic d(M)LLMs, have demonstrated performance comparable to their autoregressive counterparts, while achieving up to 10x acceleration in inference speed. The advancement of discrete diffusion LLMs and MLLMs has been largely driven by progress in two domains. The first is the development of autoregressive LLMs and MLLMs, which has accumulated vast amounts of data, benchmarks, and foundational infrastructure for training and inference. The second contributing domain is the evolution of the mathematical models underlying discrete diffusion. Together, these advancements have catalyzed a surge in dLLMs and dMLLMs research in early 2025. In this work, we present a comprehensive overview of the research in the dLLM and dMLLM domains. We trace the historical development of dLLMs and dMLLMs, formalize the underlying mathematical frameworks, and categorize representative models. We further analyze key techniques for training and inference, and summarize emerging applications across language, vision-language, and biological domains. We conclude by discussing future directions for research and deployment. Paper collection: https://github.com/LiQiiiii/DLLM-Survey

When one ball strikes another, then another, video models should predict the consequences of each bounce. In controlled experiments on multi-ball hard-sphere dynamics, we find that the performance of standard bidirectional video diffusion degrades as the causal chain lengthens, even when provided more denoising steps. In a length-matched single-ball control, where ball-ball interactions are absent, the degradation largely disappears, isolating dependent-event structure rather than video length as the cause. Across intervention studies, methods that increase effective serial computation improve performance disproportionately, including autoregressive/blockwise generation and architectural depth. We identify this pattern as the seriality gap: a mismatch between tasks requiring growing serial computation and video diffusion models whose denoising loop does not provide scalable serial compute. We then prove that, for deterministic video prediction, denoising steps do not add serial computation beyond the backbone, indicating a structural obstacle for video diffusion on serial reasoning and simulation tasks.

The memory-bound nature of the decoding stage of large language model (LLM) inference incurs significant latency. Layer-wise training-free network pruning approaches guided by the Hessian have been a prominent solution to this problem, as pruning reduces the number of nonzero parameters read from memory during decoding. Nevertheless, typical methods in this line compute the Hessian using pre-collected natural sequences, whereas the model is fed self-generated tokens during decoding, creating a distribution shift between the two sequences. The Hessian calculated on the natural sequence is different from that calculated on the generated sequence. We observe that this discrepancy causes the activation distribution during generation to deviate from that used for pruning, further hurting the pruned model performance. Moreover, most existing LLM pruning methods that bring actual speedup primarily target the sparse matrix-matrix (SpMM) multiplication, providing limited support for the sparse matrix-vector (SpMV) operations, which dominate decoding. To solve these problems, we introduce SparseDecoding, a principled decoding-aware pruning framework tailored for accurate and efficient LLM decoding. Specifically, at the algorithmic axis, SparseDecoding constructs calibration matrices from layer-wise activations collected during the dense-model autoregressive generation, excluding prefill, thereby aligning the pruning objective with the decoding activations. At the system axis, we develop an optimized N:M sparse matrix-vector kernel with bitmask indexing and fixed-step traversal. Substantial empirical results on representative LLMs (Llama-3.1-8B, Llama-3.3-70B, Qwen3-14B / 32B) demonstrate that our method consistently outperforms standard fixed-text calibration on the long-form generation benchmarks while achieving up to 1.48x end-to-end wall-clock decoding speedup on A100 GPUs.

With the success of Diffusion Models for image generation, the technologies also have revolutionized the aesthetic Quick Response (QR) code generation. Despite significant improvements in visual attractiveness for the beautified codes, their scannabilities are usually sacrificed and thus hinder their practical uses in real-world scenarios. To address this issue, we propose a novel training-free Diffusion-based QR Code generator (DiffQRCoder) to effectively craft both scannable and visually pleasing QR codes. The proposed approach introduces Scanning-Robust Perceptual Guidance (SRPG), a new diffusion guidance for Diffusion Models to guarantee the generated aesthetic codes to obey the ground-truth QR codes while maintaining their attractiveness during the denoising process. Additionally, we present another post-processing technique, Scanning Robust Manifold Projected Gradient Descent (SR-MPGD), to further enhance their scanning robustness through iterative latent space optimization. With extensive experiments, the results demonstrate that our approach not only outperforms other compared methods in Scanning Success Rate (SSR) with better or comparable CLIP aesthetic score (CLIP-aes.) but also significantly improves the SSR of the ControlNet-only approach from 60% to 99%. The subjective evaluation indicates that our approach achieves promising visual attractiveness to users as well. Finally, even with different scanning angles and the most rigorous error tolerance settings, our approach robustly achieves over 95% SSR, demonstrating its capability for real-world applications. Our project page is available at https://jwliao1209.github.io/DiffQRCoder.

Discrete diffusion models with absorbing processes have shown promise in language modeling. The key quantities to be estimated are the ratios between the marginal probabilities of two transitive states at all timesteps, called the concrete score. In this paper, we reveal that the concrete score in absorbing diffusion can be expressed as conditional probabilities of clean data, multiplied by a time-dependent scalar in an analytic form. Motivated by this finding, we propose reparameterized absorbing discrete diffusion (RADD), a dedicated diffusion model without time-condition that characterizes the time-independent conditional probabilities. Besides its simplicity, RADD can reduce the number of function evaluations (NFEs) by caching the output of the time-independent network when the noisy sample remains unchanged in a sampling interval. Empirically, RADD is up to 3.5 times faster while achieving similar performance with the strongest baseline. Built upon the new perspective of conditional distributions, we further unify absorbing discrete diffusion and any-order autoregressive models (AO-ARMs), showing that the upper bound on the negative log-likelihood for the diffusion model can be interpreted as an expected negative log-likelihood for AO-ARMs. Further, our RADD models achieve SOTA performance among diffusion models on 5 zero-shot language modeling benchmarks (measured by perplexity) at the GPT-2 scale. Our code is available at https://github.com/ML-GSAI/RADD.

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithm converges to the global optimal solution and that the L_2 distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

State-of-the-art neural algorithmic reasoners make use of message passing in graph neural networks (GNNs). But typical GNNs blur the distinction between the definition and invocation of the message function, forcing a node to send messages to its neighbours at every layer, synchronously. When applying GNNs to learn to execute dynamic programming algorithms, however, on most steps only a handful of the nodes would have meaningful updates to send. One, hence, runs the risk of inefficiencies by sending too much irrelevant data across the graph -- with many intermediate GNN steps having to learn identity functions. In this work, we explicitly separate the concepts of node state update and message function invocation. With this separation, we obtain a mathematical formulation that allows us to reason about asynchronous computation in both algorithms and neural networks.

Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an O(n^3) Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form h_θ(A_q),R, where A_q is a normalized magnetic operator, h_θ a learnable scalar spectral response, and R a block of random probes. Because the PE is a matrix function of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that k = O(log(1/varepsilon)) block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with 1/s Monte-Carlo error, and the undirected q{=}0 case improves heterophilous benchmarks over no-PE and polynomial baselines.

Diffusion models have demonstrated strong generative performance when using guidance methods such as classifier-free guidance (CFG), which enhance output quality by modifying the sampling trajectory. These methods typically improve a target output by intentionally degrading another, often the unconditional output, using heuristic perturbation functions such as identity mixing or blurred conditions. However, these approaches lack a principled foundation and rely on manually designed distortions. In this work, we propose Adversarial Sinkhorn Attention Guidance (ASAG), a novel method that reinterprets attention scores in diffusion models through the lens of optimal transport and intentionally disrupt the transport cost via Sinkhorn algorithm. Instead of naively corrupting the attention mechanism, ASAG injects an adversarial cost within self-attention layers to reduce pixel-wise similarity between queries and keys. This deliberate degradation weakens misleading attention alignments and leads to improved conditional and unconditional sample quality. ASAG shows consistent improvements in text-to-image diffusion, and enhances controllability and fidelity in downstream applications such as IP-Adapter and ControlNet. The method is lightweight, plug-and-play, and improves reliability without requiring any model retraining.

In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. We present interactive shuffle protocols for stochastic convex optimization. Our protocols rely on a new noninteractive protocol for summing vectors of bounded ell_2 norm. By combining this sum subroutine with mini-batch stochastic gradient descent, accelerated gradient descent, and Nesterov's smoothing method, we obtain loss guarantees for a variety of convex loss functions that significantly improve on those of the local model and sometimes match those of the central model.

Auto-regressive Large Language Models (LLMs) achieve strong performance on coding tasks, but incur high memory and inference costs. Diffusion-based language models (d-LLMs) offer bounded inference cost via iterative denoising, but their behavior under post-training quantization (PTQ) has been sparsely explored. We investigate the application and robustness of PTQ techniques, specifically GPTQ and a modified Hessian-Aware Quantization (HAWQ) algorithm, on a diffusion-based coding LLM (CoDA) and observe that these methods applied to CoDA exhibit greater robustness at low bitwidths compared to Qwen3-1.7B, its auto-regressive counterpart, under a standardized evaluation pipeline. We find that in our setup, CoDA exhibits greater robustness at low bitwidths (2-4 bits), with smaller accuracy degradation across HumanEval and MBPP benchmarks. Additionally, mixed-precision configurations derived from HAWQ provide smooth trade-offs across accuracy, latency, and memory. The results suggest that diffusion LLMs may offer advantages for efficient deployment due to more quantization-resilience.

Reinforcement Learning with Verifiable Rewards (RLVR) has achieved remarkable success in improving autoregressive models, especially in domains requiring correctness like mathematical reasoning and code generation. However, directly applying such paradigms to Diffusion Large Language Models (dLLMs) is fundamentally hindered by the intractability of exact likelihood computation, which forces existing methods to rely on high-variance approximations. To bridge this gap, we propose Likelihood-Free Policy Optimization (LFPO), a native framework that maps the concept of vector field flow matching to the discrete token space. Specifically, LFPO formulates alignment as geometric velocity rectification, which directly optimizes denoising logits via contrastive updates. This design effectively bypasses the errors inherent in likelihood approximation, yielding the precise gradient estimation. Furthermore, LFPO enforce consistency by predicting final solutions from intermediate steps, effectively straightening the probability flow to enable high-quality generation with significantly fewer iterations. Extensive experiments demonstrate that LFPO not only outperforms state-of-the-art baselines on code and reasoning benchmarks but also accelerates inference by approximately 20% through reduced diffusion steps.

Block Diffusion Models (BDMs) support parallel generation, flexible-length output, and KV caching, making them promising for efficient document parsing. However, existing BDMs bind denoising and cache commitment to fixed block boundaries: parallelism shrinks during intra-block denoising, while generated tokens cannot be cached until the whole block is completed. Moreover, intra-block bidirectional denoising conflicts with inter-block autoregression, creating inconsistent information flow that can challenge structure-sensitive recognition. We propose the Prefix-Adaptive Block Diffusion Model (PA-BDM), which replaces intra-block bidirectional denoising with causal denoising from prefix to suffix and treats the block size as a maximum candidate range rather than a fixed commitment unit. PA-BDM uses Confidence-gated Structural Loss (CSL) to build low-entropy prefixes before extending training to longer continuations. During inference, Progressive Prefix Commitment (PPC) then dynamically commits the longest reliable prefix into the KV cache and resets the next candidate range from the updated prefix, restoring a large parallel decoding space at each step. Experiments show that the 3B PA-BDM achieves higher recognition scores on several benchmarks and improves inference throughput by 71.6\% over the 2.5B MinerU-Diffusion.

We propose a new method for solving imaging inverse problems using text-to-image latent diffusion models as general priors. Existing methods using latent diffusion models for inverse problems typically rely on simple null text prompts, which can lead to suboptimal performance. To address this limitation, we introduce a method for prompt tuning, which jointly optimizes the text embedding on-the-fly while running the reverse diffusion process. This allows us to generate images that are more faithful to the diffusion prior. In addition, we propose a method to keep the evolution of latent variables within the range space of the encoder, by projection. This helps to reduce image artifacts, a major problem when using latent diffusion models instead of pixel-based diffusion models. Our combined method, called P2L, outperforms both image- and latent-diffusion model-based inverse problem solvers on a variety of tasks, such as super-resolution, deblurring, and inpainting.

Various common deep learning architectures, such as LSTMs, GRUs, Resnets and Highway Networks, employ state passthrough connections that support training with high feed-forward depth or recurrence over many time steps. These "Passthrough Networks" architectures also enable the decoupling of the network state size from the number of parameters of the network, a possibility has been studied by Sak2014 with their low-rank parametrization of the LSTM. In this work we extend this line of research, proposing effective, low-rank and low-rank plus diagonal matrix parametrizations for Passthrough Networks which exploit this decoupling property, reducing the data complexity and memory requirements of the network while preserving its memory capacity. This is particularly beneficial in low-resource settings as it supports expressive models with a compact parametrization less susceptible to overfitting. We present competitive experimental results on several tasks, including language modeling and a near state of the art result on sequential randomly-permuted MNIST classification, a hard task on natural data.

Recent works on compression of large language models (LLM) using quantization considered reparameterizing the architecture such that weights are distributed on the sphere. This demonstratively improves the ability to quantize by increasing the mathematical notion of coherence, resulting in fewer weight outliers without affecting the network output. In this work, we aim to further exploit this spherical geometry of the weights when performing quantization by considering Pyramid Vector Quantization (PVQ) for large language models. Arranging points evenly on the sphere is notoriously difficult, especially in high dimensions, and in case approximate solutions exists, representing points explicitly in a codebook is typically not feasible due to its additional memory cost. Instead, PVQ uses a fixed integer lattice on the sphere by projecting points onto the 1-sphere, which allows for efficient encoding and decoding without requiring an explicit codebook in memory. To obtain a practical algorithm, we propose to combine PVQ with scale quantization for which we derive theoretically optimal quantizations, under empirically verified assumptions. Further, we extend pyramid vector quantization to use Hessian information to minimize quantization error under expected feature activations, instead of only relying on weight magnitudes. Experimentally, we achieves state-of-the-art quantization performance with pareto-optimal trade-off between performance and bits per weight and bits per activation, compared to compared methods. On weight-only, we find that we can quantize a Llama-3 70B model to 3.25 bits per weight and retain 98\% accuracy on downstream tasks.

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by min{mathcal{O}(R^2/ε^2), mathcal{O}(m)} to obtain an ε-approximation of the PPR vector, where m denotes the number of edges in the graph and R is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only mathcal{O}(1/sqrtα) such linear systems, where α is the damping factor. When 1/ε^2ll m, this implies the existence of an algorithm that computes an epsilon -approximation of the PPR vector with an overall time complexity of mathcal{O}left(R^2 / (sqrtαε^2)right), independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.

Diffusion models have emerged as a promising alternative to autoregressive models in modeling discrete categorical data. Yet diffusion models that directly work on discrete data space do not fully exploit the power of iterative refinement, as the signals are lost during the transition between discrete states. Existing continuous diffusion models for discrete data have limited performance compared to discrete approaches, and the unclear link between them restricts the development of diffusion models for discrete data. In this work, we propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution. We establish a connection between the discrete diffusion and continuous flow on the statistical manifold, and building on the analogy, we introduce a simple design for the diffusion process that generalizes previous discrete diffusion models. We further propose a simulation-free training framework based on radial symmetry and a simple technique to address the high dimensionality of the manifold. Comprehensive experiments on language modeling benchmarks and other modalities show that our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models. Codes available at https://github.com/harryjo97/RDLM{https://github.com/harryjo97/RDLM}.

Despite the recent advancements, conditional image generation still faces challenges of cost, generalizability, and the need for task-specific training. In this paper, we propose Manifold Preserving Guided Diffusion (MPGD), a training-free conditional generation framework that leverages pretrained diffusion models and off-the-shelf neural networks with minimal additional inference cost for a broad range of tasks. Specifically, we leverage the manifold hypothesis to refine the guided diffusion steps and introduce a shortcut algorithm in the process. We then propose two methods for on-manifold training-free guidance using pre-trained autoencoders and demonstrate that our shortcut inherently preserves the manifolds when applied to latent diffusion models. Our experiments show that MPGD is efficient and effective for solving a variety of conditional generation applications in low-compute settings, and can consistently offer up to 3.8x speed-ups with the same number of diffusion steps while maintaining high sample quality compared to the baselines.

Discrete diffusion models for categorical generation are defined by a corruption kernel, which determines the intermediate state space and the associated reverse prediction problem. We study uniform discrete diffusion and ask whether its training objective and reverse transitions can be enriched without changing the underlying categorical corruption process. We introduce Simplax, an exact Dirichlet--categorical augmentation that couples each corrupted categorical state with an auxiliary simplex-valued variable while preserving the original uniform diffusion process as its categorical marginal. This augmentation yields a tractable Rao--Blackwellized reverse-bridge objective and a corresponding stochastic reverse sampler, while retaining the corrupted categorical state as the denoiser input. Empirically, Simplax improves the generative perplexity--entropy tradeoff on unconditional OpenWebText generation. On Sudoku, a model trained exclusively on 30-clue puzzles achieves the highest accuracy among the compared methods across all evaluated clue densities, including the minimum uniquely solvable 17-clue regime, and also achieves the highest validity in unconditional generation.