Haggai Maron

Omri Puny, Derek Lim, Bobak T. Kiani et al. · mit, nvidia

Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this hierarchy has propelled significant advances in GNN analysis and architecture developments, it suffers from several significant limitations. These include a complex definition that lacks direct guidance for model improvement and a WL hierarchy that is too coarse to study current GNNs. This paper introduces an alternative expressive power hierarchy based on the ability of GNNs to calculate equivariant polynomials of a certain degree. As a first step, we provide a full characterization of all equivariant graph polynomials by introducing a concrete basis, significantly generalizing previous results. Each basis element corresponds to a specific multi-graph, and its computation over some graph data input corresponds to a tensor contraction problem. Second, we propose algorithmic tools for evaluating the expressiveness of GNNs using tensor contraction sequences, and calculate the expressive power of popular GNNs. Finally, we enhance the expressivity of common GNN architectures by adding polynomial features or additional operations / aggregations inspired by our theory. These enhanced GNNs demonstrate state-of-the-art results in experiments across multiple graph learning benchmarks.

Ben Finkelshtein, Chaim Baskin, Haggai Maron et al. · nvidia

Equivariance to permutations and rigid motions is an important inductive bias for various 3D learning problems. Recently it has been shown that the equivariant Tensor Field Network architecture is universal -- it can approximate any equivariant function. In this paper we suggest a much simpler architecture, prove that it enjoys the same universality guarantees and evaluate its performance on Modelnet40. The code to reproduce our experiments is available at \url{https://github.com/simpleinvariance/UniversalNetwork}

Aviv Navon, Aviv Shamsian, Idan Achituve et al. · nvidia

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

Eli A. Meirom, Haggai Maron, Shie Mannor et al. · nvidia

Quantum Computing (QC) stands to revolutionize computing, but is currently still limited. To develop and test quantum algorithms today, quantum circuits are often simulated on classical computers. Simulating a complex quantum circuit requires computing the contraction of a large network of tensors. The order (path) of contraction can have a drastic effect on the computing cost, but finding an efficient order is a challenging combinatorial optimization problem. We propose a Reinforcement Learning (RL) approach combined with Graph Neural Networks (GNN) to address the contraction ordering problem. The problem is extremely challenging due to the huge search space, the heavy-tailed reward distribution, and the challenging credit assignment. We show how a carefully implemented RL-agent that uses a GNN as the basic policy construct can address these challenges and obtain significant improvements over state-of-the-art techniques in three varieties of circuits, including the largest scale networks used in contemporary QC.

Sohir Maskey, Ali Parviz, Maximilian Thiessen et al. · mit, nvidia

Graph neural networks (GNNs) are the primary tool for processing graph-structured data. Unfortunately, the most commonly used GNNs, called Message Passing Neural Networks (MPNNs) suffer from several fundamental limitations. To overcome these limitations, recent works have adapted the idea of positional encodings to graph data. This paper draws inspiration from the recent success of Laplacian-based positional encoding and defines a novel family of positional encoding schemes for graphs. We accomplish this by generalizing the optimization problem that defines the Laplace embedding to more general dissimilarity functions rather than the 2-norm used in the original formulation. This family of positional encodings is then instantiated by considering p-norms. We discuss a method for calculating these positional encoding schemes, implement it in PyTorch and demonstrate how the resulting positional encoding captures different properties of the graph. Furthermore, we demonstrate that this novel family of positional encodings can improve the expressive power of MPNNs. Lastly, we present preliminary experimental results.

Fabrizio Frasca, Beatrice Bevilacqua, Michael M. Bronstein et al. · nvidia

Subgraph GNNs are a recent class of expressive Graph Neural Networks (GNNs) which model graphs as collections of subgraphs. So far, the design space of possible Subgraph GNN architectures as well as their basic theoretical properties are still largely unexplored. In this paper, we study the most prominent form of subgraph methods, which employs node-based subgraph selection policies such as ego-networks or node marking and deletion. We address two central questions: (1) What is the upper-bound of the expressive power of these methods? and (2) What is the family of equivariant message passing layers on these sets of subgraphs?. Our first step in answering these questions is a novel symmetry analysis which shows that modelling the symmetries of node-based subgraph collections requires a significantly smaller symmetry group than the one adopted in previous works. This analysis is then used to establish a link between Subgraph GNNs and Invariant Graph Networks (IGNs). We answer the questions above by first bounding the expressive power of subgraph methods by 3-WL, and then proposing a general family of message-passing layers for subgraph methods that generalises all previous node-based Subgraph GNNs. Finally, we design a novel Subgraph GNN dubbed SUN, which theoretically unifies previous architectures while providing better empirical performance on multiple benchmarks.

Moshe Eliasof, Fabrizio Frasca, Beatrice Bevilacqua et al. · nvidia

Two main families of node feature augmentation schemes have been explored for enhancing GNNs: random features and spectral positional encoding. Surprisingly, however, there is still no clear understanding of the relation between these two augmentation schemes. Here we propose a novel family of positional encoding schemes which draws a link between the above two approaches and improves over both. The new approach, named Random Feature Propagation (RFP), is inspired by the power iteration method and its generalizations. It concatenates several intermediate steps of an iterative algorithm for computing the dominant eigenvectors of a propagation matrix, starting from random node features. Notably, these propagation steps are based on graph-dependent propagation operators that can be either predefined or learned. We explore the theoretical and empirical benefits of RFP. First, we provide theoretical justifications for using random features, for incorporating early propagation steps, and for using multiple random initializations. Then, we empirically demonstrate that RFP significantly outperforms both spectral PE and random features in multiple node classification and graph classification benchmarks.

Yam Eitan, Yoav Gelberg, Guy Bar-Shalom et al.

Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes. Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains. In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology. In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs. To the best of our knowledge, this is the first work to study the expressivity of TDL from a \emph{topological} perspective. In the second part of the paper, we develop two new classes of architectures -- multi-cellular networks (MCN) and scalable MCN (SMCN) -- which draw inspiration from expressive GNNs. MCN can reach full expressivity, but scaling it to large data objects can be computationally expansive. Designed as a more scalable alternative, SMCN still mitigates many of HOMP's expressivity limitations. Finally, we create new benchmarks for evaluating models based on their ability to learn topological properties of complexes. We then evaluate SMCN on these benchmarks and on real-world graph datasets, demonstrating improvements over both HOMP baselines and expressive graph methods, highlighting the value of expressively leveraging topological information. Code and data are available at https://github.com/yoavgelberg/SMCN.

Xiaolong Han, Zehong Wang, Bo Zhao et al.

Neural network weights are typically viewed as the end product of training, while most deep learning research focuses on data, features, and architectures. However, recent advances show that the set of all possible weight values (weight space) itself contains rich structure: pretrained models form organized distributions, exhibit symmetries, and can be embedded, compared, or even generated. Understanding such structures has tremendous impact on how neural networks are analyzed and compared, and on how knowledge is transferred across models, beyond individual training instances. This emerging research direction, which we refer to as Weight Space Learning (WSL), treats neural weights as a meaningful domain for analysis and modeling. This survey provides the first unified taxonomy of WSL. We categorize existing methods into three core dimensions: Weight Space Understanding (WSU), which studies the geometry and symmetries of weights; Weight Space Representation (WSR), which learns embeddings over model weights; and Weight Space Generation (WSG), which synthesizes new weights through hypernetworks or generative models. We further show how these developments enable practical applications, including model retrieval, continual and federated learning, neural architecture search, and data-free reconstruction. By consolidating fragmented progress under a coherent framework, this survey highlights weight space as a learnable, structured domain with growing impact across model analysis, transferring, and weight generation. We release an accompanying resource at https://github.com/Zehong-Wang/Awesome-Weight-Space-Learning.

Yaniv Galron, Hadar Sinai, Haggai Maron et al.

Graph Neural Networks (GNNs) have demonstrated remarkable success in various domains such as social networks, molecular chemistry, and more. A crucial component of GNNs is the pooling procedure, in which the node features calculated by the model are combined to form an informative final descriptor to be used for the downstream task. However, previous graph pooling schemes rely on the last GNN layer features as an input to the pooling or classifier layers, potentially under-utilizing important activations of previous layers produced during the forward pass of the model, which we regard as historical graph activations. This gap is particularly pronounced in cases where a node's representation can shift significantly over the course of many graph neural layers, and worsened by graph-specific challenges such as over-smoothing in deep architectures. To bridge this gap, we introduce HISTOGRAPH, a novel two-stage attention-based final aggregation layer that first applies a unified layer-wise attention over intermediate activations, followed by node-wise attention. By modeling the evolution of node representations across layers, our HISTOGRAPH leverages both the activation history of nodes and the graph structure to refine features used for final prediction. Empirical results on multiple graph classification benchmarks demonstrate that HISTOGRAPH offers strong performance that consistently improves traditional techniques, with particularly strong robustness in deep GNNs.

Ali Taghibakhshi, Mingyuan Ma, Ashwath Aithal et al. · nvidia

Cross-device user matching is a critical problem in numerous domains, including advertising, recommender systems, and cybersecurity. It involves identifying and linking different devices belonging to the same person, utilizing sequence logs. Previous data mining techniques have struggled to address the long-range dependencies and higher-order connections between the logs. Recently, researchers have modeled this problem as a graph problem and proposed a two-tier graph contextual embedding (TGCE) neural network architecture, which outperforms previous methods. In this paper, we propose a novel hierarchical graph neural network architecture (HGNN), which has a more computationally efficient second level design than TGCE. Furthermore, we introduce a cross-attention (Cross-Att) mechanism in our model, which improves performance by 5% compared to the state-of-the-art TGCE method.

Aviv Shamsian, David W. Zhang, Aviv Navon et al.

Learning in weight spaces, where neural networks process the weights of other deep neural networks, has emerged as a promising research direction with applications in various fields, from analyzing and editing neural fields and implicit neural representations, to network pruning and quantization. Recent works designed architectures for effective learning in that space, which takes into account its unique, permutation-equivariant, structure. Unfortunately, so far these architectures suffer from severe overfitting and were shown to benefit from large datasets. This poses a significant challenge because generating data for this learning setup is laborious and time-consuming since each data sample is a full set of network weights that has to be trained. In this paper, we address this difficulty by investigating data augmentations for weight spaces, a set of techniques that enable generating new data examples on the fly without having to train additional input weight space elements. We first review several recently proposed data augmentation schemes %that were proposed recently and divide them into categories. We then introduce a novel augmentation scheme based on the Mixup method. We evaluate the performance of these techniques on existing benchmarks as well as new benchmarks we generate, which can be valuable for future studies.

Aviv Navon, Aviv Shamsian, Ethan Fetaya et al.

Permutation symmetries of deep networks make basic operations like model merging and similarity estimation challenging. In many cases, aligning the weights of the networks, i.e., finding optimal permutations between their weights, is necessary. Unfortunately, weight alignment is an NP-hard problem. Prior research has mainly focused on solving relaxed versions of the alignment problem, leading to either time-consuming methods or sub-optimal solutions. To accelerate the alignment process and improve its quality, we propose a novel framework aimed at learning to solve the weight alignment problem, which we name Deep-Align. To that end, we first prove that weight alignment adheres to two fundamental symmetries and then, propose a deep architecture that respects these symmetries. Notably, our framework does not require any labeled data. We provide a theoretical analysis of our approach and evaluate Deep-Align on several types of network architectures and learning setups. Our experimental results indicate that a feed-forward pass with Deep-Align produces better or equivalent alignments compared to those produced by current optimization algorithms. Additionally, our alignments can be used as an effective initialization for other methods, leading to improved solutions with a significant speedup in convergence.

Dvir Samuel, Rami Ben-Ari, Nir Darshan et al.

Text-to-image diffusion models show great potential in synthesizing a large variety of concepts in new compositions and scenarios. However, the latent space of initial seeds is still not well understood and its structure was shown to impact the generation of various concepts. Specifically, simple operations like interpolation and finding the centroid of a set of seeds perform poorly when using standard Euclidean or spherical metrics in the latent space. This paper makes the observation that, in current training procedures, diffusion models observed inputs with a narrow range of norm values. This has strong implications for methods that rely on seed manipulation for image generation, with applications to few-shot and long-tail learning tasks. To address this issue, we propose a novel method for interpolating between two seeds and demonstrate that it defines a new non-Euclidean metric that takes into account a norm-based prior on seeds. We describe a simple yet efficient algorithm for approximating this interpolation procedure and use it to further define centroids in the latent seed space. We show that our new interpolation and centroid techniques significantly enhance the generation of rare concept images. This further leads to state-of-the-art performance on few-shot and long-tail benchmarks, improving prior approaches in terms of generation speed, image quality, and semantic content.

Yewei Liu, Xiyuan Wang, Yansheng Mao et al.

We propose SHINE (Scalable Hyper In-context NEtwork), a scalable hypernetwork that can map diverse meaningful contexts into high-quality LoRA adapters for large language models (LLM). By reusing the frozen LLM's own parameters in an in-context hypernetwork design and introducing architectural innovations, SHINE overcomes key limitations of prior hypernetworks and achieves strong expressive power with a relatively small number of parameters. We introduce a pretraining and instruction fine-tuning pipeline, and train our hypernetwork to generate high quality LoRA adapters from diverse meaningful contexts in a single forward pass. It updates LLM parameters without any fine-tuning, and immediately enables complex question answering tasks related to the context without directly accessing the context, effectively transforming in-context knowledge to in-parameter knowledge in one pass. Our work achieves outstanding results on various tasks, greatly saves time, computation and memory costs compared to SFT-based LLM adaptation, and shows great potential for scaling. Our code is available at https://github.com/Yewei-Liu/SHINE

Guy Bar-Shalom, Ami Tavory, Itay Evron et al.

Weight-space models learn directly from the parameters of neural networks, enabling tasks such as predicting their accuracy on new datasets. Naive methods -- like applying MLPs to flattened parameters -- perform poorly, making the design of better weight-space architectures a central challenge. While prior work leveraged permutation symmetries in standard networks to guide such designs, no analogous analysis or tailored architecture yet exists for Kolmogorov-Arnold Networks (KANs). In this work, we show that KANs share the same permutation symmetries as MLPs, and propose the KAN-graph, a graph representation of their computation. Building on this, we develop WS-KAN, the first weight-space architecture that learns on KANs, which naturally accounts for their symmetry. We analyze WS-KAN's expressive power, showing it can replicate an input KAN's forward pass - a standard approach for assessing expressiveness in weight-space architectures. We construct a comprehensive ``zoo'' of trained KANs spanning diverse tasks, which we use as benchmarks to empirically evaluate WS-KAN. Across all tasks, WS-KAN consistently outperforms structure-agnostic baselines, often by a substantial margin. Our code is available at https://github.com/BarSGuy/KAN-Graph-Metanetwork.

Beatrice Bevilacqua, Moshe Eliasof, Eli Meirom et al.

Subgraph GNNs are provably expressive neural architectures that learn graph representations from sets of subgraphs. Unfortunately, their applicability is hampered by the computational complexity associated with performing message passing on many subgraphs. In this paper, we consider the problem of learning to select a small subset of the large set of possible subgraphs in a data-driven fashion. We first motivate the problem by proving that there are families of WL-indistinguishable graphs for which there exist efficient subgraph selection policies: small subsets of subgraphs that can already identify all the graphs within the family. We then propose a new approach, called Policy-Learn, that learns how to select subgraphs in an iterative manner. We prove that, unlike popular random policies and prior work addressing the same problem, our architecture is able to learn the efficient policies mentioned above. Our experimental results demonstrate that Policy-Learn outperforms existing baselines across a wide range of datasets.

Hila Manor, Rinon Gal, Haggai Maron et al.

Visual analogy learning enables image manipulation through demonstration rather than textual description, allowing users to specify complex transformations difficult to articulate in words. Given a triplet $\{\mathbf{a}$, $\mathbf{a}'$, $\mathbf{b}\}$, the goal is to generate $\mathbf{b}'$ such that $\mathbf{a} : \mathbf{a}' :: \mathbf{b} : \mathbf{b}'$. Recent methods adapt text-to-image models to this task using a single Low-Rank Adaptation (LoRA) module, but they face a fundamental limitation: attempting to capture the diverse space of visual transformations within a fixed adaptation module constrains generalization capabilities. Inspired by recent work showing that LoRAs in constrained domains span meaningful, interpolatable semantic spaces, we propose LoRWeB, a novel approach that specializes the model for each analogy task at inference time through dynamic composition of learned transformation primitives, informally, choosing a point in a "space of LoRAs". We introduce two key components: (1) a learnable basis of LoRA modules, to span the space of different visual transformations, and (2) a lightweight encoder that dynamically selects and weighs these basis LoRAs based on the input analogy pair. Comprehensive evaluations demonstrate our approach achieves state-of-the-art performance and significantly improves generalization to unseen visual transformations. Our findings suggest that LoRA basis decompositions are a promising direction for flexible visual manipulation. Code and data are in https://research.nvidia.com/labs/par/lorweb

Derek Lim, Joshua Robinson, Stefanie Jegelka et al. · nvidia

Recent work has shown the utility of developing machine learning models that respect the structure and symmetries of eigenvectors. These works promote sign invariance, since for any eigenvector v the negation -v is also an eigenvector. However, we show that sign invariance is theoretically limited for tasks such as building orthogonally equivariant models and learning node positional encodings for link prediction in graphs. In this work, we demonstrate the benefits of sign equivariance for these tasks. To obtain these benefits, we develop novel sign equivariant neural network architectures. Our models are based on a new analytic characterization of sign equivariant polynomials and thus inherit provable expressiveness properties. Controlled synthetic experiments show that our networks can achieve the theoretically predicted benefits of sign equivariant models. Code is available at https://github.com/cptq/Sign-Equivariant-Nets.

Dvir Samuel, Barak Meiri, Haggai Maron et al.

Diffusion inversion is the problem of taking an image and a text prompt that describes it and finding a noise latent that would generate the exact same image. Most current deterministic inversion techniques operate by approximately solving an implicit equation and may converge slowly or yield poor reconstructed images. We formulate the problem by finding the roots of an implicit equation and devlop a method to solve it efficiently. Our solution is based on Newton-Raphson (NR), a well-known technique in numerical analysis. We show that a vanilla application of NR is computationally infeasible while naively transforming it to a computationally tractable alternative tends to converge to out-of-distribution solutions, resulting in poor reconstruction and editing. We therefore derive an efficient guided formulation that fastly converges and provides high-quality reconstructions and editing. We showcase our method on real image editing with three popular open-sourced diffusion models: Stable Diffusion, SDXL-Turbo, and Flux with different deterministic schedulers. Our solution, Guided Newton-Raphson Inversion, inverts an image within 0.4 sec (on an A100 GPU) for few-step models (SDXL-Turbo and Flux.1), opening the door for interactive image editing. We further show improved results in image interpolation and generation of rare objects.

Guy Bar-Shalom, Fabrizio Frasca, Derek Lim et al.

The automated detection of hallucinations and training data contamination is pivotal to the safe deployment of Large Language Models (LLMs). These tasks are particularly challenging in settings where no access to model internals is available. Current approaches in this setup typically leverage only the probabilities of actual tokens in the text, relying on simple task-specific heuristics. Crucially, they overlook the information contained in the full sequence of next-token probability distributions. We propose to go beyond hand-crafted decision rules by learning directly from the complete observable output of LLMs -- consisting not only of next-token probabilities, but also the full sequence of next-token distributions. We refer to this as the LLM Output Signature (LOS), and treat it as a reference data type for detecting hallucinations and data contamination. To that end, we introduce LOS-Net, a lightweight attention-based architecture trained on an efficient encoding of the LOS, which can provably approximate a broad class of existing techniques for both tasks. Empirically, LOS-Net achieves superior performance across diverse benchmarks and LLMs, while maintaining extremely low detection latency. Furthermore, it demonstrates promising transfer capabilities across datasets and LLMs. Full code is available at https://github.com/BarSGuy/Beyond-next-token-probabilities.

Guy Bar-Shalom, Fabrizio Frasca, Yaniv Galron et al.

Detecting hallucinations in Large Language Model-generated text is crucial for their safe deployment. While probing classifiers show promise, they operate on isolated layer-token pairs and are LLM-specific, limiting their effectiveness and hindering cross-LLM applications. In this paper, we introduce a novel approach to address these shortcomings. We build on the natural sequential structure of activation data in both axes (layers $\times$ tokens) and advocate treating full activation tensors akin to images. We design ACT-ViT, a Vision Transformer-inspired model that can be effectively and efficiently applied to activation tensors and supports training on data from multiple LLMs simultaneously. Through comprehensive experiments encompassing diverse LLMs and datasets, we demonstrate that ACT-ViT consistently outperforms traditional probing techniques while remaining extremely efficient for deployment. In particular, we show that our architecture benefits substantially from multi-LLM training, achieves strong zero-shot performance on unseen datasets, and can be transferred effectively to new LLMs through fine-tuning. Full code is available at https://github.com/BarSGuy/ACT-ViT.

Guy Bar-Shalom, Yam Eitan, Fabrizio Frasca et al.

Subgraph GNNs enhance message-passing GNNs expressivity by representing graphs as sets of subgraphs, demonstrating impressive performance across various tasks. However, their scalability is hindered by the need to process large numbers of subgraphs. While previous approaches attempted to generate smaller subsets of subgraphs through random or learnable sampling, these methods often yielded suboptimal selections or were limited to small subset sizes, ultimately compromising their effectiveness. This paper introduces a new Subgraph GNN framework to address these issues. Our approach diverges from most previous methods by associating subgraphs with node clusters rather than with individual nodes. We show that the resulting collection of subgraphs can be viewed as the product of coarsened and original graphs, unveiling a new connectivity structure on which we perform generalized message passing. Crucially, controlling the coarsening function enables meaningful selection of any number of subgraphs. In addition, we reveal novel permutation symmetries in the resulting node feature tensor, characterize associated linear equivariant layers, and integrate them into our Subgraph GNN. We also introduce novel node marking strategies and provide a theoretical analysis of their expressive power and other key aspects of our approach. Extensive experiments on multiple graph learning benchmarks demonstrate that our method is significantly more flexible than previous approaches, as it can seamlessly handle any number of subgraphs, while consistently outperforming baseline approaches. Our code is available at https://github.com/BarSGuy/Efficient-Subgraph-GNNs.

Derek Lim, Joshua Robinson, Lingxiao Zhao et al.

We introduce SignNet and BasisNet -- new neural architectures that are invariant to two key symmetries displayed by eigenvectors: (i) sign flips, since if $v$ is an eigenvector then so is $-v$; and (ii) more general basis symmetries, which occur in higher dimensional eigenspaces with infinitely many choices of basis eigenvectors. We prove that under certain conditions our networks are universal, i.e., they can approximate any continuous function of eigenvectors with the desired invariances. When used with Laplacian eigenvectors, our networks are provably more expressive than existing spectral methods on graphs; for instance, they subsume all spectral graph convolutions, certain spectral graph invariants, and previously proposed graph positional encodings as special cases. Experiments show that our networks significantly outperform existing baselines on molecular graph regression, learning expressive graph representations, and learning neural fields on triangle meshes. Our code is available at https://github.com/cptq/SignNet-BasisNet .

Yoav Gelberg, Yam Eitan, Michael Bronstein et al.

Long-context language modeling is increasingly constrained by the Key-Value (KV) cache, whose memory and decode-time access costs scale linearly with the prefix length. This bottleneck has motivated a range of context-compression methods, from token-level summarization to recent optimization-based KV compression methods. These post-hoc methods operate on the KV cache of a fixed pretrained model, so their effectiveness is fundamentally limited by how well the model's internal representations can be compressed. In this work, we formalize the notion of KV compressibility and show that it is a property of the learned representations, rather than of the context alone. We prove that almost any sequence-to-vector function admits both highly compressible and inherently non-compressible transformer implementations, highlighting the need to guide transformers toward compressible representations during training. Motivated by this, we propose KV-Compression Aware Training (KV-CAT), a continued pretraining procedure that incentivizes the emergence of compressible representations. We introduce a train-time KV sparsification policy that masks KV slots during training. This forces the model to use fewer KV slots and encourages it to learn representations amenable to post-hoc compression. Empirically, we show that KV-CAT improves the quality-budget tradeoff of downstream compression methods across retrieval, long-context question answering, and perplexity-based evaluation of compressed-prefix continuation.

Derek Lim, Haggai Maron, Marc T. Law et al. · nvidia

Neural networks efficiently encode learned information within their parameters. Consequently, many tasks can be unified by treating neural networks themselves as input data. When doing so, recent studies demonstrated the importance of accounting for the symmetries and geometry of parameter spaces. However, those works developed architectures tailored to specific networks such as MLPs and CNNs without normalization layers, and generalizing such architectures to other types of networks can be challenging. In this work, we overcome these challenges by building new metanetworks - neural networks that take weights from other neural networks as input. Put simply, we carefully build graphs representing the input neural networks and process the graphs using graph neural networks. Our approach, Graph Metanetworks (GMNs), generalizes to neural architectures where competing methods struggle, such as multi-head attention layers, normalization layers, convolutional layers, ResNet blocks, and group-equivariant linear layers. We prove that GMNs are expressive and equivariant to parameter permutation symmetries that leave the input neural network functions unchanged. We validate the effectiveness of our method on several metanetwork tasks over diverse neural network architectures.

Aviv Shamsian, Aviv Navon, David W. Zhang et al.

Learning in deep weight spaces (DWS), where neural networks process the weights of other neural networks, is an emerging research direction, with applications to 2D and 3D neural fields (INRs, NeRFs), as well as making inferences about other types of neural networks. Unfortunately, weight space models tend to suffer from substantial overfitting. We empirically analyze the reasons for this overfitting and find that a key reason is the lack of diversity in DWS datasets. While a given object can be represented by many different weight configurations, typical INR training sets fail to capture variability across INRs that represent the same object. To address this, we explore strategies for data augmentation in weight spaces and propose a MixUp method adapted for weight spaces. We demonstrate the effectiveness of these methods in two setups. In classification, they improve performance similarly to having up to 10 times more data. In self-supervised contrastive learning, they yield substantial 5-10% gains in downstream classification.

Yoni Kasten, Wuyue Lu, Haggai Maron · nvidia

This paper addresses the long-standing challenge of reconstructing 3D structures from videos with dynamic content. Current approaches to this problem were not designed to operate on casual videos recorded by standard cameras or require a long optimization time. Aiming to significantly improve the efficiency of previous approaches, we present TracksTo4D, a learning-based approach that enables inferring 3D structure and camera positions from dynamic content originating from casual videos using a single efficient feed-forward pass. To achieve this, we propose operating directly over 2D point tracks as input and designing an architecture tailored for processing 2D point tracks. Our proposed architecture is designed with two key principles in mind: (1) it takes into account the inherent symmetries present in the input point tracks data, and (2) it assumes that the movement patterns can be effectively represented using a low-rank approximation. TracksTo4D is trained in an unsupervised way on a dataset of casual videos utilizing only the 2D point tracks extracted from the videos, without any 3D supervision. Our experiments show that TracksTo4D can reconstruct a temporal point cloud and camera positions of the underlying video with accuracy comparable to state-of-the-art methods, while drastically reducing runtime by up to 95\%. We further show that TracksTo4D generalizes well to unseen videos of unseen semantic categories at inference time.

Guy Bar-Shalom, Beatrice Bevilacqua, Haggai Maron

In the realm of Graph Neural Networks (GNNs), two exciting research directions have recently emerged: Subgraph GNNs and Graph Transformers. In this paper, we propose an architecture that integrates both approaches, dubbed Subgraphormer, which combines the enhanced expressive power, message-passing mechanisms, and aggregation schemes from Subgraph GNNs with attention and positional encodings, arguably the most important components in Graph Transformers. Our method is based on an intriguing new connection we reveal between Subgraph GNNs and product graphs, suggesting that Subgraph GNNs can be formulated as Message Passing Neural Networks (MPNNs) operating on a product of the graph with itself. We use this formulation to design our architecture: first, we devise an attention mechanism based on the connectivity of the product graph. Following this, we propose a novel and efficient positional encoding scheme for Subgraph GNNs, which we derive as a positional encoding for the product graph. Our experimental results demonstrate significant performance improvements over both Subgraph GNNs and Graph Transformers on a wide range of datasets.

Christopher Morris, Fabrizio Frasca, Nadav Dym et al. · nvidia

Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite their practical success, our theoretical understanding of the properties of GNNs remains highly incomplete. Recent theoretical advancements primarily focus on elucidating the coarse-grained expressive power of GNNs, predominantly employing combinatorial techniques. However, these studies do not perfectly align with practice, particularly in understanding the generalization behavior of GNNs when trained with stochastic first-order optimization techniques. In this position paper, we argue that the graph machine learning community needs to shift its attention to developing a balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

Moshe Eliasof, Beatrice Bevilacqua, Carola-Bibiane Schönlieb et al.

In recent years, significant efforts have been made to refine the design of Graph Neural Network (GNN) layers, aiming to overcome diverse challenges, such as limited expressive power and oversmoothing. Despite their widespread adoption, the incorporation of off-the-shelf normalization layers like BatchNorm or InstanceNorm within a GNN architecture may not effectively capture the unique characteristics of graph-structured data, potentially reducing the expressive power of the overall architecture. Moreover, existing graph-specific normalization layers often struggle to offer substantial and consistent benefits. In this paper, we propose GRANOLA, a novel graph-adaptive normalization layer. Unlike existing normalization layers, GRANOLA normalizes node features by adapting to the specific characteristics of the graph, particularly by generating expressive representations of its neighborhood structure, obtained by leveraging the propagation of Random Node Features (RNF) in the graph. We present theoretical results that support our design choices. Our extensive empirical evaluation of various graph benchmarks underscores the superior performance of GRANOLA over existing normalization techniques. Furthermore, GRANOLA emerges as the top-performing method among all baselines within the same time complexity of Message Passing Neural Networks (MPNNs).

Jingchu Gai, Yiheng Du, Bohang Zhang et al. · nvidia

Graph spectra are an important class of structural features on graphs that have shown promising results in enhancing Graph Neural Networks (GNNs). Despite their widespread practical use, the theoretical understanding of the power of spectral invariants -- particularly their contribution to GNNs -- remains incomplete. In this paper, we address this fundamental question through the lens of homomorphism expressivity, providing a comprehensive and quantitative analysis of the expressive power of spectral invariants. Specifically, we prove that spectral invariant GNNs can homomorphism-count exactly a class of specific tree-like graphs which we refer to as parallel trees. We highlight the significance of this result in various contexts, including establishing a quantitative expressiveness hierarchy across different architectural variants, offering insights into the impact of GNN depth, and understanding the subgraph counting capabilities of spectral invariant GNNs. In particular, our results significantly extend Arvind et al. (2024) and settle their open questions. Finally, we generalize our analysis to higher-order GNNs and answer an open question raised by Zhang et al. (2024).

Joshua Southern, Yam Eitan, Guy Bar-Shalom et al.

Subgraph GNNs have emerged as promising architectures that overcome the expressiveness limitations of Graph Neural Networks (GNNs) by processing bags of subgraphs. Despite their compelling empirical performance, these methods are afflicted by a high computational complexity: they process bags whose size grows linearly in the number of nodes, hindering their applicability to larger graphs. In this work, we propose an effective and easy-to-implement approach to dramatically alleviate the computational cost of Subgraph GNNs and unleash broader applications thereof. Our method, dubbed HyMN, leverages walk-based centrality measures to sample a small number of relevant subgraphs and drastically reduce the bag size. By drawing a connection to perturbation analysis, we highlight the strength of the proposed centrality-based subgraph sampling, and further prove that these walk-based centralities can be additionally used as Structural Encodings for improved discriminative power. A comprehensive set of experimental results demonstrates that HyMN provides an effective synthesis of expressiveness, efficiency, and downstream performance, unlocking the application of Subgraph GNNs to dramatically larger graphs. Not only does our method outperform more sophisticated subgraph sampling approaches, it is also competitive, and sometimes better, than other state-of-the-art approaches for a fraction of their runtime.

Fabrizio Frasca, Fabian Jogl, Moshe Eliasof et al.

To develop a preliminary understanding towards Graph Foundation Models, we study the extent to which pretrained Graph Neural Networks can be applied across datasets, an effort requiring to be agnostic to dataset-specific features and their encodings. We build upon a purely structural pretraining approach and propose an extension to capture feature information while still being feature-agnostic. We evaluate pretrained models on downstream tasks for varying amounts of training samples and choices of pretraining datasets. Our preliminary results indicate that embeddings from pretrained models improve generalization only with enough downstream data points and in a degree which depends on the quantity and properties of pretraining data. Feature information can lead to improvements, but currently requires some similarities between pretraining and downstream feature spaces.

Vignesh Balaji, Christos Kozyrakis, Gal Chechik et al.

Graph Neural Networks (GNNs) enable learning on realworld graphs and mini-batch training has emerged as the de facto standard for training GNNs because it can scale to very large graphs and improve convergence. Current mini-batch construction policies largely ignore efficiency considerations of GNN training. Specifically, existing mini-batching techniques employ randomization schemes to improve accuracy and convergence. However, these randomization schemes are often agnostic to the structural properties of the graph (for eg. community structure), resulting in highly irregular memory access patterns during GNN training that make suboptimal use of on-chip GPU caches. On the other hand, while deterministic mini-batching based solely on graph structure delivers fast runtime performance, the lack of randomness compromises both the final model accuracy and training convergence speed. In this paper, we present Community-structure-aware Randomized Mini-batching (COMM-RAND), a novel methodology that bridges the gap between the above extremes. COMM-RAND allows practitioners to explore the space between pure randomness and pure graph structural awareness during mini-batch construction, leading to significantly more efficient GNN training with similar accuracy. We evaluated COMM-RAND across four popular graph learning benchmarks. COMM-RAND cuts down GNN training time by up to 2.76x (1.8x on average) while achieving an accuracy that is within 1.79% points (0.42% on average) compared to popular random mini-batching approaches.

Yoav Gelberg, Yam Eitan, Aviv Navon et al.

Gradients of neural networks encode valuable information for optimization, editing, and analysis of models. Therefore, practitioners often treat gradients as inputs to task-specific algorithms, e.g. for pruning or optimization. Recent works explore learning algorithms that operate directly on gradients but use architectures that are not specifically designed for gradient processing, limiting their applicability. In this paper, we present a principled approach for designing architectures that process gradients. Our approach is guided by three principles: (1) equivariant design that preserves neuron permutation symmetries, (2) processing sets of gradients across multiple data points to capture curvature information, and (3) efficient gradient representation through rank-1 decomposition. Based on these principles, we introduce GradMetaNet, a novel architecture for learning on gradients, constructed from simple equivariant blocks. We prove universality results for GradMetaNet, and show that previous approaches cannot approximate natural gradient-based functions that GradMetaNet can. We then demonstrate GradMetaNet's effectiveness on a diverse set of gradient-based tasks on MLPs and transformers, such as learned optimization, INR editing, and estimating loss landscape curvature.

Yaniv Galron, Fabrizio Frasca, Haggai Maron et al.

Temporal graph learning has applications in recommendation systems, traffic forecasting, and social network analysis. Although multiple architectures have been introduced, progress in positional encoding for temporal graphs remains limited. Extending static Laplacian eigenvector approaches to temporal graphs through the supra-Laplacian has shown promise, but also poses key challenges: high eigendecomposition costs, limited theoretical understanding, and ambiguity about when and how to apply these encodings. In this paper, we address these issues by (1) offering a theoretical framework that connects supra-Laplacian encodings to per-time-slice encodings, highlighting the benefits of leveraging additional temporal connectivity, (2) introducing novel methods to reduce the computational overhead, achieving up to 56x faster runtimes while scaling to graphs with 50,000 active nodes, and (3) conducting an extensive experimental study to identify which models, tasks, and datasets benefit most from these encodings. Our findings reveal that while positional encodings can significantly boost performance in certain scenarios, their effectiveness varies across different models.

Adir Dayan, Yam Eitan, Haggai Maron

Weight-space learning studies neural architectures that operate directly on the parameters of other neural networks. Motivated by the growing availability of pretrained models, recent work has demonstrated the effectiveness of weight-space networks across a wide range of tasks. SOTA weight-space networks rely on permutation-equivariant designs to improve generalization. However, this may negatively affect expressive power, warranting theoretical investigation. Importantly, unlike other structured domains, weight-space learning targets maps operating on both weight and function spaces, making expressivity analysis particularly subtle. While a few prior works provide partial expressivity results, a comprehensive characterization is still missing. In this work, we address this gap by developing a systematic theory for expressivity of weight-space networks. We first prove that all prominent permutation-equivariant networks are equivalent in expressive power. We then establish universality in both weight- and function-space settings under mild, natural assumptions on the input weights, and characterize the edge-case regimes where universality no longer holds. Together, these results provide a strong and unified foundation for the expressivity of weight-space networks.

Yam Eitan, Moshe Eliasof, Yoav Gelberg et al.

Despite significant advances in Graph Neural Networks (GNNs), their limited expressivity remains a fundamental challenge. Research on GNN expressivity has produced many expressive architectures, leading to architecture hierarchies with models of increasing expressive power. Separately, derivatives of GNNs with respect to node features have been widely studied in the context of the oversquashing and over-smoothing phenomena, GNN explainability, and more. To date, these derivatives remain unexplored as a means to enhance GNN expressivity. In this paper, we show that these derivatives provide a natural way to enhance the expressivity of GNNs. We introduce High-Order Derivative GNN (HOD-GNN), a novel method that enhances the expressivity of Message Passing Neural Networks (MPNNs) by leveraging high-order node derivatives of the base model. These derivatives generate expressive structure-aware node embeddings processed by a second GNN in an end-to-end trainable architecture. Theoretically, we show that the resulting architecture family's expressive power aligns with the WL hierarchy. We also draw deep connections between HOD-GNN, Subgraph GNNs, and popular structural encoding schemes. For computational efficiency, we develop a message-passing algorithm for computing high-order derivatives of MPNNs that exploits graph sparsity and parallelism. Evaluations on popular graph learning benchmarks demonstrate HOD-GNN's strong performance on popular graph learning tasks.

Fabrizio Frasca, Guy Bar-Shalom, Yftah Ziser et al.

Large Language Models (LLMs) often generate incorrect or unsupported content, known as hallucinations. Existing detection methods rely on heuristics or simple models over isolated computational traces such as activations, or attention maps. We unify these signals by representing them as attributed graphs, where tokens are nodes, edges follow attentional flows, and both carry features from attention scores and activations. Our approach, CHARM, casts hallucination detection as a graph learning task and tackles it by applying GNNs over the above attributed graphs. We show that CHARM provably subsumes prior attention-based heuristics and, experimentally, it consistently outperforms other leading approaches across diverse benchmarks. Our results shed light on the relevant role played by the graph structure and on the benefits of combining computational traces, whilst showing CHARM exhibits promising zero-shot performance on cross-dataset transfer.

Ran Elbaz, Guy Bar-Shalom, Yam Eitan et al.

Permutation equivariant neural networks employing parameter-sharing schemes have emerged as powerful models for leveraging a wide range of data symmetries, significantly enhancing the generalization and computational efficiency of the resulting models. Recently, Kolmogorov-Arnold Networks (KANs) have demonstrated promise through their improved interpretability and expressivity compared to traditional architectures based on MLPs. While equivariant KANs have been explored in recent literature for a few specific data types, a principled framework for applying them to data with permutation symmetries in a general context remains absent. This paper introduces Function Sharing KAN (FS-KAN), a principled approach to constructing equivariant and invariant KA layers for arbitrary permutation symmetry groups, unifying and significantly extending previous work in this domain. We derive the basic construction of these FS-KAN layers by generalizing parameter-sharing schemes to the Kolmogorov-Arnold setup and provide a theoretical analysis demonstrating that FS-KANs have the same expressive power as networks that use standard parameter-sharing layers, allowing us to transfer well-known and important expressivity results from parameter-sharing networks to FS-KANs. Empirical evaluations on multiple data types and symmetry groups show that FS-KANs exhibit superior data efficiency compared to standard parameter-sharing layers, by a wide margin in certain cases, while preserving the interpretability and adaptability of KANs, making them an excellent architecture choice in low-data regimes.

Harel Mendelman, Haggai Maron, Ronen Talmon

Graph Neural Networks (GNNs) excel at analyzing graph-structured data but struggle on heterophilic graphs, where connected nodes often belong to different classes. While this challenge is commonly addressed with specialized GNN architectures, graph rewiring remains an underexplored strategy in this context. We provide theoretical foundations linking edge homophily, GNN embedding smoothness, and node classification performance, motivating the need to enhance homophily. Building on this insight, we introduce a rewiring framework that increases graph homophily using a reference graph, with theoretical guarantees on the homophily of the rewired graph. To broaden applicability, we propose a label-driven diffusion approach for constructing a homophilic reference graph from node features and training labels. Through extensive simulations, we analyze how the homophily of both the original and reference graphs influences the rewired graph homophily and downstream GNN performance. We evaluate our method on 11 real-world heterophilic datasets and show that it outperforms existing rewiring techniques and specialized GNNs for heterophilic graphs, achieving improved node classification accuracy while remaining efficient and scalable to large graphs.

Ran Elbaz, Gilad Yehudai, Meirav Galun et al. · nvidia

Reconstructing training data from trained neural networks is an active area of research with significant implications for privacy and explainability. Recent advances have demonstrated the feasibility of this process for several data types. However, reconstructing data from group-invariant neural networks poses distinct challenges that remain largely unexplored. This paper addresses this gap by first formulating the problem and discussing some of its basic properties. We then provide an experimental evaluation demonstrating that conventional reconstruction techniques are inadequate in this scenario. Specifically, we observe that the resulting data reconstructions gravitate toward symmetric inputs on which the group acts trivially, leading to poor-quality results. Finally, we propose two novel methods aiming to improve reconstruction in this setup and present promising preliminary experimental results. Our work sheds light on the complexities of reconstructing data from group invariant neural networks and offers potential avenues for future research in this domain.

Bohang Zhang, Lingxiao Zhao, Haggai Maron

Incorporating spectral information to enhance Graph Neural Networks (GNNs) has shown promising results but raises a fundamental challenge due to the inherent ambiguity of eigenvectors. Various architectures have been proposed to address this ambiguity, referred to as spectral invariant architectures. Notable examples include GNNs and Graph Transformers that use spectral distances, spectral projection matrices, or other invariant spectral features. However, the potential expressive power of these spectral invariant architectures remains largely unclear. The goal of this work is to gain a deep theoretical understanding of the expressive power obtainable when using spectral features. We first introduce a unified message-passing framework for designing spectral invariant GNNs, called Eigenspace Projection GNN (EPNN). A comprehensive analysis shows that EPNN essentially unifies all prior spectral invariant architectures, in that they are either strictly less expressive or equivalent to EPNN. A fine-grained expressiveness hierarchy among different architectures is also established. On the other hand, we prove that EPNN itself is bounded by a recently proposed class of Subgraph GNNs, implying that all these spectral invariant architectures are strictly less expressive than 3-WL. Finally, we discuss whether using spectral features can gain additional expressiveness when combined with more expressive GNNs.

Aviv Navon, Aviv Shamsian, Idan Achituve et al.

In Multi-task learning (MTL), a joint model is trained to simultaneously make predictions for several tasks. Joint training reduces computation costs and improves data efficiency; however, since the gradients of these different tasks may conflict, training a joint model for MTL often yields lower performance than its corresponding single-task counterparts. A common method for alleviating this issue is to combine per-task gradients into a joint update direction using a particular heuristic. In this paper, we propose viewing the gradients combination step as a bargaining game, where tasks negotiate to reach an agreement on a joint direction of parameter update. Under certain assumptions, the bargaining problem has a unique solution, known as the Nash Bargaining Solution, which we propose to use as a principled approach to multi-task learning. We describe a new MTL optimization procedure, Nash-MTL, and derive theoretical guarantees for its convergence. Empirically, we show that Nash-MTL achieves state-of-the-art results on multiple MTL benchmarks in various domains.

Or Litany, Haggai Maron, David Acuna et al.

Standard Federated Learning (FL) techniques are limited to clients with identical network architectures. This restricts potential use-cases like cross-platform training or inter-organizational collaboration when both data privacy and architectural proprietary are required. We propose a new FL framework that accommodates heterogeneous client architecture by adopting a graph hypernetwork for parameter sharing. A property of the graph hyper network is that it can adapt to various computational graphs, thereby allowing meaningful parameter sharing across models. Unlike existing solutions, our framework does not limit the clients to share the same architecture type, makes no use of external data and does not require clients to disclose their model architecture. Compared with distillation-based and non-graph hypernetwork baselines, our method performs notably better on standard benchmarks. We additionally show encouraging generalization performance to unseen architectures.

Christopher Morris, Yaron Lipman, Haggai Maron et al.

In recent years, algorithms and neural architectures based on the Weisfeiler--Leman algorithm, a well-known heuristic for the graph isomorphism problem, have emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm's use in a machine-learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph and node representation learning, discuss recent extensions, and outline the algorithm's connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research.

Beatrice Bevilacqua, Fabrizio Frasca, Derek Lim et al.

Message-passing neural networks (MPNNs) are the leading architecture for deep learning on graph-structured data, in large part due to their simplicity and scalability. Unfortunately, it was shown that these architectures are limited in their expressive power. This paper proposes a novel framework called Equivariant Subgraph Aggregation Networks (ESAN) to address this issue. Our main observation is that while two graphs may not be distinguishable by an MPNN, they often contain distinguishable subgraphs. Thus, we propose to represent each graph as a set of subgraphs derived by some predefined policy, and to process it using a suitable equivariant architecture. We develop novel variants of the 1-dimensional Weisfeiler-Leman (1-WL) test for graph isomorphism, and prove lower bounds on the expressiveness of ESAN in terms of these new WL variants. We further prove that our approach increases the expressive power of both MPNNs and more expressive architectures. Moreover, we provide theoretical results that describe how design choices such as the subgraph selection policy and equivariant neural architecture affect our architecture's expressive power. To deal with the increased computational cost, we propose a subgraph sampling scheme, which can be viewed as a stochastic version of our framework. A comprehensive set of experiments on real and synthetic datasets demonstrates that our framework improves the expressive power and overall performance of popular GNN architectures.

Rinon Gal, Or Patashnik, Haggai Maron et al.

Can a generative model be trained to produce images from a specific domain, guided by a text prompt only, without seeing any image? In other words: can an image generator be trained "blindly"? Leveraging the semantic power of large scale Contrastive-Language-Image-Pre-training (CLIP) models, we present a text-driven method that allows shifting a generative model to new domains, without having to collect even a single image. We show that through natural language prompts and a few minutes of training, our method can adapt a generator across a multitude of domains characterized by diverse styles and shapes. Notably, many of these modifications would be difficult or outright impossible to reach with existing methods. We conduct an extensive set of experiments and comparisons across a wide range of domains. These demonstrate the effectiveness of our approach and show that our shifted models maintain the latent-space properties that make generative models appealing for downstream tasks.

Dror Moran, Hodaya Koslowsky, Yoni Kasten et al.

Existing deep methods produce highly accurate 3D reconstructions in stereo and multiview stereo settings, i.e., when cameras are both internally and externally calibrated. Nevertheless, the challenge of simultaneous recovery of camera poses and 3D scene structure in multiview settings with deep networks is still outstanding. Inspired by projective factorization for Structure from Motion (SFM) and by deep matrix completion techniques, we propose a neural network architecture that, given a set of point tracks in multiple images of a static scene, recovers both the camera parameters and a (sparse) scene structure by minimizing an unsupervised reprojection loss. Our network architecture is designed to respect the structure of the problem: the sought output is equivariant to permutations of both cameras and scene points. Notably, our method does not require initialization of camera parameters or 3D point locations. We test our architecture in two setups: (1) single scene reconstruction and (2) learning from multiple scenes. Our experiments, conducted on a variety of datasets in both internally calibrated and uncalibrated settings, indicate that our method accurately recovers pose and structure, on par with classical state of the art methods. Additionally, we show that a pre-trained network can be used to reconstruct novel scenes using inexpensive fine-tuning with no loss of accuracy.