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Abstract:Many of the most popular graph neural networks fall into the category of message-passing neural networks (MPNNs). Famously, MPNNs' ability to distinguish between graphs is limited to graphs separable by the Weisfeiler-Lemann (WL) graph isomorphism test, and the strongest MPNNs, in terms of separation power, are WL-equivalent. Recently, it was shown that the quality of separation provided by standard WL-equivalent MPNN can be very low, resulting in WL-separable graphs being mapped to very similar, hardly distinguishable features. This paper addresses this issue by seeking bi-Lipschitz continuity guarantees for MPNNs. We demonstrate that, in contrast with standard summation-based MPNNs, which lack bi-Lipschitz properties, our proposed model provides a bi-Lipschitz graph embedding with respect to two standard graph metrics. Empirically, we show that our MPNN is competitive with standard MPNNs for several graph learning tasks and is far more accurate in over-squashing long-range tasks.

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Abstract:This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches, which address these problems using parameter-sharing, we consider an alternative methodology based on irreducible representations and Schur's lemma. Using this methodology, we obtain an alternative derivation for existing models like DeepSets, 2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks. The derivation for DWS networks is significantly simpler than that of previous results. Next, we extend our approach to unaligned symmetric sets, where equivariance to the wreath product of groups is required. Previous works have addressed this problem in a rather restrictive setting, in which almost all wreath equivariant layers are Siamese. In contrast, we give a full characterization of layers in this case and show that there is a vast number of additional non-Siamese layers in some settings. We also show empirically that these additional non-Siamese layers can improve performance in tasks like graph anomaly detection, weight space alignment, and learning Wasserstein distances. Our code is available at \href{https://github.com/yonatansverdlov/Irreducible-Representations-of-Deep-Weight-Spaces}{GitHub}.

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Abstract: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-equivalent 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-equivalent 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.

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Abstract:Famously, multiset neural networks based on sum-pooling can separate all distinct multisets, and as a result can be used by message passing neural networks (MPNNs) to separate all pairs of graphs that can be separated by the 1-WL graph isomorphism test. However, the quality of this separation may be very weak, to the extent that the embeddings of "separable" multisets and graphs might even be considered identical when using fixed finite precision. In this work, we propose to fully analyze the separation quality of multiset models and MPNNs via a novel adaptation of Lipschitz and H\"{o}lder continuity to parametric functions. We prove that common sum-based models are lower-H\"{o}lder continuous, with a H\"{o}lder exponent that decays rapidly with the network's depth. Our analysis leads to adversarial examples of graphs which can be separated by three 1-WL iterations, but cannot be separated in practice by standard maximally powerful MPNNs. To remedy this, we propose two novel MPNNs with improved separation quality, one of which is lower Lipschitz continuous. We show these MPNNs can easily classify our adversarial examples, and compare favorably with standard MPNNs on standard graph learning tasks.

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Abstract:We present the $\textit{Sliced Wasserstein Embedding}$ $\unicode{x2014}$ a novel method to embed multisets and distributions over $\mathbb{R}^d$ into Euclidean space. Our embedding is injective and approximately preserves the Sliced Wasserstein distance. Moreover, when restricted to multisets, it is bi-Lipschitz. We also prove that it is $\textit{impossible}$ to embed distributions over $\mathbb{R}^d$ into a Euclidean space in a bi-Lipschitz manner, even under the assumption that their support is bounded and finite. We demonstrate empirically that our embedding offers practical advantage in learning tasks over existing methods for handling multisets.

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Abstract:Semi-algebraic priors are ubiquitous in signal processing and machine learning. Prevalent examples include a) linear models where the signal lies in a low-dimensional subspace; b) sparse models where the signal can be represented by only a few coefficients under a suitable basis; and c) a large family of neural network generative models. In this paper, we prove a transversality theorem for semi-algebraic sets in orthogonal or unitary representations of groups: with a suitable dimension bound, a generic translate of any semi-algebraic set is transverse to the orbits of the group action. This, in turn, implies that if a signal lies in a low-dimensional semi-algebraic set, then it can be recovered uniquely from measurements that separate orbits. As an application, we consider the implications of the transversality theorem to the problem of recovering signals that are translated by random group actions from their second moment. As a special case, we discuss cryo-EM: a leading technology to constitute the spatial structure of biological molecules, which serves as our prime motivation. In particular, we derive explicit bounds for recovering a molecular structure from the second moment under a semi-algebraic prior and deduce information-theoretic implications. We also obtain information-theoretic bounds for three additional applications: factoring Gram matrices, multi-reference alignment, and phase retrieval. Finally, we deduce bounds for designing permutation invariant separators in machine learning.

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Abstract:Canonicalization provides an architecture-agnostic method for enforcing equivariance, with generalizations such as frame-averaging recently gaining prominence as a lightweight and flexible alternative to equivariant architectures. Recent works have found an empirical benefit to using probabilistic frames instead, which learn weighted distributions over group elements. In this work, we provide strong theoretical justification for this phenomenon: for commonly-used groups, there is no efficiently computable choice of frame that preserves continuity of the function being averaged. In other words, unweighted frame-averaging can turn a smooth, non-symmetric function into a discontinuous, symmetric function. To address this fundamental robustness problem, we formally define and construct \emph{weighted} frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of $SO(2)$, $SO(3)$, and $S_n$ on point clouds.

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Abstract:The $k$-Weifeiler-Leman ($k$-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs). Recently, the $2$-WL test was proven to be complete on weighted graphs which encode $3\mathrm{D}$ point cloud data. Consequently, GNNs whose expressive power is equivalent to the $2$-WL test are provably universal on point clouds. Yet, this result is limited to invariant continuous functions on point clouds. In this paper we extend this result in three ways: Firstly, we show that $2$-WL tests can be extended to point clouds which include both positions and velocity, a scenario often encountered in applications. Secondly, we show that PPGN (Maron et al., 2019) can simulate $2$-WL uniformly on all point clouds with low complexity. Finally, we show that a simple modification of this PPGN architecture can be used to obtain a universal equivariant architecture that can approximate all continuous equivariant functions uniformly. Building on our results, we develop our WeLNet architecture, which can process position-velocity pairs, compute functions fully equivariant to permutations and rigid motions, and is provably complete and universal. Remarkably, WeLNet is provably complete precisely in the setting in which it is implemented in practice. Our theoretical results are complemented by experiments showing WeLNet sets new state-of-the-art results on the N-Body dynamics task and the GEOM-QM9 molecular conformation generation task.

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Authors:Christopher Morris, Nadav Dym, Haggai Maron, İsmail İlkan Ceylan, Fabrizio Frasca, Ron Levie, Derek Lim, Michael Bronstein, Martin Grohe, Stefanie Jegelka

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Abstract: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 more balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

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Abstract:The key ingredient to retrieving a signal from its Fourier magnitudes, namely, to solve the phase retrieval problem, is an effective prior on the sought signal. In this paper, we study the phase retrieval problem under the prior that the signal lies in a semi-algebraic set. This is a very general prior as semi-algebraic sets include linear models, sparse models, and ReLU neural network generative models. The latter is the main motivation of this paper, due to the remarkable success of deep generative models in a variety of imaging tasks, including phase retrieval. We prove that almost all signals in R^N can be determined from their Fourier magnitudes, up to a sign, if they lie in a (generic) semi-algebraic set of dimension N/2. The same is true for all signals if the semi-algebraic set is of dimension N/4. We also generalize these results to the problem of signal recovery from the second moment in multi-reference alignment models with multiplicity free representations of compact groups. This general result is then used to derive improved sample complexity bounds for recovering band-limited functions on the sphere from their noisy copies, each acted upon by a random element of SO(3).

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