Zhengchao Wan

Mitchell Black, Zhengchao Wan, Amir Nayyeri et al.

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.

Gal Mishne, Zhengchao Wan, Yusu Wang et al.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincaré ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincaré ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincaré ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Samantha Chen, Sunhyuk Lim, Facundo Mémoli et al.

In this paper, we present a novel interpretation of the so-called Weisfeiler-Lehman (WL) distance, introduced by Chen et al. (2022), using concepts from stochastic processes. The WL distance aims at comparing graphs with node features, has the same discriminative power as the classic Weisfeiler-Lehman graph isomorphism test and has deep connections to the Gromov-Wasserstein distance. This new interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive. We further explore the connections between the WL distance and certain Message Passing Neural Networks, and discuss the implications of the WL distance for understanding the Lipschitz property and the universal approximation results for these networks.

Tristan Brugère, Zhengchao Wan, Yusu Wang

(Directed) graphs with node attributes are a common type of data in various applications and there is a vast literature on developing metrics and efficient algorithms for comparing them. Recently, in the graph learning and optimization communities, a range of new approaches have been developed for comparing graphs with node attributes, leveraging ideas such as the Optimal Transport (OT) and the Weisfeiler-Lehman (WL) graph isomorphism test. Two state-of-the-art representatives are the OTC distance proposed in (O'Connor et al., 2022) and the WL distance in (Chen et al., 2022). Interestingly, while these two distances are developed based on different ideas, we observe that they both view graphs as Markov chains, and are deeply connected. Indeed, in this paper, we propose a unified framework to generate distances for Markov chains (thus including (directed) graphs with node attributes), which we call the Optimal Transport Markov (OTM) distances, that encompass both the OTC and the WL distances. We further introduce a special one-parameter family of distances within our OTM framework, called the discounted WL distance. We show that the discounted WL distance has nice theoretical properties and can address several limitations of the existing OTC and WL distances. Furthermore, contrary to the OTC and the WL distances, our new discounted WL distance can be differentiated after a entropy-regularization similar to the Sinkhorn distance, making it suitable to use in learning frameworks, e.g., as the reconstruction loss in a graph generative model.

Mitchell Black, Zhengchao Wan, Gal Mishne et al.

The distinguishing power of graph transformers is closely tied to the choice of positional encoding: features used to augment the base transformer with information about the graph. There are two primary types of positional encoding: absolute positional encodings (APEs) and relative positional encodings (RPEs). APEs assign features to each node and are given as input to the transformer. RPEs instead assign a feature to each pair of nodes, e.g., graph distance, and are used to augment the attention block. A priori, it is unclear which method is better for maximizing the power of the resulting graph transformer. In this paper, we aim to understand the relationship between these different types of positional encodings. Interestingly, we show that graph transformers using APEs and RPEs are equivalent in terms of distinguishing power. In particular, we demonstrate how to interchange APEs and RPEs while maintaining their distinguishing power in terms of graph transformers. Based on our theoretical results, we provide a study on several APEs and RPEs (including the resistance distance and the recently introduced stable and expressive positional encoding (SPE)) and compare their distinguishing power in terms of transformers. We believe our work will help navigate the huge number of choices of positional encoding and will provide guidance on the future design of positional encodings for graph transformers.

Sawyer Robertson, Zhengchao Wan, Alexander Cloninger

The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond. In this article we put forth a bold claim: that the two fields should be understood as one and the same, up to a choice of $p$. We make this claim precise by introducing the parameterized family of $p$-Beckmann distances for probability measures on graphs and relate them sharply to certain Wasserstein distances. Then, we break open a suite of results including explicit connections to optimal stopping times and random walks on graphs, graph Sobolev spaces, and a Benamou-Brenier type formula for $2$-Beckmann distance. We further explore empirical implications in the world of unsupervised learning for graph data and propose further study of the usage of these metrics where Wasserstein distance may produce computational bottlenecks.

Sawyer Jack Robertson, Chester Holtz, Zhengchao Wan et al.

We study the problem of semi-supervised learning on graphs in the regime where data labels are scarce or possibly corrupted. We propose an approach called $p$-conductance learning that generalizes the $p$-Laplace and Poisson learning methods by introducing an objective reminiscent of $p$-Laplacian regularization and an affine relaxation of the label constraints. This leads to a family of probability measure mincut programs that balance sparse edge removal with accurate distribution separation. Our theoretical analysis connects these programs to well-known variational and probabilistic problems on graphs (including randomized cuts, effective resistance, and Wasserstein distance) and provides motivation for robustness when labels are diffused via the heat kernel. Computationally, we develop a semismooth Newton-conjugate gradient algorithm and extend it to incorporate class-size estimates when converting the continuous solutions into label assignments. Empirical results on computer vision and citation datasets demonstrate that our approach achieves state-of-the-art accuracy in low label-rate, corrupted-label, and partial-label regimes.

Zhengchao Wan, Qingsong Wang, Gal Mishne et al.

Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Samantha Chen, Sunhyuk Lim, Facundo Mémoli et al.

The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and compares subtle structures of the underlying LMMCs and, as a consequence of this, it is more discriminating than the distance between graphs used for defining the state-of-the-art Wasserstein Weisfeiler-Lehman graph kernel. Inspired by the structure of the WL distance we identify a neural network architecture on LMMCs which turns out to be universal w.r.t. continuous functions defined on the space of all LMMCs (which includes all graphs) endowed with the WL distance. Finally, the WL distance turns out to be stable w.r.t. a natural variant of the Gromov-Wasserstein (GW) distance for comparing metric Markov chains that we identify. Hence, the WL distance can also be construed as a polynomial time lower bound for the GW distance which is in general NP-hard to compute.

Kun Jin, Facundo Mémoli, Zhengchao Wan

We introduce the Gaussian transform (GT), an optimal transport inspired iterative method for denoising and enhancing latent structures in datasets. Under the hood, GT generates a new distance function (GT distance) on a given dataset by computing the $\ell^2$-Wasserstein distance between certain Gaussian density estimates obtained by localizing the dataset to individual points. Our contribution is twofold: (1) theoretically, we establish firstly that GT is stable under perturbations and secondly that in the continuous case, each point possesses an asymptotically ellipsoidal neighborhood with respect to the GT distance; (2) computationally, we accelerate GT both by identifying a strategy for reducing the number of matrix square root computations inherent to the $\ell^2$-Wasserstein distance between Gaussian measures, and by avoiding redundant computations of GT distances between points via enhanced neighborhood mechanisms. We also observe that GT is both a generalization and a strengthening of the mean shift (MS) method, and it is also a computationally efficient specialization of the recently proposed Wasserstein Transform (WT) method. We perform extensive experimentation comparing their performance in different scenarios.

Facundo Mémoli, Zane Smith, Zhengchao Wan

We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces. The construction draws inspiration from Optimal Transportation ideas. We establish precise connections with the mean shift family of algorithms and establish the stability of both our method and mean shift under data perturbation.