An Optimal Transport Approach for Network Regression

TL;DR

We address network regression where the response is a graph and covariates are Euclidean vectors. The method represents graphs as multivariate Gaussians via the Laplacian pseudoinverse and performs regression in the Wasserstein space through Fréchet means, reducing the problem to a weighted Wasserstein barycenter computation. The approach uses entropy-regularized fixed-point iterations and covariance shifts to handle degeneracies, with convergence evidence across synthetic and real data and clear improvements over Frobenius-based regression in prediction accuracy and scalability. This work motivates further development of convergence theory for Fréchet means in Wasserstein spaces and extensions to Gromov-Wasserstein distances for graphs of varying sizes.

Abstract

We study the problem of network regression, where one is interested in how the topology of a network changes as a function of Euclidean covariates. We build upon recent developments in generalized regression models on metric spaces based on Fréchet means and propose a network regression method using the Wasserstein metric. We show that when representing graphs as multivariate Gaussian distributions, the network regression problem requires the computation of a Riemannian center of mass (i.e., Fréchet means). Fréchet means with non-negative weights translates into a barycenter problem and can be efficiently computed using fixed point iterations. Although the convergence guarantees of fixed-point iterations for the computation of Wasserstein affine averages remain an open problem, we provide evidence of convergence in a large number of synthetic and real-data scenarios. Extensive numerical results show that the proposed approach improves existing procedures by accurately accounting for graph size, topology, and sparsity in synthetic experiments. Additionally, real-world experiments using the proposed approach result in higher Coefficient of Determination ($R^{2}$) values and lower mean squared prediction error (MSPE), cementing improved prediction capabilities in practice.

Figures (15)

• Figure 1: We train our global network regression models over $\{X_i,G_i\}_{i = 1}^4$ pairs where $G_i$ is the response and $X_i$ is the predictor. Then, we predict the graphs with predictor $x = 5$. The Frobenius regressor produces a graph (top) with thirty times the error than our Wasserstein regressor (bottom).
• Figure 2: The Frobenius distance between the predicted graph for $x=5$ and ground truth trained on cycle graphs with an increasing number of nodes for Wasserstein, Frobenius, and Entropic Wasserstein regressors -- an extension of experiment in Figure \ref{['fig:1']}. This demonstrates that the Wasserstein-based regressors outperform the Frobenius across networks of varying sizes. Moreover, the error growth is slower for the Wasserstein regressors, suggesting their superior performance over large-scale networks. Note that the Wasserstein and Entropic Wasserstein outputs are indistinguishable.
• Figure 3: We train regressors over 5 input and response pairs $\{X_i, G_i\}_{i = 1}^5$ respectively. The covariate $X_i$ is an integer from 1 to 5, and $G_i$ is a named graph in the legend ordered by increasing connectivity, e.g., $X_1 = 1$ and $G_1$ is the path graph. We output the graphs predicted with the Wasserstein regressor for each 0.5 step between 1 and 5 on the x-axis. Each line plots the error between a named graph and interpolations over 0.1 steps for the Frobenius (top) and Wasserstein (bottom) regressors.
• Figure 4: Additional interpolated graphs from the Wasserstein regressor in Fig \ref{['fig:3']}, with inputs of 2.2, 3.2, 3.7, and 4.2.
• Figure 5: Heatmaps representing the distance from the Wasserstein regressor to true graphs for cycle (a), star (b), wheel (c), and complete (d), with minimums for Frobenius (yellow), Wasserstein (magenta), and ground truth (red) as points. For all of these heatmaps, the $x$-axis is the value of the second smallest eigenvalue, and the $y$-axis is the value of the 3rd smallest eigenvalue, both being logarithmic and ranging from $1$ to $100$

Theorems & Definitions (3)

• Remark 1
• Theorem 1: Theorem 4.2 in barycenterEquation
• Proposition 1: Propositon 3.4 in haasler2023bureswasserstein