Network Learning with Semi-relaxed Gromov-Wasserstein

Inferring network topology from noisy node observations is a central problem in graph signal processing. In this paper, we consider Laplacian-constrained graph estimation for Gaussian Markov random fields, focusing on the underdetermined regime in which the number of samples is smaller than the number of graph nodes. Existing approaches often formulate the problem as a sparsity-regularized maximum-likelihood estimation problem. While effective, such methods typically require iterative optimization and are often computationally demanding, particularly under Laplacian constraints. Instead, we propose a non-iterative estimator of graph Laplacians that uses effective resistance for regularization, and evaluate the method using a simple sparsification procedure. Experiments show that with some trade-off in edge and weight recovery on the considered dataset, computational cost for moderately sized graphs can be substantially reduced.

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.

Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emph{Distance-Matrix Wasserstein} (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples $n$ points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasserstein error of approximating each original measure with $n$ samples. Hence population DMW converges to GW as sampled subspaces become dense. We further give finite-sample bounds, including intrinsic-dimensional rates that depend on the data manifold rather than the ambient matrix dimension $\binom n2$. For scalable computation, we introduce sliced and multi-scale DMW; for $p=1$, the sliced multi-scale dissimilarity yields positive-definite exponential kernels. Experiments on synthetic metric spaces, scalability benchmarks, graph classification, and two-sample testing validate the theory and demonstrate an interpretable GW-style proxy for structural comparison.