Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms

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.

Lloyd's $K$-means algorithm, also known as naïve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $ρ$-approximate algorithm for weighted $k$-means and any $ε>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/Γ)ρ+O(ε)$ times the optimal integral fair cost, where $Γ\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(ε)$, improving the previous factor of $5+O(ε)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.