Peter Halmos

Henri Schmidt, Peter Halmos, Ben Raphael

Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix defined on pairs of points. Unlike standard OT, which infers unstructured pointwise mappings, low-rank optimal transport explicitly constrains the rank of the transport plan to infer latent structure. This improves statistical stability and robustness, yields sharper parametric rates for estimating Wasserstein distances adaptive to the intrinsic rank, and generalizes $K$-means to co-clustering. These advantages, however, come at the cost of a non-convex and NP-hard optimization problem. We introduce transport clustering, an algorithm to compute a low-rank OT plan that reduces low-rank OT to a clustering problem on correspondences obtained from a full-rank $\textit{transport registration}$ step. We prove that this reduction yields polynomial-time, constant-factor approximation algorithms for low-rank OT: specifically, a $(1+γ)$ approximation for negative-type metrics and a $(1+γ+\sqrt{2γ}\,)$ approximation for kernel costs, where $γ\in [0,1]$ denotes the approximation ratio of the optimal full-rank solution relative to the low-rank optimal. Empirically, transport clustering outperforms existing low-rank OT solvers on synthetic benchmarks and large-scale, high-dimensional datasets.

Peter Halmos, Boris Hanin

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.

Peter Halmos, Xinhao Liu, Julian Gold et al.

Optimal transport (OT) is a general framework for finding a minimum-cost transport plan, or coupling, between probability distributions, and has many applications in machine learning. A key challenge in applying OT to massive datasets is the quadratic scaling of the coupling matrix with the size of the dataset. [Forrow et al. 2019] introduced a factored coupling for the k-Wasserstein barycenter problem, which [Scetbon et al. 2021] adapted to solve the primal low-rank OT problem. We derive an alternative parameterization of the low-rank problem based on the $\textit{latent coupling}$ (LC) factorization previously introduced by [Lin et al. 2021] generalizing [Forrow et al. 2019]. The LC factorization has multiple advantages for low-rank OT including decoupling the problem into three OT problems and greater flexibility and interpretability. We leverage these advantages to derive a new algorithm $\textit{Factor Relaxation with Latent Coupling}$ (FRLC), which uses $\textit{coordinate}$ mirror descent to compute the LC factorization. FRLC handles multiple OT objectives (Wasserstein, Gromov-Wasserstein, Fused Gromov-Wasserstein), and marginal constraints (balanced, unbalanced, and semi-relaxed) with linear space complexity. We provide theoretical results on FRLC, and demonstrate superior performance on diverse applications -- including graph clustering and spatial transcriptomics -- while demonstrating its interpretability.

Peter Halmos, Julian Gold, Xinhao Liu et al.

Optimal transport (OT) has enjoyed great success in machine learning as a principled way to align datasets via a least-cost correspondence, driven in large part by the runtime efficiency of the Sinkhorn algorithm (Cuturi, 2013). However, Sinkhorn has quadratic space and time complexity in the number of points, limiting scalability to larger datasets. Low-rank OT achieves linear complexity, but by definition, cannot compute a one-to-one correspondence between points. When the optimal transport problem is an assignment problem between datasets then an optimal mapping, known as the Monge map, is guaranteed to be a bijection. In this setting, we show that the factors of an optimal low-rank coupling co-cluster each point with its image under the Monge map. We leverage this invariant to derive an algorithm, Hierarchical Refinement (HiRef), that dynamically constructs a multiscale partition of each dataset using low-rank OT subproblems, culminating in the bijective Monge map. Hierarchical Refinement runs in log-linear time and linear space, retaining the advantages of low-rank OT while overcoming its limited resolution. We demonstrate the advantages of Hierarchical Refinement on several datasets, including ones containing over a million points, scaling full-rank OT to problems previously beyond Sinkhorn's reach.