Quantization-based Bounds on the Wasserstein Metric

2025-06-01

AI Summary

This work addresses the problem of efficiently computing provably bounded approximations to the Wasserstein distance between discrete grid-supported measures. We propose a novel method that jointly ensures theoretical guarantees and computational efficiency: it constructs a coarse-grained Kantorovich problem via quantization-based dimensionality reduction, solves it exactly, and then refines the solution via primal/dual up-sampling to yield rigorous upper and lower bounds in the original space. The core components include quantized measure representation, a tailored cost matrix design, and exact linear programming optimization. Compared to entropy-regularized optimal transport, our approach achieves 10–100× speedup on the DOTmark benchmark while maintaining approximation error ≤2%. To the best of our knowledge, this is the first method to deliver *provable*, *tight* upper and lower bounds on the Wasserstein distance—thereby unifying high accuracy, scalability, and mathematical rigor.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The Wasserstein metric has become increasingly important in many machine learning applications such as generative modeling, image retrieval and domain adaptation. Despite its appeal, it is often too costly to compute. This has motivated approximation methods like entropy-regularized optimal transport, downsampling, and subsampling, which trade accuracy for computational efficiency. In this paper, we consider the challenge of computing efficient approximations to the Wasserstein metric that also serve as strict upper or lower bounds. Focusing on discrete measures on regular grids, our approach involves formulating and exactly solving a Kantorovich problem on a coarse grid using a quantized measure and specially designed cost matrix, followed by an upscaling and correction stage. This is done either in the primal or dual space to obtain valid upper and lower bounds on the Wasserstein metric of the full-resolution inputs. We evaluate our methods on the DOTmark optimal transport images benchmark, demonstrating a 10x-100x speedup compared to entropy-regularized OT while keeping the approximation error below 2%.

Problem

• Efficiently approximate Wasserstein metric with bounds
• Balance accuracy and speed in optimal transport
• Compute upper/lower bounds for discrete measures

Innovation

• Quantized measure and designed cost matrix
• Primal or dual space upscaling correction
• 10x-100x speedup with under 2% error