Convex relaxation approaches for high dimensional optimal transport

2025-11-17

AI Summary

High-dimensional optimal transport (OT) suffers from prohibitive computational and sample complexity. To address this, we propose a convex optimization framework that relaxes OT via marginal moments and cluster-wise moments, leveraging distributional locality and sparse correlation structure to approximate the high-dimensional coupling using low-order marginal statistics and intra-cluster moments. Our method integrates semidefinite programming, sparse covariance modeling, and convex relaxation techniques. We theoretically establish that, under sparse-correlation Gaussian assumptions, both computational and sample complexities are significantly reduced; empirically, the approach demonstrates robustness beyond Gaussianity. Key contributions include: (i) the first provably complexity-reducing lower-bound estimator for high-dimensional OT, and (ii) an analytically derived, interpretable, nonparametric transport map extracted from this estimator—offering a principled alternative to neural-network-based generative modeling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations that exploit locality and correlative sparsity in the distributions. These methods approximate high-dimensional couplings using low-order marginals and sparse moment statistics, yielding semidefinite programs that provide lower bounds on the OT cost with greatly reduced complexity. For Gaussian distributions with sparse correlations, we prove reductions in both computational and sample complexity, and experiments show the approach also works well for non-Gaussian cases. In addition, we demonstrate how to extract transport maps from our relaxations, offering a simpler and interpretable alternative to neural networks in generative modeling. Our results suggest that convex relaxations can provide a promising path for dimension reduction in high-dimensional OT.

Problem

• Convex relaxation reduces computational complexity in high-dimensional optimal transport
• Methods approximate couplings using sparse moment statistics and low-order marginals
• Approach extracts interpretable transport maps as neural network alternative

Innovation

• Convex relaxation with marginal and cluster moments
• Approximates couplings using low-order sparse statistics
• Extracts transport maps as neural network alternative