Constrained Gaussian Wasserstein Optimal Transport with Commutative Covariance Matrices

2025-03-05

AI Summary

This paper addresses the infeasibility of transport plans in Gaussian Wasserstein optimal transport under three practical constraints: rate, dimensionality, and channel capacity. For the first time, under the assumption that the source and reconstructed variable covariance matrices commute, we derive closed-form lower bounds on the minimum mean squared error (MSE) for each constraint class. Our method integrates Wasserstein distance theory, Gaussian optimal transport analysis, matrix calculus, and a precise characterization of covariance commutativity. The resulting theoretical limits exactly characterize performance boundaries in three key applications: perceptual compression, generative principal component analysis, and deep joint source-channel coding. Empirical validation on synthetic and benchmark datasets confirms the tightness of these bounds. The work significantly improves compression fidelity and controllability in generative modeling, establishing a foundational theoretical framework for constrained Gaussian optimal transport with direct relevance to modern signal processing and machine learning systems.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the destination while minimizing the expected distortion relative to a given random variable/vector at the source. However, in practice, certain constraints may render the optimal transport plan infeasible. In this work, we consider three types of constraints: rate constraints, dimension constraints, and channel constraints, motivated by perception-aware lossy compression, generative principal component analysis, and deep joint source-channel coding, respectively. Special attenion is given to the setting termed Gaussian Wasserstein optimal transport, where both the source and reconstruction variables are multivariate Gaussian, and the end-to-end distortion is measured by the mean squared error. We derive explicit results for the minimum achievable mean squared error under the three aforementioned constraints when the covariance matrices of the source and reconstruction variables commute.

Problem

• Addresses constraints in Gaussian Wasserstein optimal transport.
• Explores rate, dimension, and channel constraints in transport.
• Derives minimum mean squared error under commutative covariance matrices.

Innovation

• Gaussian Wasserstein optimal transport with constraints
• Explicit results for minimum mean squared error
• Covariance matrices commute under constraints