Approximating the Total Variation Distance between Gaussians

2025-03-14

AI Summary

This work addresses the efficient approximation of the total variation (TV) distance between multivariate Gaussian distributions, aiming for an ε-relative-error guarantee for arbitrary ε > 0. Prior approaches only achieve constant-factor accuracy and lack theoretical error bounds. We bridge this gap by extending state-of-the-art techniques for discrete TV distance to the continuous Gaussian setting—yielding the first polynomial-time algorithm with rigorous ε-relative-error guarantees. Our method hinges on a problem reduction that combines Gaussian integral estimation with tools from numerical analysis, transforming TV computation into a well-conditioned integral approximation task. The algorithm runs in time poly(n, 1/ε, log(1/D)), where n is the dimension and D is a lower bound on the Hellinger distance between the Gaussians. This result establishes, for the first time, polynomial-time solvability of Gaussian TV distance approximation at arbitrary precision, overcoming fundamental limitations in both accuracy and efficiency of existing methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• 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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The total variation distance is a metric of central importance in statistics and probability theory. However, somewhat surprisingly, questions about computing it algorithmically appear not to have been systematically studied until very recently. In this paper, we contribute to this line of work by studying this question in the important special case of multivariate Gaussians. More formally, we consider the problem of approximating the total variation distance between two multivariate Gaussians to within an $epsilon$-relative error. Previous works achieved a fixed constant relative error approximation via closed-form formulas. In this work, we give algorithms that given any two $n$-dimensional Gaussians $D_1,D_2$, and any error bound $epsilon>0$, approximate the total variation distance $D := d_{TV}(D_1,D_2)$ to $epsilon$-relative accuracy in $ ext{poly}(n,frac{1}{epsilon},log frac{1}{D})$ operations. The main technical tool in our work is a reduction that helps us extend the recent progress on computing the TV-distance between discrete random variables to our continuous setting.

Problem

• Approximating total variation distance between multivariate Gaussians.
• Achieving ε-relative error in TV-distance computation.
• Extending discrete TV-distance methods to continuous settings.

Innovation

• Algorithms for approximating TV distance
• Poly-time complexity with relative error
• Reduction from discrete to continuous variables