ITSPACE: Monotone Gaussian Optimal Transport Updates

2026-06-29

AI Summary

This work addresses the challenge of aligning covariance matrices in domain adaptation and Gaussian embeddings via optimal transport by proposing ITSPACE, a novel method that achieves, for the first time, monotonic descent optimization toward the exact Bures-Wasserstein (BW) objective. Built upon a square-root factorization, ITSPACE employs a majorization-minimization framework combined with a polar decomposition approximation to enable closed-form iterative updates under computational constraints, while preserving positive semi-definiteness and rank constraints. Experimental results demonstrate that ITSPACE significantly outperforms BW gradient descent, alternative covariance-geometry approaches, and entropy-regularized sample-based optimal transport baselines on real-world covariance alignment tasks, rapidly converging to solutions with low BW discrepancy.

Abstract

Covariance matrices serve as compact descriptors of feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings. Under a centered Gaussian approximation, the unregularized Wasserstein-2 optimal-transport (OT) discrepancy admits a closed form on covariances given by the Bures-Wasserstein (BW) objective on the symmetric positive definite (SPD) cone. We propose ITSPACE (Iterative Transport for Stable Proximal Alignment of Covariance Embeddings), a proximal majorization-minimization method that directly optimizes this exact BW objective through closed-form updates in a square-root factorization. In exact arithmetic, each iteration satisfies a sufficient-decrease inequality for the BW objective; under inexact polar computations, we provide an explicit certificate-gap bound controlling deviations from exact descent. The resulting iterations preserve PSD structure by construction and naturally support rank-restricted factors, making ITSPACE well-suited as a lightweight inner-loop primitive in settings where adaptation must be performed from unlabeled target batches under strict step and compute budgets. Across real-world covariance-alignment benchmarks, ITSPACE reaches low-BW-gap solutions substantially faster than BW-gradient descent, methods based on other covariance geometries, and entropically regularized sample-OT baselines.

Problem

• covariance alignment
• Bures-Wasserstein
• optimal transport
• domain adaptation
• SPD matrices

Innovation

• Bures-Wasserstein
• optimal transport
• covariance alignment
• proximal majorization-minimization
• SPD manifold