ITSPACE: Monotone Gaussian Optimal Transport Updates
2026-06-29 • Machine Learning
Machine Learning
AI summaryⓘ
The authors focus on comparing and aligning covariance matrices, which are used to summarize data in machine learning tasks like adapting models to new data. They introduce ITSPACE, a new method to optimize a mathematical distance called the Bures-Wasserstein objective exactly and efficiently. Their approach updates the matrix factors iteratively, keeping important properties intact and working well even with limited computational resources. Experiments show ITSPACE finds better solutions faster than existing methods for covariance alignment.
Covariance matrixWasserstein distanceOptimal transportBures-Wasserstein distanceSymmetric positive definite matricesProximal majorization-minimizationPolar decompositionGaussian approximationDomain adaptation
Authors
Woojoo Na, Jennifer Dy
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.