ITSPACE: Monotone Gaussian Optimal Transport Updates

Abstract

Covariance matrices compactly summarize feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings. Under a Gaussian approximation, the unregularized Wasserstein-2 *optimal transport* (OT) discrepancy reduces to the *Bures--Wasserstein* (BW) distance between *symmetric positive definite* (SPD) covariance matrices. We introduce **ITSPACE** (Iterative Transport for Stable Proximal Alignment of Covariance Embeddings), a lightweight *few-step* method for covariance alignment under tight compute and memory budgets: it maintains a low-rank representation and produces a valid covariance estimate at every iteration through simple closed-form updates. **ITSPACE** is designed for the rank-budgeted, anytime regime relevant to covariance-based domain adaptation and test-time moment matching: under exact computations, each step provably decreases the *exact* BW distance, and under approximate linear-algebra steps we provide a computable certificate bound that quantifies any deviation from monotone descent. Empirically, in the strict few-step regime **ITSPACE** reaches the same BW distance thresholds faster than BW-targeting gradient descent under a common rank budget, and is more stable than Euclidean, alternative-geometry, and entropically regularized baselines.