ITSPACE: Monotone Gaussian Optimal Transport Updates

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.