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.

Neural Information Processing Systems http://nips.cc/

The covariance matrix adaptation evolution strategy (CMA-ES) is arguably one of the most powerful real-valued derivative-free optimization algorithms, finding many applications in machine learning. The CMA-ES is a Monte Carlo method, sampling from a sequence of multi-variate Gaussian distributions. Given the function values at the sampled points, updating and storing the covariance matrix dominates the time and space complexity in each iteration of the algorithm. We propose a numerically stable quadratic-time covariance matrix update scheme with minimal memory requirements based on maintaining triangular Cholesky factors. This requires a modification of the cumulative step-size adaption (CSA) mechanism in the CMA-ES, in which we replace the inverse of the square root of the covariance matrix by the inverse of the triangular Cholesky factor. Because the triangular Cholesky factor changes smoothly with the matrix square root, this modification does not change the behavior of the CMA-ES in terms of required objective function evaluations as verified empirically. Thus, the described algorithm can and should replace the standard CMA-ES if updating and storing the covariance matrix matters.

The following theorem is a more general version of Theorem 5.1. Assume that Assumptions 1 to 3 hold. Note that the only difference between Theorem B.1 and Theorem 5.1 lies in That is, the "oldest" response used to update By Jensen's inequality and L -smoothness, we have null f In order for the paper to be self-contained, we restate the proof here. The following lemma is slightly modified from Lemma 8 in [18]. By Lemma B.1, we have B Combining Appendix B.3.1 and Appendix B.3.2, we have B.4 Deriving the convergence bound In this subsection, we obtain Theorem B.1 based on the descent lemma.