ITSPACE用闭式更新直接优化Bures-Wasserstein距离，在协方差对齐任务上比梯度下降快得多，适合资源受限的域适应场景。
ITSPACE（Iterative Transport for Stable Proximal Alignment of Covariance Embeddings）是一种直接优化Bures-Wasserstein目标函数的近端主要化-最小化方法，通过平方根分解中的闭式更新实现。每次迭代在精确算术下满足目标函数的充分下降不等式，在非精确极分解下提供显式认证间隙边界。该方法保持PSD结构且支持秩限制因子，适用于无标注目标批次下的严格步数和计算预算场景。在多个真实协方差对齐基准测试中，ITSPACE达到低BW间隙解的速度显著快于BW梯度下降、其他协方差几何方法以及熵正则化样本OT基线。
ITSPACE: Monotone Gaussian Optimal Transport Updates
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.