# ITSPACE: Monotone Gaussian Optimal Transport Updates

arXiv: https://arxiv.org/abs/2606.30523 · HTML: https://mlsanity.com/paper/2606.30523/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Recognition in Terra Incognita](https://mlsanity.com/paper/1807.04975/) (2018)
- [VisDA: The Visual Domain Adaptation Challenge](https://mlsanity.com/paper/1710.06924/) (2017)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)

## Abstract (arXiv)

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.