# Statistical Inference for Bures-Wasserstein Flows

arXiv: https://arxiv.org/abs/2310.13764 · HTML: https://mlsanity.com/paper/2310.13764/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2023.

## Built on by (most cited first, 1 of 1)

- [Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence](https://mlsanity.com/paper/2606.17196/) (2026)

## Abstract (arXiv)

We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry. We make use of the Riemmanian-like structure induced by this metric to define a notion of mean and covariance of a random flow, and develop an associated Karhunen-Lo`eve expansion. We then treat the problem of estimation and construction of functional principal components from a finite collection of covariance flows, observed fully or irregularly. Our theoretical results are motivated by modern problems in functional data analysis, where one observes operator-valued random processes -- for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. We illustrate our methodology by means of simulations and data analyses.