# Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

arXiv: https://arxiv.org/abs/1801.01990 · HTML: https://mlsanity.com/paper/1801.01990/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2018.

## Built on by (most cited first, 19 of 19)

- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [Duality of Bures and Shape Distances with Implications for Comparing Neural Representations](https://mlsanity.com/paper/2311.11436/) (2023)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss](https://mlsanity.com/paper/2303.03027/) (2023)
- [Collaborative Bayesian Optimization via Wasserstein Barycenters](https://mlsanity.com/paper/2504.10770/) (2025)
- [Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings](https://mlsanity.com/paper/2101.01429/) (2021)
- [Estimation of Riemannian distances between covariance operators and Gaussian processes](https://mlsanity.com/paper/2108.11683/) (2021)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [Geometric Stability: The Missing Axis of Representations](https://mlsanity.com/paper/2601.09173/) (2026)
- [Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals](https://mlsanity.com/paper/2102.03267/) (2021)
- [Two derivations of Principal Component Analysis on datasets of distributions](https://mlsanity.com/paper/2306.13503/) (2023)
- [Measuring Time-Series Dataset Similarity using Wasserstein Distance](https://mlsanity.com/paper/2507.22189/) (2025)
- [Finite sample approximations of exact and entropic Wasserstein distances between covariance operators and Gaussian processes](https://mlsanity.com/paper/2104.12368/) (2021)
- [Computationally-efficient initialisation of GPs: The generalised variogram method](https://mlsanity.com/paper/2210.05394/) (2022)
- [Clustering-based Meta Bayesian Optimization with Theoretical Guarantee](https://mlsanity.com/paper/2503.06093/) (2025)
- [Weighted Wasserstein Barycenter of Gaussian Processes for exotic Bayesian Optimization tasks](https://mlsanity.com/paper/2602.09181/) (2026)
- [The Geometric Canary: Predicting Steerability and Detecting Drift via Representational Stability](https://mlsanity.com/paper/2604.17698/) (2026)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## Abstract (arXiv)

Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen--Lo`eve expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality of these operators. In this paper, we describe the manifold geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric. We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a complete description of those aspects of this manifold geometry that are important in terms of statistical inference, and establish key properties of the Fr'echet mean of a random sample of covariances, as well as generative models that are canonical for such metrics and link with the problem of registration of functional data.