An Accelerated Proximal Gradient Method for Bures-Wasserstein Space
Abstract
We study composite optimization over the Bures--Wasserstein (BW) space of Gaussian measures. Our main observation is that a useful class of BW problems can be lifted into the Euclidean space of affine transport maps, such that the lifted problems can be solved efficiently using accelerated proximal gradient methods. For objectives whose smooth and nonsmooth components admit convex smooth and convex proximable affine lifts, respectively, our algorithm BW-FISTA achieves minimax-optimal objective gap O(k−2) after k iterations. We verify these regularity conditions for potential and interaction energies together with common spectral regularizers, and we treat stochastic gradients under a suitable noise model. As a principal application, we consider Gaussian variational inference with a log-concave, log-smooth target. Here, the resulting entropy term is handled by an efficient, exact proximal step, and we improve the previous rate of O(k−1) to O(k−2). We numerically verify this predicted boost with synthetic experiments.
Related
- Wasserstein accelerated proximal gradient algorithms
- Accelerated Proximal Gradient Method for Non-Strongly Convex Time-Varying Optimization
- Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size
- Accelerated Bregman Proximal Gradient Methods from Dual Geometric Perspectives
- The Convergence Rate of Accelerated Proximal Gradient Methods with General Inertia Term for Convex Composite Multiobjective Optimization
BibTeX
@misc{nylof2026accelerated,
title = {{An Accelerated Proximal Gradient Method for Bures-Wasserstein Space}},
author = {Jakob Nylöf and Sloan Nietert and Julien Pallage and Giancarlo Ferrari-Trecate and Daniel Kuhn and John Lygeros},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08468},
eprint = {2611.08468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08468},
}
RIS
TY - UNPB
TI - An Accelerated Proximal Gradient Method for Bures-Wasserstein Space
AU - Jakob Nylöf
AU - Sloan Nietert
AU - Julien Pallage
AU - Giancarlo Ferrari-Trecate
AU - Daniel Kuhn
AU - John Lygeros
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08468
UR - https://arxiv.org/abs/2611.08468
AB - We study composite optimization over the Bures--Wasserstein (BW) space of Gaussian measures. Our main observation is that a useful class of BW problems can be lifted into the Euclidean space of affine transport maps, such that the lifted problems can be solved efficiently using accelerated proximal gradient methods. For objectives whose smooth and nonsmooth components admit convex smooth and convex proximable affine lifts, respectively, our algorithm BW-FISTA achieves minimax-optimal objective gap O(k^−2) after k iterations. We verify these regularity conditions for potential and interaction energies together with common spectral regularizers, and we treat stochastic gradients under a suitable noise model. As a principal application, we consider Gaussian variational inference with a log-concave, log-smooth target. Here, the resulting entropy term is handled by an efficient, exact proximal step, and we improve the previous rate of O(k^−1) to O(k^−2). We numerically verify this predicted boost with synthetic experiments.
ER -