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The p-Bures-Wasserstein distance via 3×3 block matrices (with application to p-Bures-Wasserstein barycenters)
Abstract
We study the p-Bures-Wasserstein distance Dp(A,B)=(tr Ap+tr Bp−2|A1/2B1/2|pp)1/2 on positive semidefinite matrices; it equals the Bures-Wasserstein distance for p=1, and the Hilbert-Schmidt distance for p=2. We prove that Dp is a metric for every 1≤ p≤2, by proving a stronger 2014 conjecture of Lin and van den Driessche on 3×3 block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for Dp by a fixed-point equation and prove geometric convergence of the associated iteration.
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BibTeX
@misc{sra2026pbureswasserstein,
title = {{The $p$-Bures-Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures-Wasserstein barycenters)}},
author = {Suvrit Sra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15889},
eprint = {2610.15889},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15889},
}
RIS
TY - UNPB
TI - The p-Bures-Wasserstein distance via 3×3 block matrices (with application to p-Bures-Wasserstein barycenters)
AU - Suvrit Sra
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15889
UR - https://arxiv.org/abs/2610.15889
AB - We study the p-Bures-Wasserstein distance D_p(A,B)=(tr A^p+tr B^p−2|A^1/2B^1/2|_p^p)^1/2 on positive semidefinite matrices; it equals the Bures-Wasserstein distance for p=1, and the Hilbert-Schmidt distance for p=2. We prove that D_p is a metric for every 1≤ p≤2, by proving a stronger 2014 conjecture of Lin and van den Driessche on 3×3 block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for D_p by a fixed-point equation and prove geometric convergence of the associated iteration.
ER -