Wasserstein approximation of measures via orthogonal frames: statistical and computational complexities
Abstract
Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it. Selecting a basis in which the samples are sparsely represented is a recurring task in unsupervised learning, addressed by methods in factor analysis, dictionary learning, and subspace clustering. We pose this task as Wasserstein approximation of the data distribution by measures supported on the axes of an orthogonal frame, which, in turn, reduces to a nonsmooth optimization problem over the orthogonal group O(d). We show that, under mild assumptions, both empirical risk minimizers and a locally initialized stochastic gradient method recover the population-optimal frame at the near-parametric rate Õd(n−1/2), where n is the sample size. The main difficulty is that, although the population objective is smooth, the empirical objective is not-this mismatch precludes standard covering arguments. We overcome this issue by replacing Lipschitz continuity of the empirical gradient with a coarsely Lipschitz bound, a tool that may be of independent interest. Numerical experiments support our theoretical findings.
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BibTeX
@misc{diaz2026wasserstein,
title = {{Wasserstein approximation of measures via orthogonal frames: statistical and computational complexities}},
author = {Mateo Díaz and Nicolás García Trillos and Daniel López-Castaño and Congwei Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36581},
eprint = {2610.36581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36581},
}
RIS
TY - UNPB
TI - Wasserstein approximation of measures via orthogonal frames: statistical and computational complexities
AU - Mateo Díaz
AU - Nicolás García Trillos
AU - Daniel López-Castaño
AU - Congwei Yang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36581
UR - https://arxiv.org/abs/2610.36581
AB - Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it. Selecting a basis in which the samples are sparsely represented is a recurring task in unsupervised learning, addressed by methods in factor analysis, dictionary learning, and subspace clustering. We pose this task as Wasserstein approximation of the data distribution by measures supported on the axes of an orthogonal frame, which, in turn, reduces to a nonsmooth optimization problem over the orthogonal group O(d). We show that, under mild assumptions, both empirical risk minimizers and a locally initialized stochastic gradient method recover the population-optimal frame at the near-parametric rate Õ_d(n^−1/2), where n is the sample size. The main difficulty is that, although the population objective is smooth, the empirical objective is not-this mismatch precludes standard covering arguments. We overcome this issue by replacing Lipschitz continuity of the empirical gradient with a coarsely Lipschitz bound, a tool that may be of independent interest. Numerical experiments support our theoretical findings.
ER -