Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity
Abstract
We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of L2-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.
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BibTeX
@misc{mallery2026sparse,
title = {{Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity}},
author = {Brendan Mallery and Shuchin Aeron and James Murphy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12924},
eprint = {2610.12924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12924},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity
AU - Brendan Mallery
AU - Shuchin Aeron
AU - James Murphy
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12924
UR - https://arxiv.org/abs/2610.12924
AB - We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of L^2-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.
ER -