Improved Generalization Bound and Learning of Sparsity Patterns for Data-Driven Low-Rank Approximation
Abstract
Learning sketching matrices for fast and accurate low-rank approximation (LRA) has gained increasing attention. Recently, Bartlett, Indyk, and Wagner (COLT 2022) presented a generalization bound for the learning-based LRA. Specifically, for rank-$k$ approximation using an $m \times n$ learned sketching matrix with $s$ non-zeros in each column, they proved an $\tilde O(nsm)$ bound on the fat shattering dimension ($\tilde O$ hides logarithmic factors). We build on their work and make two contributions. (1) We present a better $\tilde O(nsk)$ bound ($k \le m$). En route to obtaining this result, we give a low-complexity Goldberg–Jerrum algorithm for computing pseudo-inverse matrices, which would be of independent interest. (2) We alleviate an assumption of the previous study that sketching matrices have a fixed sparsity pattern. We prove that learning positions of non-zeros increases the fat shattering dimension only by $O(ns\log n)$. In addition, experiments confirm the practical benefit of learning sparsity patterns.
BibTeX
@InProceedings{pmlr-v206-sakaue23a,
title = {Improved Generalization Bound and Learning of Sparsity Patterns for Data-Driven Low-Rank Approximation},
author = {Sakaue, Shinsaku and Oki, Taihei},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {1--10},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/sakaue23a/sakaue23a.pdf},
url = {https://proceedings.mlr.press/v206/sakaue23a.html},
abstract = {Learning sketching matrices for fast and accurate low-rank approximation (LRA) has gained increasing attention. Recently, Bartlett, Indyk, and Wagner (COLT 2022) presented a generalization bound for the learning-based LRA. Specifically, for rank-$k$ approximation using an $m \times n$ learned sketching matrix with $s$ non-zeros in each column, they proved an $\tilde O(nsm)$ bound on the fat shattering dimension ($\tilde O$ hides logarithmic factors). We build on their work and make two contributions. (1) We present a better $\tilde O(nsk)$ bound ($k \le m$). En route to obtaining this result, we give a low-complexity Goldberg–Jerrum algorithm for computing pseudo-inverse matrices, which would be of independent interest. (2) We alleviate an assumption of the previous study that sketching matrices have a fixed sparsity pattern. We prove that learning positions of non-zeros increases the fat shattering dimension only by $O(ns\log n)$. In addition, experiments confirm the practical benefit of learning sparsity patterns.}
}