Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix

Insu Han, Haim Avron, Jinwoo Shin
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:3984-3993, 2020.

Abstract

This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(Aij)] explicitly. The main idea underlying our method is to combine a polynomial approximation of f with the existing tensor sketch scheme for approximating monomials of entries of A. To balance the errors of the two approximation components in an optimal manner, we propose a novel regression formula to find polynomial coefficients given A and f. In particular, we utilize a coreset-based regression with a rigorous approximation guarantee. Finally, we demonstrate the applicability and superiority of the proposed scheme under various machine learning tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-han20a, title = {Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix}, author = {Han, Insu and Avron, Haim and Shin, Jinwoo}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {3984--3993}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/han20a/han20a.pdf}, url = {https://proceedings.mlr.press/v119/han20a.html}, abstract = {This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(Aij)] explicitly. The main idea underlying our method is to combine a polynomial approximation of f with the existing tensor sketch scheme for approximating monomials of entries of A. To balance the errors of the two approximation components in an optimal manner, we propose a novel regression formula to find polynomial coefficients given A and f. In particular, we utilize a coreset-based regression with a rigorous approximation guarantee. Finally, we demonstrate the applicability and superiority of the proposed scheme under various machine learning tasks.} }
Endnote
%0 Conference Paper %T Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix %A Insu Han %A Haim Avron %A Jinwoo Shin %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-han20a %I PMLR %P 3984--3993 %U https://proceedings.mlr.press/v119/han20a.html %V 119 %X This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(Aij)] explicitly. The main idea underlying our method is to combine a polynomial approximation of f with the existing tensor sketch scheme for approximating monomials of entries of A. To balance the errors of the two approximation components in an optimal manner, we propose a novel regression formula to find polynomial coefficients given A and f. In particular, we utilize a coreset-based regression with a rigorous approximation guarantee. Finally, we demonstrate the applicability and superiority of the proposed scheme under various machine learning tasks.
APA
Han, I., Avron, H. & Shin, J.. (2020). Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:3984-3993 Available from https://proceedings.mlr.press/v119/han20a.html.

Related Material