Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time

David Woodruff, Amir Zandieh
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:23933-23964, 2022.

Abstract

We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the important special case of the q-fold self-tensoring of a dataset, which is the feature matrix of the degree-q polynomial kernel, the leading term of our method’s runtime is proportional to the size of the dataset and has no dependence on q. Previous techniques either incur a poly(q) factor slowdown in their runtime or remove the dependence on q at the expense of having sub-optimal target dimension, and depend quadratically on the number of data-points in their runtime. Our sampling technique relies on a collection of q partially correlated random projections which can be simultaneously applied to a dataset X in total time that only depends on the size of X, and at the same time their q-fold Kronecker product acts as a near-isometry for any fixed vector in the column span of X⊗q. We also show that our sampling methods generalize to other classes of kernels beyond polynomial, such as Gaussian and Neural Tangent kernels.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-woodruff22a, title = {Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time}, author = {Woodruff, David and Zandieh, Amir}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {23933--23964}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/woodruff22a/woodruff22a.pdf}, url = {https://proceedings.mlr.press/v162/woodruff22a.html}, abstract = {We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the important special case of the q-fold self-tensoring of a dataset, which is the feature matrix of the degree-q polynomial kernel, the leading term of our method’s runtime is proportional to the size of the dataset and has no dependence on q. Previous techniques either incur a poly(q) factor slowdown in their runtime or remove the dependence on q at the expense of having sub-optimal target dimension, and depend quadratically on the number of data-points in their runtime. Our sampling technique relies on a collection of q partially correlated random projections which can be simultaneously applied to a dataset X in total time that only depends on the size of X, and at the same time their q-fold Kronecker product acts as a near-isometry for any fixed vector in the column span of $X^{\otimes q}$. We also show that our sampling methods generalize to other classes of kernels beyond polynomial, such as Gaussian and Neural Tangent kernels.} }
Endnote
%0 Conference Paper %T Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time %A David Woodruff %A Amir Zandieh %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-woodruff22a %I PMLR %P 23933--23964 %U https://proceedings.mlr.press/v162/woodruff22a.html %V 162 %X We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the important special case of the q-fold self-tensoring of a dataset, which is the feature matrix of the degree-q polynomial kernel, the leading term of our method’s runtime is proportional to the size of the dataset and has no dependence on q. Previous techniques either incur a poly(q) factor slowdown in their runtime or remove the dependence on q at the expense of having sub-optimal target dimension, and depend quadratically on the number of data-points in their runtime. Our sampling technique relies on a collection of q partially correlated random projections which can be simultaneously applied to a dataset X in total time that only depends on the size of X, and at the same time their q-fold Kronecker product acts as a near-isometry for any fixed vector in the column span of $X^{\otimes q}$. We also show that our sampling methods generalize to other classes of kernels beyond polynomial, such as Gaussian and Neural Tangent kernels.
APA
Woodruff, D. & Zandieh, A.. (2022). Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:23933-23964 Available from https://proceedings.mlr.press/v162/woodruff22a.html.

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