Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time

TL;DR

This paper introduces a fast, input sparsity time sampling algorithm for approximating Gram matrices of tensor product matrices, significantly improving efficiency over previous methods, especially for polynomial kernels.

Contribution

The authors develop a novel sampling technique using correlated random projections that achieves near-optimal sample complexity and runtime, with special efficiency for polynomial kernel feature matrices.

Findings

Achieves spectral approximation with nearly optimal sample size.

Runtime for polynomial kernel matrices depends only on dataset size, not on tensor degree.

Generalizes to other kernels like Gaussian and Neural Tangent kernels.

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 input dataset and has no dependence on $q$. Previous techniques either incur poly$(q)$ slowdowns 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.