Fast Sampling Based Sketches for Tensors

TL;DR

This paper presents a novel sampling-based sketching method for tensors that enables efficient computation of $ ext{l}_0$ sampling and $ ext{l}_1$ embeddings, significantly reducing computational complexity for rank-one tensors.

Contribution

The paper introduces a fast convolution-based sampling technique for tensor sketches, improving efficiency for classical sampling and embedding problems in tensor analysis.

Findings

Achieves tensor sketches with time scaling with $d$ for rank-one tensors.

Enables efficient $ ext{l}_0$ sampling and $ ext{l}_1$ embeddings.

Reduces computational complexity from $d^2$ or $d^3$ to $d$ for specific tensor operations.

Abstract

We introduce a new approach for applying sampling-based sketches to two and three mode tensors. We illustrate our technique to construct sketches for the classical problems of $\ell_0$ sampling and producing $\ell_1$ embeddings. In both settings we achieve sketches that can be applied to a rank one tensor in $(\mathbb{R}^d)^{\otimes q}$ (for $q=2,3$) in time scaling with $d$ rather than $d^2$ or $d^3$. Our main idea is a particular sampling construction based on fast convolution which allows us to quickly compute sums over sufficiently random subsets of tensor entries.