The GaussianSketch for Almost Relative Error Kernel Distance

TL;DR

This paper introduces the GaussianSketch, a new method for embedding Gaussian kernels into Euclidean space with near-relative error, using truncated expansions and recursive tensor sketches, improving efficiency and accuracy.

Contribution

The paper presents two novel GaussianSketch variants that efficiently approximate Gaussian kernel distances with near-relative error, leveraging truncated expansions and recursive tensor sketches.

Findings

Achieves almost (1+ε)-relative error with small additive α

First variant has poly-logarithmic dependence on 1/α and polynomial on d

Second variant has poly-logarithmic dependence on 1/α and linear on d

Abstract

We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration and approximation properties of these sketches, we use them to approximate the kernel distance between points sets. These sketches yield almost $(1+\varepsilon)$-relative error, but with a small additive $\alpha$ term. In the first variants the dependence on $1/\alpha$ is poly-logarithmic, but has higher degree of polynomial dependence on the original dimension $d$. In the second variant, the dependence on $1/\alpha$ is still poly-logarithmic, but the dependence on $d$ is linear.