Locality Sensitive Hashing for p-Exponential Kernels with Applications to Density Estimation

2026-10-04

AI Summary

This study addresses the absence of Locality-Sensitive Hashing (LSH) schemes for p-exponential kernels (1 < p ≤ 2) in high-dimensional spaces, where existing methods are restricted to p = 1. For the first time, this work extends LSH admissibility theory to this parameter range, theoretically proving that p-exponential kernels over bounded domains admit LSH. Methodologically, it introduces a "Mosaic LSH" framework based on Poisson hyperplane processes and constructs a novel hashing mechanism leveraging l1-biased p-stable vectors to enable efficient sampling. These contributions yield effective LSH encoding for p-exponential kernels and derive a new density estimation method tailored to such kernels, providing both theoretical foundations and algorithmic support for high-dimensional similarity search.

Abstract

A kernel $k(x,y)$ is LSHable if there exists a locality sensitive hashing scheme $H$ such that $k(x,y)=\Pr_{h\sim H}[h(x)=h(y)]$ for all $x,y$. This notion plays a key role in efficient kernel methods in high dimensions. In this work, we show that the $p$-exponential kernel $k(x,y)=\exp(-\lVert x-y \rVert_p)$ is LSHable in bounded regions for all $1<p\leq2$. Previously, this was known only for $p=1$. Our new"mosaic LSH"scheme is based on a Poisson hyperplane process with hyperplanes sampled as $\ell_1$-biased $p$-stable vectors, for which we develop efficient sampling procedures. As applications, our results yield new and efficient density estimation methods based on LSHability for those $p$-exponential kernels.

Problem

• Locality Sensitive Hashing
• p-Exponential Kernels
• LSHable
• Density Estimation
• Kernel Methods

Innovation

• Locality Sensitive Hashing
• p-Exponential Kernels
• Mosaic LSH
• Poisson Hyperplane Process
• Density Estimation