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

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.

Data Structures and Algorithms (cs.DS); Computational Geometry (cs.CG); Machine Learning (cs.LG)

arXiv:2610.05174 [cs.DS]

(or arXiv:2610.05174v1 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2610.05174 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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