Kernel Approximation on Algebraic Varieties
Name
21m1425050.pdf
Description
Published version
Size
1.64 MB
Format
Adobe PDF
Checksum (MD5)
dfe92b86aab30b727a45f6ae0295a617
Author(s)
Altschuler, Jason M
Parrilo, Pablo A
Date Issued
March 2023
Journal
SIAM Journal on Applied Algebra and Geometry
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that significantly better approximations are obtainable in this setting: the rank required to achieve a given error depends on the variety’s dimension rather than the ambient dimension, which is typically much larger. This is true in both high-precision and high-dimensional regimes. Our results are presented for smooth isotropic kernels, the predominant class of kernels used in applications. Our main technical insight is to approximate smooth kernels by polynomial kernels and to leverage two key properties of polynomial kernels that hold when they are restricted to a variety. First, their ranks decrease exponentially in the variety’s co-dimension. Second, their maximum values are governed by their values over a small set of points. Together, our results provide a general approach for exploiting (approximate) “algebraic structure” in datasets in order to efficiently solve large-scale data science problems.
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/21M1425050
