Oblivious sketching of high-degree polynomial kernels

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Documents

  • Thomas D. Ahle
  • Michael Kapralov
  • Jakob B.T. Knudsen
  • Rasmus Pagh
  • Ameya Velingker
  • David P. Woodruff
  • Amir Zandieh

Kernel methods are fundamental tools in machine learning that allow detection of non-linear dependencies between data without explicitly constructing feature vectors in high dimensional spaces. A major disadvantage of kernel methods is their poor scalability: primitives such as kernel PCA or kernel ridge regression generally take prohibitively large quadratic space and (at least) quadratic time, as kernel matrices are usually dense. Some methods for speeding up kernel linear algebra are known, but they all invariably take time exponential in either the dimension of the input point set (e.g., fast multipole methods suffer from the curse of dimensionality) or in the degree of the kernel function. Oblivious sketching has emerged as a powerful approach to speeding up numerical linear algebra over the past decade, but our understanding of oblivious sketching solutions for kernel matrices has remained quite limited, suffering from the aforementioned exponential dependence on input parameters. Our main contribution is a general method for applying sketching solutions developed in numerical linear algebra over the past decade to a tensoring of data points without forming the tensoring explicitly. This leads to the first oblivious sketch for the polynomial kernel with a target dimension that is only polynomially dependent on the degree of the kernel function, as well as the first oblivious sketch for the Gaussian kernel on bounded datasets that does not suffer from an exponential dependence on the dimensionality of input data points.

Original languageEnglish
Title of host publication31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
EditorsShuchi Chawla
Number of pages20
PublisherAssociation for Computing Machinery
Publication date2020
Pages141-160
ISBN (Electronic)9781611975994
Publication statusPublished - 2020
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States
Duration: 5 Jan 20208 Jan 2020

Conference

Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
LandUnited States
BySalt Lake City
Periode05/01/202008/01/2020
SponsorACM Special Interest Group on Algorithms and Computation Theory (SIGACT), SIAM Activity Group on Discrete Mathematics

ID: 258720675