## Abstract

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.

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 language | English |
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Title of host publication | SODA '20: Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms |

Number of pages | 20 |

Publisher | Association for Computing Machinery |

Publication date | 2020 |

ISBN (Electronic) | 978-1-61197-599-4 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Machine Learning
- Kernel Methods
- Scalability
- Oblivious Sketching
- Numerical Linear Algebra