Fourier Sparse Leverage Scores and Approximate Kernel Learning

06/12/2020
by   Tamás Erdélyi, et al.
0

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study s-sparse functions of the form f(x) = ∑_j=1^s a_j e^i λ_j x for coefficients a_j ∈ℂ and frequencies λ_j ∈ℝ. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the statistical dimension of the approximated kernel matrix. It is the first "oblivious sketching method" with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.

READ FULL TEXT

Authors

page 13

04/26/2018

Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees

Random Fourier features is one of the most popular techniques for scalin...
09/21/2011

Explicit Approximations of the Gaussian Kernel

We investigate training and using Gaussian kernel SVMs by approximating ...
03/20/2019

On Sampling Random Features From Empirical Leverage Scores: Implementation and Theoretical Guarantees

Random features provide a practical framework for large-scale kernel app...
06/05/2021

Sparsification for Sums of Exponentials and its Algorithmic Applications

Many works in signal processing and learning theory operate under the as...
03/21/2020

Scaling up Kernel Ridge Regression via Locality Sensitive Hashing

Random binning features, introduced in the seminal paper of Rahimi and R...
11/20/2019

Random Fourier Features via Fast Surrogate Leverage Weighted Sampling

In this paper, we propose a fast surrogate leverage weighted sampling st...
02/04/2021

Wind Field Reconstruction with Adaptive Random Fourier Features

We investigate the use of spatial interpolation methods for reconstructi...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.