Orthogonal Random Features: Explicit Forms and Sharp Inequalities

Nizar Demni, Hachem Kadri

Abstract: Random features have been introduced to scale up kernel methods via randomization techniques. In particular, random Fourier features and orthogonal random features were used to approximate the popular Gaussian kernel. Random Fourier features are built in this case using a random Gaussian matrix. In this work, we analyze the bias and the variance of the kernel approximation based on orthogonal random features which makes use of Haar orthogonal matrices. We provide explicit expressions for these quantities using normalized Bessel functions, showing that orthogonal random features does not approximate the Gaussian kernel but a Bessel kernel. We also derive sharp exponential bounds supporting the view that orthogonal random features are less dispersed than random Fourier features.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission: Camera-ready version: we have addressed the points raised by the reviewers and the action editor.

Assigned Action Editor: ~Bruno_Loureiro1

Submission Number: 2654