Orthogonal Random Features: Explicit Forms and Sharp Inequalities

TL;DR

This paper analyzes orthogonal random features for kernel approximation, revealing they approximate a Bessel kernel rather than Gaussian, with explicit bias and variance expressions and bounds showing less dispersion than Fourier features.

Contribution

It provides explicit formulas for bias and variance of orthogonal random features and demonstrates they approximate a Bessel kernel, not Gaussian, with sharper bounds on dispersion.

Findings

Orthogonal random features approximate a Bessel kernel.

Explicit bias and variance formulas are derived.

Orthogonal features are less dispersed than Fourier features.

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.