Orthogonal Random Features: Explicit Forms and Sharp Inequalities
This work addresses kernel method scalability for machine learning practitioners, but it is incremental as it builds on existing random feature techniques.
The paper tackled the problem of approximating the Gaussian kernel using orthogonal random features, revealing that they actually approximate a Bessel kernel and provide less dispersion than random Fourier features, with explicit expressions for bias and variance derived using normalized Bessel functions.
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.