Recycling Randomness with Structure for Sublinear time Kernel Expansions

TL;DR

This paper introduces a unified framework for recycling Gaussian random vectors into structured matrices, enabling efficient approximation of kernel functions in sublinear time, extending existing methods like Fastfood to a broader family of matrices.

Contribution

It generalizes the Fastfood approach to include Circulant, Toeplitz, Hankel, and low-displacement matrices, providing theoretical guarantees and empirical validation.

Findings

Structured matrices can approximate kernels efficiently.

Controlling structure and randomness reduces variance.

Empirical results support broader structured matrix use.

Abstract

We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features.