Manifold Random Features

TL;DR

This paper introduces Manifold Random Features (MRFs), a novel method leveraging graph discretization and random features to approximate functions on manifolds, with theoretical analysis and experimental validation.

Contribution

The paper proposes a new paradigm for creating random features on manifolds using graph discretization and connects discrete graph-based features with continuous kernel approximations.

Findings

MRFs provide positive, bounded features for low-variance approximation.

Deep asymptotic connection established between GRFs and continuous random features.

Re-discovery of Gaussian kernel approximation mechanism for improving linear-attention Transformers.

Abstract

We present a new paradigm for creating random features to approximate bi-variate functions (in particular, kernels) defined on general manifolds. This new mechanism of Manifold Random Features (MRFs) leverages discretization of the manifold and the recently introduced technique of Graph Random Features (GRFs) to learn continuous fields on manifolds. Those fields are used to find continuous approximation mechanisms that otherwise, in general scenarios, cannot be derived analytically. MRFs provide positive and bounded features, a key property for accurate, low-variance approximation. We show deep asymptotic connection between GRFs, defined on discrete graph objects, and continuous random features used for regular kernels. As a by-product of our method, we re-discover recently introduced mechanism of Gaussian kernel approximation applied in particular to improve linear-attention Transformers, considering simple random walks on graphs and by-passing original complex mathematical computations. We complement our algorithm with a rigorous theoretical analysis and verify in thorough experimental studies.