Scalable Bayesian inference for heat kernel Gaussian processes on manifolds

TL;DR

This paper introduces a scalable Bayesian inference method for heat kernel Gaussian processes on manifolds, enabling efficient analysis of large-scale data like fMRI in the Human Connectome Project while preserving data geometry.

Contribution

Develops a novel, efficient estimation technique for heat kernel Gaussian processes that reduces computational complexity from cubic to linear time.

Findings

Method handles large sample sizes effectively.

Significantly reduces computational complexity.

Demonstrates improved accuracy in manifold learning tasks.

Abstract

We establish a scalable manifold learning method and theory, motivated by the problem of estimating fMRI activation manifolds in the Human Connectome Project (HCP). Our primary contribution is the development of an efficient estimation technique for heat kernel Gaussian processes in the exponential family model. This approach handles large sample sizes $n$, preserves the intrinsic geometry of data, and significantly reduces computational complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n)$ via a novel reduced-rank approximation of the graph Laplacian's transition matrix and a Truncated Singular Value Decomposition for the eigenpair computation. The numerical experiments demonstrate the scalability and improved accuracy of our method for manifold learning tasks involving complex large-scale data.