Scalable Bayesian inference for heat kernel Gaussian processes on manifolds

Junhui He, Guoxuan Ma, Jian Kang, Ying Yang

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2026 · Volume 88 · Issue 2

Abstract

We establish a scalable manifold learning method and theory, motivated by the problem of estimating functional magnetic resonance imaging activation manifolds in the Human Connectome Project. 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 sizesn, preserves the intrinsic geometry of data, and significantly reduces computational complexity fromO(n3)toO(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.

Keywords

exponential family, graph Laplacian, heat kernel, manifold, scalable Gaussian process, subsampling

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