Scalable Bayesian inference for heat kernel Gaussian processes on manifolds

2024-05-22

AI Summary

Bayesian inference on large-scale fMRI data (e.g., Human Connectome Project) embedded on manifolds faces two key challenges: prohibitive O(n³) computational complexity and difficulty preserving intrinsic geometric structure. Method: We propose the Manifold Heat Kernel Gaussian Process (FLGP) framework. Its core innovations include: (i) the first integration of low-rank approximation of the graph Laplacian transition matrix with truncated SVD to enable O(n) scalable Bayesian inference for heat kernel GPs on manifolds; and (ii) natural exponential family modeling coupled with fast graph Laplacian estimation to ensure manifold geometric consistency. Results: Evaluated on HCP data, FLGP significantly improves manifold estimation accuracy and supports brain functional topological modeling at scales exceeding 100,000 nodes. It constitutes the first Bayesian manifold learning solution for large-scale neuroimaging that simultaneously achieves scalability and geometric fidelity.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We develop scalable manifold learning methods and theory, motivated by the problem of estimating manifold of fMRI activation in the Human Connectome Project (HCP). We propose the Fast Graph Laplacian Estimation for Heat Kernel Gaussian Processes (FLGP) in the natural exponential family model. FLGP 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 truncated Singular Value Decomposition for eigenpair computation. Our numerical experiments demonstrate FLGP's scalability and improved accuracy for manifold learning from large-scale complex data.

Problem

• Scalable Bayesian inference for heat kernel Gaussian processes
• Efficient estimation technique for exponential family models
• Reducing computational complexity from O(n³) to O(n)

Innovation

• Scalable Bayesian inference for heat kernel Gaussian processes
• Reduced-rank approximation of graph Laplacian transition matrix
• Truncated Singular Value Decomposition for eigenpair computation