Projected Bayesian Spatial Factor Models

Abstract

Factor models balance flexibility, identifiability, and computational efficiency, with Bayesian spatial factor models particularly prone to identifiability challenges and scaling limitations. This work introduces Projected Bayesian Spatial Factor (PBSF) models, a new class of models designed to achieve scalability and robust identifiability for spatial factor analysis. PBSF models are defined through a novel Markov chain Monte Carlo construction, Projected MCMC (ProjMC$^2$), which leverages conditional conjugacy and projection to improve posterior stability and mixing by constraining factor sampling to a scaled Stiefel manifold. Theoretical results establish convergence of ProjMC$^2$ irrespective of initialisation. By integrating scalable univariate spatial modelling, PBSF provides a flexible and interpretable framework for low-dimensional spatial representation learning of massive spatial data. Simulation studies demonstrate substantial efficiency and robustness gains, and an application to human kidney spatial transcriptomics data highlights the practical utility of the proposed methodology for improving interpretability in spatial omics.