Projected Bayesian Spatial Factor Models

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.

💡 Research Summary

This paper tackles two longstanding challenges in Bayesian spatial factor analysis: the inherent non‑identifiability of factor loadings and latent factors, and the prohibitive computational cost of Gaussian‑process (GP) priors when the number of spatial locations is large. The authors introduce the Projected Bayesian Spatial Factor (PBSF) model together with a novel Markov chain Monte Carlo scheme called Projected MCMC (ProjMC²). The key idea is to project the n × K latent factor matrix F onto a scaled Stiefel manifold by first centering each column, then extracting the Q‑factor of a thin QR decomposition and scaling by √(n‑1). This projection removes rotation, scaling, and permutation ambiguities, collapsing all equivalent representations of the product ΛᵀF into a single point on the manifold.

ProjMC² operates within a blocked Gibbs sampler. At each iteration it (1) samples the regression coefficients and residual covariance (γ, Σ) from their matrix‑normal‑inverse‑Wishart conditional posterior using the projected factor matrix ˜F, (2) draws the latent factors F and the GP hyper‑parameters ψ from their respective conditionals (the former a multivariate normal derived from the GP covariance, the latter from the GP prior), and (3) re‑projects the newly sampled F back onto the Stiefel manifold to obtain the updated ˜F. The authors prove that the resulting transition kernel admits a unique stationary distribution and that the chain converges in total variation from any starting point, guaranteeing that PBSF is a valid posterior target irrespective of initialization.

To achieve scalability, each factor fₖ is modeled with a Nearest‑Neighbor Gaussian Process (NNGP) rather than a full GP. NNGP approximates the covariance using only a small set of nearest neighbours, reducing both memory and computational complexity from O(n³) to near‑linear in n while preserving local and global spatial structure. The framework also naturally accommodates Bayesian hyper‑parameter learning and missing data within the same Gibbs updates.

Simulation studies with n = 10,000 locations, q = 50 outcomes, and K = 5 factors demonstrate that PBSF with ProjMC² attains 5–10× speed‑ups over standard Bayesian spatial factor models and over NNGP‑based factor models without projection. Moreover, posterior variance of the loadings and factors is markedly reduced, indicating that the projection effectively resolves the non‑identifiability that otherwise hampers mixing.

The methodology is applied to a human kidney spatial transcriptomics dataset comprising ~10,000 spots and 30 genes. PBSF’s low‑dimensional representation is used for spatial domain identification and compared against two state‑of‑the‑art auto‑encoder methods, STAGATE and GraphST. PBSF achieves comparable or superior Adjusted Rand Index scores while providing interpretable factor loadings that highlight biologically meaningful gene sets associated with specific tissue structures (e.g., glomeruli, proximal tubules). Unlike the black‑box deep‑learning baselines, PBSF delivers full posterior uncertainty quantification and clear mechanistic insight.

In summary, the paper delivers a principled Bayesian solution that simultaneously resolves identifiability issues and scales to massive spatial datasets. By projecting latent factors onto a Stiefel manifold and leveraging NNGP approximations, the Projected Bayesian Spatial Factor model offers a powerful, interpretable, and computationally feasible tool for modern spatial omics and other high‑dimensional spatial applications. Future extensions may include non‑linear factor structures, dynamic spatio‑temporal extensions, and integration with other probabilistic deep‑learning frameworks.