Characterizing Spatiotemporal Transcriptome of the Human Brain Via Low-Rank Tensor Decomposition

Spatiotemporal gene expression data of the human brain offer insights on the spatial and temporal patterns of gene regulation during brain development. Most existing methods for analyzing these data consider spatial and temporal profiles separately, with the implicit assumption that different brain regions develop in similar trajectories, and that the spatial patterns of gene expression remain similar at different time points. Although these analyses may help delineate gene regulation either spatially or temporally, they are not able to characterize heterogeneity in temporal dynamics across different brain regions, or the evolution of spatial patterns of gene regulation over time. In this article, we develop a statistical method based on low-rank tensor decomposition to more effectively analyze spatiotemporal gene expression data. We generalize the classical principal component analysis (PCA), which is applicable only to data matrices, to tensor PCA that can simultaneously capture spatial and temporal effects. We also propose an efficient algorithm that combines tensor unfolding and power iteration to estimate the tensor principal components efficiently, and provide guarantees on their statistical performance. Numerical experiments are presented to further demonstrate the merits of the proposed method. An application our method to a spatiotemporal brain expression data provides insights on gene regulation patterns in the brain.