LRSPAT: A low-rank framework for spatial omics statistics

We describe LRSPAT (low-rank spatial toolkit), a fast and memory-efficient framework for approximating measures of spatial association for high-dimensional data. While LRSPAT can be applied to any multivariate spatial dataset, development was motivated by the computational challenge of identifying spatially variable genes in high-resolution spatial transcriptomics (ST) data generated by technologies such as 10x Visium HD, Xenium and Atera. LRSPAT leverages a truncated SVD of the expression data and a thresholded spatial weights matrix to perform reduced-rank reconstruction of spatial statistics in the quadratic form family, including global and local versions of Moran's I, Geary's C, and Getis-Ord G. A regularization approach is leveraged to account for the inflated null distribution of spatial statistics computed on latent variables. By performing key operations on the low-dimensional embeddings, LRSPAT is orders of magnitude faster than standard implementations with significantly lower memory requirements. Because the low-rank approach denoises and desparsifies ST data, LRSPAT is also more accurate than standard techniques at identifying genes with true spatial expression patterns. The dramatic improvements in execution time and memory consumption enable the genome-wide analysis of spatially variable genes (SVGs) and exploration of the full range of hyperparameters including spatial scale, distance metric, and embedding rank. This preprint outlines the background and mathematical details of the approach with limited preliminary results and a short conclusion.