Nonnegative spatial factorization

Gaussian processes are widely used for the analysis of spatial data due to their nonparametric flexibility and ability to quantify uncertainty, and recently developed scalable approximations have facilitated application to massive datasets. For multivariate outcomes, linear models of coregionalization combine dimension reduction with spatial correlation. However, their real-valued latent factors and loadings are difficult to interpret because, unlike nonnegative models, they do not recover a parts-based representation. We present nonnegative spatial factorization (NSF), a spatially-aware probabilistic dimension reduction model that naturally encourages sparsity. We compare NSF to real-valued spatial factorizations such as MEFISTO and nonspatial dimension reduction methods using simulations and high-dimensional spatial transcriptomics data. NSF identifies generalizable spatial patterns of gene expression. Since not all patterns of gene expression are spatial, we also propose a hybrid extension of NSF that combines spatial and nonspatial components, enabling quantification of spatial importance for both observations and features. A TensorFlow implementation of NSF is available from https://github.com/willtownes/nsf-paper .