Inverse Problems in Biology, Deconvolution of Mixed Signals in Spatial Transcriptomics Data, and How to Use Matrix Factorization for Nearly Everything

Inverse problems arise when we want to use data to extract an insight into the inner workings of a system. Such problems are often ill-posed, meaning that there are multiple explanations compatible with the observations and it is therefore necessary to constrain the problem to arrive at a concrete solution. High-throughput high-resolution genome-wide spatial transcriptomics data is a recent breakthrough technology presenting great promise for gaining insights into cellular interactions and tissue-level systems biology. Extracting biologically useful knowledge from this new data modality comes with a novel set of computational challenges. One such challenge is figuring out how much different cell types have contributed to each spatial point measured from the tissue. We will go step-by-step over a simple, intuitive and interpretable solution (manifested by a matrix factorization) that utilizes expert-annotated reference data of cell types to constrain this ill-posed inverse problem.