Imputing single cell RNA sequencing data: Mathematical, statistical and computational challenges

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Novel single cell RNA sequencing (scRNA-seq) technologies can simultaneously measure the expression levels of all 30,000 genes over thousands to millions of individual cells. The analysis of scRNA-seq data has already led to fundamental advances in biology, including discovery of new cell types, detection of subtle differences between similar cells, and reconstruction of cellular developmental trajectories. Single- cell measurements involve amplification of tiny amounts of RNA and result in extremely sparse data matrices with many zeros, While some of these zeros are due to missing data (dropouts), others represent true biological inactivity. Yet, many scRNA-seq imputation methods treat all observed zero entries identically, leading to imputed matrices that often overestimate transcriptional activity. Other methods that do attempt to distinguish biological zeros from dropouts lack rigorous theoretical guarantees. The goals of this proposal are to develop models, supporting mathematical theory, and computational tools that explicitly take the existence of true biological zeros into account. Matrix imputation under this constraint involves both computational challenges as well as theoretical questions in random matrix theory and high dimensional statistics. These include rank estimation and low rank sparse matrix recovery from partially observed data, and biclustering in the presence of dropouts and zeros, We plan to develop novel approaches based on non-smooth continuous optimization, and derive accompanying statistical guarantees, We also plan to develop ensemble learning approaches that cleverly combine the outputs of multiple imputation algorithms. Finally, we hope to gain important insights regarding recovery from such data via a study of minimax rates and information lower bounds. To address these challenges, we will build on our promising preliminary results and the joint expertise of the investigators in spectral methods, high dimensional statistics, matrix analysis, numerical optimization, and genomics.