RESUMEN
Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these 2 kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multistable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.
RESUMEN
Single-cell gene expression studies promise to reveal rare cell types and cryptic states, but the high variability of single-cell RNA-seq measurements frustrates efforts to assay transcriptional differences between cells. We introduce the Census algorithm to convert relative RNA-seq expression levels into relative transcript counts without the need for experimental spike-in controls. Analyzing changes in relative transcript counts led to dramatic improvements in accuracy compared to normalized read counts and enabled new statistical tests for identifying developmentally regulated genes. Census counts can be analyzed with widely used regression techniques to reveal changes in cell-fate-dependent gene expression, splicing patterns and allelic imbalances. We reanalyzed single-cell data from several developmental and disease studies, and demonstrate that Census enabled robust analysis at multiple layers of gene regulation. Census is freely available through our updated single-cell analysis toolkit, Monocle 2.