Convex Analysis of Mixtures for Separating Non-negative Well-grounded Sources

Blind Source Separation (BSS) is a powerful tool for analyzing composite data patterns in many areas, such as computational biology. We introduce a novel BSS method, Convex Analysis of Mixtures (CAM), for separating non-negative well-grounded sources, which learns the mixing matrix by identifying the lateral edges of the convex data scatter plot. We propose and prove a sufficient and necessary condition for identifying the mixing matrix through edge detection in the noise-free case, which enables CAM to identify the mixing matrix not only in the exact-determined and over-determined scenarios, but also in the under-determined scenario. We show the optimality of the edge detection strategy, even for cases where source well-groundedness is not strictly satisfied. The CAM algorithm integrates plug-in noise filtering using sector-based clustering, an efficient geometric convex analysis scheme, and stability-based model order selection. The superior performance of CAM against a panel of benchmark BSS techniques is demonstrated on numerically mixed gene expression data of ovarian cancer subtypes. We apply CAM to dissect dynamic contrast-enhanced magnetic resonance imaging data taken from breast tumors and time-course microarray gene expression data derived from in-vivo muscle regeneration in mice, both producing biologically plausible decomposition results.