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

TL;DR

This paper introduces Convex Analysis of Mixtures (CAM), a novel method for separating non-negative sources in blind source separation, applicable to various cases including under-determined scenarios, with demonstrated superior performance on simulated, biological, and medical data.

Contribution

CAM provides a new convex geometric approach for source separation that works under various conditions and includes noise filtering and model selection, advancing BSS techniques.

Findings

CAM accurately separates sources in simulated data.

CAM outperforms benchmark BSS methods on gene expression data.

CAM produces plausible biological decompositions in medical imaging.

Abstract

Blind Source Separation (BSS) has proven to be a powerful tool for the analysis of composite patterns in engineering and science. We introduce 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 prove a sufficient and necessary condition for identifying the mixing matrix through edge detection, which also serves as the foundation for CAM to be applied not only to the exact-determined and over-determined cases, but also to the under-determined case. 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. We demonstrate the principle of CAM on simulated data and numerically mixed natural images. The superior performance of CAM against a panel of benchmark BSS techniques is demonstrated on numerically mixed gene expression data. We then 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.