A Geometric Blind Source Separation Method Based on Facet Component Analysis

TL;DR

This paper introduces a geometric method called facet component analysis (FCA) for blind source separation of nonnegative mixtures, leveraging facet identification of the data's cone structure, and demonstrates its effectiveness on spectroscopic data.

Contribution

The paper proposes a novel FCA approach that identifies facets of the data cone for source separation, relaxing vertex-based conditions and incorporating noise handling techniques.

Findings

Effective separation of spectroscopic data demonstrated

Robustness to noise with denoising techniques

Outperforms vertex-based methods in certain scenarios

Abstract

Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed {\em facet component analysis} (FCA). The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis or VCA) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points, and develop an efficient facet identification method by combining data classification and linear regression. For noisy data, we show that denoising methods may be employed, such as the total variation technique in imaging processing, and principle component analysis. We show computational results on nuclear magnetic resonance spectroscopic data to substantiate our method.