Underdetermined Blind Identification via $k$-Sparse Component Analysis: RANSAC-driven Orthogonal Subspace Search

Two primary families of methods exist for underdetermined blind identification (UBI) based on the sparsity of the source matrix: sparse component analysis (SCA) and $k$-SCA. SCA assumes one active source at each time instant, while $k$-SCA allows for varying numbers of active sources represented by $k$. However, existing $k$-SCA methods, which claim to solve UBI problems by accommodating $k$-sparse sources, predominantly rely on $1$-sparse sources, limiting their effectiveness in real-world scenarios with high noise levels. In this paper, we propose an effective and computationally less complex approach for UBI, specifically focusing on the challenging case when the number of active sources is equal to the number of sensors minus one ($k=m-1$). Our approach overcomes limitations by using a two-step scenario: (1) estimating the orthogonal complement subspaces of the overall space and (2) identifying the mixing vectors. We present an integrated algorithm based on the Gram-Schmidt process and random sample consensus (RANSAC) method to solve both steps. Experimental results using simulated data demonstrate the superior effectiveness of our proposed method compared to existing algorithms.