$(L_r, L_r, 1)$-Decompositions, Sparse Component Analysis, and the Blind Separation of Sums of Exponentials

Nithin Govindarajan, Ethan N. Epperly, Lieven De Lathauwer

Abstract: We derive new uniqueness results for $(L_r,L_r,1)$-type block-term decompositions of third-order tensors by drawing connections to sparse component analysis. It is shown that our uniqueness results have a natural application in the context of the blind source separation problem, since they ensure uniqueness even among $(L_r,L_r,1)$-decompositions with incomparable rank profiles, allowing for stronger separation results for signals consisting of sums of exponentials in the presence of common poles among the source signals. As a byproduct, this line of ideas also suggests a new approach for computing $(L_r,L_r,1)$-decompositions, which proceeds by sequentially computing a canonical polyadic decomposition of the input tensor, followed by performing a sparse factorization on the third factor matrix.