The Pseudo Joint EigenValue Decomposition: Definition, Properties and Application to the Canonical Polyadic Decomposition

In this paper, we introduce and study an original joint diagonalization problem. The novelty of this problem lies in the relationship between the two diagonalizing matrices, one being the Moore-Penrose pseudo-inverse of the other. We first provide some theoretical results highlighting the link between the underlying decomposition and the classical joint eigenvalue decomposition. Our main theoretical result is that the problem can be solved by using any joint eigenvalue decomposition algorithm. We then derive an algorithm for the canonical polyadic decomposition of tensors robust to rank overestimation. Eventually, this new algorithm is applied to analyze real mixtures of fluorescence signals.