Least Squares Approximate Joint Diagonalization on the Orthogonal Group

The theory and derivation of a novel method for approximate joint diagonalization (AJD) on the orthogonal group of matrices are presented. The proposed algorithms are fast and simple, hence, easy to implement. We introduce a least-squares-type cost function, which is to be minimized under the constraint that the matrix to be sought for is orthogonal. A gradient flow for optimizing such cost function is derived and its stability is analyzed within the framework of differential geometry. It is proposed to numerically approximate the gradient flow by using a geodesic-based and an Euler-like update algorithms. Numerical examples about blind source separation of speech signals are illustrated to support the analysis.