Gradient based Approximate Joint Diagonalization by orthogonal transforms

Approximate Joint Diagonalization (AJD) of a set of symmetric matrices by an orthogonal transform is a popular problem in Blind Source Separation (BSS). In this paper we propose a gradient based algorithm which maximizes the sum of squares of diagonal entries of all the transformed symmetric matrices. Our main contribution is to transform the orthogonality constrained optimization problem into an unconstrained problem. This transform is performed in two steps: First by parameterizing the orthogonal transform matrix by the matrix exponential of a skew-symmetric matrix. Second, by introducing an isomorphism between the vector space of skew-symmetric matrices and the Euclidean vector space of appropriate dimension. This transform is then applied to a gradient based algorithm called GAEX to perform joint diagonalization of a set of symmetric matrices.