Diagonality measures of Hermitian positive-definite matrices with application to the approximate joint diagonalization problem

Abstract

International audienceIn this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determinant α-divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case. These diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of Hermitian positive-definite matrices. Numerical computations based on a modified Newton method are presented and commented

Approximate Joint Diagonalization According to the Natural Riemannian Distance

“…For the distance based on the Frobenius norm (corresponding to functionnal (1)) and for the log-det divergence (corresponding to (3)), the closest diagonal matrix to a matrix C is simply its diagonal part ddiag(C) [3]. Using the Riemannian distance (5), the closest diagonal matrix Λ is the unique solution to equation [3] ddiag(log(C −1 Λ)) = 0.…”

“…We refer to the resulting algorithms by acronyms FD-AJD (Frobenius distance) for (1), mFD-AJD (modified Frobenius distance) for (2) and LD-AJD (log-det) for (3). We initialize all algorithms with the inverse of the square root of the arithmetic mean of the target matrices.…”

“…Some variations of (1) have been proposed in [2,3] in order to have better invariance properties of the cost function. In [4], (1) is minimized through an indirect strategy.…”

“…Recently, a generalization of (3) based on the log-det α-divergence has been proposed in [3] showing promising results.…”

“…In section 2 the Riemannian cost function (section 2.3) is defined along with a framework to optimize it (sections 2.3 and 2.4). This new criterion stems from the Riemannian diagonality measure [3] (section 2.2) on the cone of HPD matrices [7,8] (section 2.1). In section 3 we compare the performance of our Riemannian criterion to three state of the art ones on both simulated data and on a real electroencephalographic (EEG) recording.…”