Approximate Joint Diagonalization and Geometric Mean of Symmetric Positive Definite Matrices

Abstract

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.

Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review

“…The mixing matrix depends on the dipole position and orientation in the brain, the physical properties of the head and on the position of the electrodes on the scalp. Once estimated B, the source process are estimated by (14), out of the usual scaling and permutation ambiguities [90,91]. Now, let Si and Sj be the covariance matrix of the unknown source process for any two trials.…”

“…An expanding line of research on Riemannian optimization is today at the forefront in the signal processing community and is gaining attention in biomedical engineering. The mathematical connection between the concept of Riemannian mean and the blind source separation/independent component analysis problem in EEG data has been explored in [91]. 29 Finally, it is worth mentioning that Riemannian methods (MDM and/or tangent space projection) submitted by author A.B.…”

Transmission of wireless backhaul signal in a cellular system with small moving cells

“…Instead of the proposed approach, a sub-optimal solution of (18) can be obtained by searching a unitary matrix that diagonalizesS −1 k in an average sense when Z k is unknown. The solution can be obtained by means of approximated joint diagonalization (AJD) that searches for a unitary matrix by jointly diagonalizing a large number of samples ofS −1 k (orS k ) in a Monte-Carlo manner [24]. However, the computational complexity may become extremely large as the number of samples increases.…”

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

“…In Congedo et al [28] it was shown that the Riemannian metric-based geometric mean [16,17,29] of a set of Hermitian positive-definite matrices M k , k = 1, . .…”

“…From a computational point of view the approximate joint diagonalizer approach to compute means may take advantage of the quadratic convergence displayed by some approximate joint diagonalizer algorithms [28]. In [28] it has been shown that the properties of approximation (16) inherit directly from the properties of the cost function used to find the joint diagonalizer C; using Pham's cost function the approximation verifies the determinant identity, the joint homogeneity, invariance by congruence transformation, but not the self-duality. An approximate joint diagonalization criterion satisfies the self-duality property whenever if C is an approximate joint diagonalizer of the set M 1 , .…”