Approximate joint diagonalization and geometric mean of symmetric positive definite matrices.

Marco Congedo,Alexandre Barachant,Maher Moakher,Bijan Afsari,Jesus Malo

doi:10.1371/journal.pone.0121423

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.

Highlights

The study of distance measures between symmetric positive definite (SPD) matrices and the definition of the center of mass for a number of them has recently grown very fast, driven by practical problems in radar data processing, image processing, computer vision, shape analysis, medical imaging, sensor networks, elasticity, mechanics, numerical analysis and machine learning (e.g., [1,2,3,4,5,6,7,8,9,10])
In order to perform simulations we generate sets of SPD matrices according to model (1) (see (2), and see the generation of simulated matrices in [28, 33]): a set of K matrices is generated as AT True þ where ATrue2
These three noise levels correspond approximately to typical low, medium and high noise situations according to approximate joint diagonalization (AJD) standards, that is to say, the diagonalization that can be obtained on a set generated with σ = 1 would be considered very bad for practical AJD purposes and the AJD matrix B in this case cannot be assumed well-defined

Summary

Introduction

The study of distance measures between symmetric positive definite (SPD) matrices and the definition of the center of mass for a number of them has recently grown very fast, driven by practical problems in radar data processing, image processing, computer vision, shape analysis, medical imaging (especially diffusion MRI and Brain-Computer Interface), sensor networks, elasticity, mechanics, numerical analysis and machine learning (e.g., [1,2,3,4,5,6,7,8,9,10]). The family of means given by (30) is invariant with respect to the AJD permutation indeterminacy P, for any invertible AJD solution B with inverse A and any invertible diagonal scaling matrix Δ.

Results

Conclusion