Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

In 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 $\alpha$-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

Simultaneous Matrix Diagonalization for Structural Brain Networks Classification

This paper considers the problem of brain disease classification based on connectome data. A connectome is a network representation of a human brain. The typical connectome classification problem is very challenging because of the small sample size and high dimensionality of the data. We propose to use simultaneous approximate diagonalization of adjacency matrices in order to compute their eigenstructures in more stable way. The obtained approximate eigenvalues are further used as features for classification. The proposed approach is demonstrated to be efficient for detection of Alzheimer's disease, outperforming simple baselines and competing with state-of-the-art approaches to brain disease classification

Least-Squares Joint Diagonalization of a matrix set by a congruence transformation

The approximate joint diagonalization (AJD) is an important analytic tool at the base of numerous independent component analysis (ICA) and other blind source separation (BSS) methods, thus finding more and more applications in medical imaging analysis. In this work we present a new AJD algorithm named SDIAG (Spheric Diagonalization). It imposes no constraint either on the input matrices or on the joint diagonalizer to be estimated, thus it is very general. Whereas it is well grounded on the classical leastsquares criterion, a new normalization reveals a very simple form of the solution matrix. Numerical simulations shown that the algorithm, named SDIAG (spheric diagonalization), behaves well as compared to state-of-the art AJD algorithms.Comment: 2nd Singaporean-French IPAL Symposium, Singapour : Singapour (2009