Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

[1]  L. Skovgaard.  A Riemannian geometry of the multivariate normal model , 1984 .

[2]  Christian Jutten,et al.  Classification of covariance matrices using a Riemannian-based kernel for BCI applications , 2013, Neurocomputing.

[3]  P. Michor,et al.  THE CURVATURE OF THE BOGOLIUBOV-KUBO-MORI SCALAR PRODUCT ON MATRICES , 2008 .

[4]  Rachid Deriche,et al.  DTI segmentation by statistical surface evolution , 2006, IEEE Transactions on Medical Imaging.

[5]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.

[6]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[7]  Xavier Pennec,et al.  Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging , 2010, 1009.3045.

[8]  Yong Moon Park.  CONSTRUCTION OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS , 2000 .

[9]  Daniel Rueckert,et al.  A Probabilistic Framework to Infer Brain Functional Connectivity from Anatomical Connections , 2011, IPMI.

[10]  Baba C. Vemuri,et al.  A Novel Dynamic System in the Space of SPD Matrices with Applications to Appearance Tracking , 2013, SIAM J. Imaging Sci..

[11]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[12]  P. Thomas Fletcher,et al.  Riemannian geometry for the statistical analysis of diffusion tensor data , 2007, Signal Process..

[13]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[14]  Xavier Pennec,et al.  Is Affine-Invariance Well Defined on SPD Matrices? A Principled Continuum of Metrics , 2019, GSI.