On the Fisher-Rao Information Metric in the Space of Normal Distributions

The Fisher-Rao distance between two probability distribution functions, as well as other divergence measures, is related to entropy and is in the core of the research area of information geometry. It can provide a framework and enlarge the perspective of analysis for a wide variety of domains, such as statistical inference, image processing (texture classification and inpainting), clustering processes and morphological classification. We present here a compact summary of results regarding the Fisher-Rao distance in the space of multivariate normal distributions including some historical background, closed forms in special cases, bounds, numerical approaches and references to recent applications.

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

[2]  Maher Moakher,et al.  The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data , 2011, Journal of Mathematical Imaging and Vision.

[3]  Sueli I. R. Costa,et al.  On bounds for the Fisher-Rao distance between multivariate normal distributions , 2015 .

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

[5]  Frank Nielsen,et al.  The statistical Minkowski distances: Closed-form formula for Gaussian Mixture Models , 2019, GSI.

[6]  Sueli I. Rodrigues Costa,et al.  Clustering using the fisher-rao distance , 2016, 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).

[7]  Sueli I. Rodrigues Costa,et al.  A totally geodesic submanifold of the multivariate normal distributions and bounds for the Fisher-Rao distance , 2016, 2016 IEEE Information Theory Workshop (ITW).

[8]  Jesús Angulo,et al.  Morphological Processing of Univariate Gaussian Distribution-Valued Images Based on Poincaré Upper-Half Plane Representation , 2014 .

[9]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[10]  B. Efron.  Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency) , 1975 .

[11]  J. M. Oller,et al.  AN EXPLICIT SOLUTION OF INFORMATION GEODESIC EQUATIONS FOR THE MULTIVARIATE NORMAL MODEL , 1991 .

[12]  Sueli I. Rodrigues Costa,et al.  Fisher information distance: a geometrical reading? , 2012, Discret. Appl. Math..

[13]  P. Mahalanobis.  On the generalized distance in statistics , 1936 .

[14]  Stéphane Puechmorel,et al.  On the Geodesic Distance in Shapes K-means Clustering , 2018, Entropy.

[15]  Frank Nielsen,et al.  Model centroids for the simplification of Kernel Density estimators , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[16]  N. N. Chent︠s︡ov.  Statistical decision rules and optimal inference , 1982 .

[17]  J. Burbea.  Informative Geometry of Probability Spaces , 1984 .

[18]  Frank Chongwoo Park,et al.  DTI Segmentation and Fiber Tracking Using Metrics on Multivariate Normal Distributions , 2013, Journal of Mathematical Imaging and Vision.

[19]  Josep M. Oller,et al.  A distance between multivariate normal distributions based in an embedding into the Siegel group , 1990 .

[20]  Frederic Barbaresco,et al.  Tracking quality monitoring based on information geometry and geodesic shooting , 2016, 2016 17th International Radar Symposium (IRS).

[21]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[22]  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.

[23]  Stephen Taylor,et al.  Clustering Financial Return Distributions Using the Fisher Information Metric , 2018, Entropy.

[24]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[25]  Ryad Benosman,et al.  A Fisher-Rao Metric for Paracatadioptric Images of Lines , 2012, International Journal of Computer Vision.

[26]  Shun-ichi Amari,et al.  Information Geometry and Its Applications , 2016 .

[27]  Paul Scheunders,et al.  Geodesics on the Manifold of Multivariate Generalized Gaussian Distributions with an Application to Multicomponent Texture Discrimination , 2011, International Journal of Computer Vision.

[28]  Frank Nielsen,et al.  Simplification and hierarchical representations of mixtures of exponential families , 2010 .