The Kähler Mean of Block-Toeplitz Matrices with Toeplitz Structured Blocks

Abstract:The Riemannian geometry of the space P m , of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space P m . First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches.

Section: Riemannian Geometry Of P Mmentioning

confidence: 99%

Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices

Hajri

Ilea

Said

et al. 2016

“…Let B + n,m be the set of positive definite block Teoplitz matrices composed of n × n blocks of m × m matrices (PD BT). For a stationary signal, the autocorrelation matrix R is PD BT (see [5,6,14]). Authors of [5,6,14] proposed a generalization of Verblunsky coefficients and defined a parametrization of PD BT matrices,…”

Section: Radar Datamentioning

confidence: 99%

“…Information geometry is now a standard framework in radar processing (see [4][5][6][9][10][11][12][13]). The information geometry on positive definite Teoplitz block Teoplitz matrices is directly related to the metric on the Siegel space (see [14]). Indeed, Toeplitz block Toeplitz matrices can be represented by a symmetric positive definite matrix and a point laying in a product of Siegel disks.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, Toeplitz block Toeplitz matrices can be represented by a symmetric positive definite matrix and a point laying in a product of Siegel disks. The metric considered on Toeplitz block Toeplitz matrices is induced by the product metric between a metric on the symmetric positive definite matrices and the Siegel disks metrics (see [4][5][6]9,14]). …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Kernel Density Estimation on the Siegel Space with an Application to Radar Processing

Chevallier

Forget

Barbaresco

et al. 2016

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational reasons, we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier's kernel density estimator. The computation of the kernels is made possible by the symmetric structure of the Siegel space. The method is applied to density estimation of reflection coefficients from radar observations.

“…Let the inner product We will in the following illustrate information geometry for multivariate Gaussian density [169]: This information geometry has been intensively studied for structured matrices [151][152][153][154][155][156][157][158][159][160][161][162][163][164][165][166] and in statistics [167] and is linked to the seminal work of Siegel [168] on symmetric bounded domains.…”

Section: Souriau Lie Group Model and Koszul Hessian Geometry Applied mentioning

confidence: 99%

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

Barbaresco

2016

Abstract:We introduce the symplectic structure of information geometry based on Souriau's Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.Keywords: Lie group thermodynamics; moment map; Gibbs density; Gibbs equilibrium; maximum entropy; information geometry; symplectic geometry; Cartan-Poincaré integral invariant; geometric mechanics; Euler-Poincaré equation; Fisher metric; gauge theory; affine group Lorsque le fait qu'on rencontre est en opposition avec une théorie régnante, il faut accepter le fait et abandonner la théorie, alors même que celle-ci, soutenue par de grands noms, est généralement adoptée -Claude Bernard in "Introduction à l'Étude de la Médecine Expérimentale" [1] Au départ, la théorie de la stabilité structurelle m'avait paru d'une telle ampleur et d'une telle généralité, qu'avec elle je pouvais espérer en quelque sorte remplacer la thermodynamique par la géométrie, géométriser en un certain sens la thermodynamique, éliminer des considérations thermodynamiques tous les aspects à caractère mesurable et stochastiques pour ne conserver que la caractérisation géométrique correspondante des attracteurs.-René Thom in "Logos et théorie des Catastrophes" [2] Entropy