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

Chevallier, Emmanuel; Forget, Thibault; Barbaresco, Frédéric; Angulo, Jesús

doi:10.3390/e18110396

Cited by 27 publications

(17 citation statements)

References 35 publications

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“…Such algorithms would combine efficiency with reduced computational complexity, since the main requirement for maximum likelihood estimation of Gaussian distributions is computation of Riemannian barycentres, a task for which there exists an increasing number of high-performance routines. As discussed in Section V, this is the essential advantage of the definition of Gaussian distributions adopted in the present paper, over other recent definitions, also used in designing statistical learning algorithms, such as those considered in [26]- [28].…”

Section: Introductionmentioning

confidence: 94%

“…The sum in the second term on the right hand side does not depend on σ. Therefore,xN can be found separately, by minimising this sum over the values ofx (minimisation instead of maximisation is due to the minus sign ahead of the sum). However, this is the same as minimising the empirical variance function (26). Now, the unique global minimiser of the empirical variance function is the Riemannian barycentre of x1, .…”

Section: B Maximum Likelihood Estimationmentioning

confidence: 99%

“…In recent literature [26]- [28], a fully non-parametric method for density estimation in M was developed. This is based on the method of kernel density estimation on Riemannian manifolds, first outlined in [76].…”

Section: Comparison To Kernel Density Estimationmentioning

confidence: 99%

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Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

Said

Hajri

Bombrun

et al. 2018

IEEE Trans. Inform. Theory

108

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The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. The present paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features : (1) they are completely tractable, analytically or numerically, when dealing with large covariance matrices, (2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e. Fréchet or geometric mean), (3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. The paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.

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Section: Introductionmentioning

confidence: 94%

Section: B Maximum Likelihood Estimationmentioning

confidence: 99%

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Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

Said

Hajri

Bombrun

et al. 2018

IEEE Trans. Inform. Theory

108

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“…Those presentations highlight the connection with the searches of McCullagh and Gromov. Those presentation is useful in both the theoretical statistics and the applied statistics [17,18,21,24,55,57,58].…”

Section: Two Alternative Definitionsmentioning

confidence: 99%

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, "dualistic relation of Amari" and "Riemannian or symplectic Foliations"; "Koszul geometry" and "linearization of webs"; "KV homology" and "complexity of models". Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

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“…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

Entropy

Self Cite

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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

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

Cited by 27 publications

References 35 publications

Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

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

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