The Fisher-Rao geometry of CES distributions

• URL: http://arxiv.org/abs/2310.01032v1
• Date: Mon, 2 Oct 2023 09:23:32 GMT
• Title: The Fisher-Rao geometry of CES distributions
• Authors: Florent Bouchard, Arnaud Breloy, Antoine Collas, Alexandre Renaux, Guillaume Ginolhac
• Abstract summary: The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
• Score: 50.50897590847961
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher-Rao information geometry. Interestingly, this yields a point of view that allows for leveragingmany tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cram\'er-Rao bounds, and classification using Riemannian distances.

Related papers

• Score-based pullback Riemannian geometry [10.649159213723106]
  We propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning. We produce high-quality geodesics through the data support and reliably estimates the intrinsic dimension of the data manifold. Our framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training.
  arXiv  Detail & Related papers  (2024-10-02T18:52:12Z)
• RMLR: Extending Multinomial Logistic Regression into General Geometries [64.16104856124029]
  Our framework only requires minimal geometric properties, thus exhibiting broad applicability. We develop five families of SPD MLRs under five types of power-deformed metrics. On rotation matrices we propose Lie MLR based on the popular bi-invariant metric.
  arXiv  Detail & Related papers  (2024-09-28T18:38:21Z)
• Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds [65.04845593770727]
  We present two new metrics on the Symmetric Positive Definite (SPD) manifold via the Cholesky manifold. Our metrics are easy to use, computationally efficient, and numerically stable.
  arXiv  Detail & Related papers  (2024-07-02T18:46:13Z)
• Pulling back symmetric Riemannian geometry for data analysis [0.0]
  Ideal data analysis tools for data sets should account for non-linear geometry. A rich mathematical structure to account for non-linear geometries has been shown to be able to capture the data geometry. Many standard data analysis tools initially developed for data in Euclidean space can be generalised efficiently to data on a symmetric Riemannian manifold.
  arXiv  Detail & Related papers  (2024-03-11T10:59:55Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Geometric variational inference [0.0]
  Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques are used to go beyond point estimates. This work proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation.
  arXiv  Detail & Related papers  (2021-05-21T17:18:50Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Hybrid and Generalized Bayesian Cram\'{e}r-Rao Inequalities via Information Geometry [15.33401602207049]
  This chapter summarizes the recent results which extend this framework to more general Cram'er-Rao inequalities. We apply Eguchi's theory to a generalized form of Czsisz'ar $f$-divergence.
  arXiv  Detail & Related papers  (2021-04-02T14:21:49Z)
• Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
  A principled way to model nonlinear geometric structure inherent in data is provided. However, these operations are typically computationally demanding. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
  arXiv  Detail & Related papers  (2021-02-12T17:38:04Z)
• Riemannian geometry for Compound Gaussian distributions: application to recursive change detection [11.90288071168733]
  The Fisher information metric is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient.
  arXiv  Detail & Related papers  (2020-05-20T14:51:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.