Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning

• URL: http://arxiv.org/abs/2411.02726v1
• Date: Tue, 05 Nov 2024 01:52:27 GMT
• Title: Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning
• Authors: Imen Ayadi, Florent Bouchard, Frédéric Pascal,
• Abstract summary: Two algorithms to compute the likelihood estimator (MLE) are proposed. For the $t$-Wishart distribution, the performance of the MLE and statistical learning algorithms are evaluated.
• Score: 7.499241611226476
• License:
• Abstract: This paper deals with Elliptical Wishart distributions - which generalize the Wishart distribution - in the context of signal processing and machine learning. Two algorithms to compute the maximum likelihood estimator (MLE) are proposed: a fixed point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated such as consistency, asymptotic normality and an intrinsic version of Fisher efficiency. On the statistical learning side, novel classification and clustering methods are designed. For the $t$-Wishart distribution, the performance of the MLE and statistical learning algorithms are evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.

Related papers

• Maximum Likelihood Estimation on Stochastic Blockmodels for Directed Graph Clustering [22.421702511126373]
  We formulate clustering as estimating underlying communities in the directed block model. We introduce two efficient and interpretable directed clustering algorithms, a spectral clustering algorithm and a semidefinite programming based clustering algorithm.
  arXiv  Detail & Related papers  (2024-03-28T15:47:13Z)
• Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers [143.6249073384419]
  In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.
  arXiv  Detail & Related papers  (2024-01-29T02:08:40Z)
• Anomaly Detection Under Uncertainty Using Distributionally Robust Optimization Approach [0.9217021281095907]
  Anomaly detection is defined as the problem of finding data points that do not follow the patterns of the majority. The one-class Support Vector Machines (SVM) method aims to find a decision boundary to distinguish between normal data points and anomalies. A distributionally robust chance-constrained model is proposed in which the probability of misclassification is low.
  arXiv  Detail & Related papers  (2023-12-03T06:13:22Z)
• Distributed Linear Regression with Compositional Covariates [5.085889377571319]
  We focus on the distributed sparse penalized linear log-contrast model in massive compositional data. Two distributed optimization techniques are proposed for solving the two different constrained convex optimization problems. In the decentralized topology, we introduce a distributed coordinate-wise descent algorithm for obtaining a communication-efficient regularized estimation.
  arXiv  Detail & Related papers  (2023-10-21T11:09:37Z)
• Efficient Model-Free Exploration in Low-Rank MDPs [76.87340323826945]
  Low-Rank Markov Decision Processes offer a simple, yet expressive framework for RL with function approximation. Existing algorithms are either (1) computationally intractable, or (2) reliant upon restrictive statistical assumptions. We propose the first provably sample-efficient algorithm for exploration in Low-Rank MDPs.
  arXiv  Detail & Related papers  (2023-07-08T15:41:48Z)
• Generalizing Backpropagation for Gradient-Based Interpretability [103.2998254573497]
  We show that the gradient of a model is a special case of a more general formulation using semirings. This observation allows us to generalize the backpropagation algorithm to efficiently compute other interpretable statistics.
  arXiv  Detail & Related papers  (2023-07-06T15:19:53Z)
• Representation Learning with Multi-Step Inverse Kinematics: An Efficient and Optimal Approach to Rich-Observation RL [106.82295532402335]
  Existing reinforcement learning algorithms suffer from computational intractability, strong statistical assumptions, and suboptimal sample complexity. We provide the first computationally efficient algorithm that attains rate-optimal sample complexity with respect to the desired accuracy level. Our algorithm, MusIK, combines systematic exploration with representation learning based on multi-step inverse kinematics.
  arXiv  Detail & Related papers  (2023-04-12T14:51:47Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Geometry of EM and related iterative algorithms [8.228889210180268]
  The Expectation--Maximization (EM) algorithm is a simple meta-algorithm that has been used for many years as a methodology for statistical inference. In this paper, we introduce the $em$ algorithm, an information geometric formulation of the EM algorithm, and its extensions and applications to various problems.
  arXiv  Detail & Related papers  (2022-09-03T00:23:23Z)
• Estimating leverage scores via rank revealing methods and randomization [50.591267188664666]
  We study algorithms for estimating the statistical leverage scores of rectangular dense or sparse matrices of arbitrary rank. Our approach is based on combining rank revealing methods with compositions of dense and sparse randomized dimensionality reduction transforms.
  arXiv  Detail & Related papers  (2021-05-23T19:21:55Z)
• Statistical Outlier Identification in Multi-robot Visual SLAM using Expectation Maximization [18.259478519717426]
  This paper introduces a novel and distributed method for detecting inter-map loop closure outliers in simultaneous localization and mapping (SLAM) The proposed algorithm does not rely on a good initialization and can handle more than two maps at a time.
  arXiv  Detail & Related papers  (2020-02-07T06:34:44Z)

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.