Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix

• URL: http://arxiv.org/abs/2406.14269v1
• Date: Thu, 20 Jun 2024 12:50:54 GMT
• Title: Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix
• Authors: The Tien Mai,
• Abstract summary: This paper provides theoretical results for the tempered posterior with the fully specified horseshoe prior in high-dimensional settings. We also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Precision matrices are crucial in many fields such as social networks, neuroscience, and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off-diagonal position of the precision matrix indicates conditional independence between nodes. In high-dimensional settings where the dimension of the precision matrix $p$ exceeds the sample size $n$ and the matrix is sparse, methods like graphical Lasso, graphical SCAD, and CLIME are popular for estimating GGMs. While frequentist methods are well-studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate by \citet{li2019graphical}, applying the global-local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high-dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior.

Related papers

• DNNLasso: Scalable Graph Learning for Matrix-Variate Data [2.7195102129095003]
  We introduce a diagonally non-negative graphical lasso model for estimating the Kronecker-sum structured precision matrix. DNNLasso outperforms the state-of-the-art methods by a large margin in both accuracy and computational time.
  arXiv  Detail & Related papers  (2024-03-05T02:49:00Z)
• Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions [18.95928707619676]
  We propose a convex formulation with well-established properties for MGGD parameters. The proposed framework is flexible as it combines a variety of regularizations for the precision matrix, the mean and perturbations. Experiments show a more accurate precision and covariance matrix estimation with similar performance for the mean vector parameter.
  arXiv  Detail & Related papers  (2023-12-12T18:08:04Z)
• Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing [28.91482208876914]
  We consider the problem of parameter estimation in a high-dimensional generalized linear model. Despite their wide use, a rigorous performance characterization, as well as a principled way to preprocess the data, are available only for unstructured designs.
  arXiv  Detail & Related papers  (2023-08-28T11:49:23Z)
• Large-scale gradient-based training of Mixtures of Factor Analyzers [67.21722742907981]
  This article contributes both a theoretical analysis as well as a new method for efficient high-dimensional training by gradient descent. We prove that MFA training and inference/sampling can be performed based on precision matrices, which does not require matrix inversions after training is completed. Besides the theoretical analysis and matrices, we apply MFA to typical image datasets such as SVHN and MNIST, and demonstrate the ability to perform sample generation and outlier detection.
  arXiv  Detail & Related papers  (2023-08-26T06:12:33Z)
• 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)
• Graph Polynomial Convolution Models for Node Classification of Non-Homophilous Graphs [52.52570805621925]
  We investigate efficient learning from higher-order graph convolution and learning directly from adjacency matrix for node classification. We show that the resulting model lead to new graphs and residual scaling parameter. We demonstrate that the proposed methods obtain improved accuracy for node-classification of non-homophilous parameters.
  arXiv  Detail & Related papers  (2022-09-12T04:46:55Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Estimation of sparse Gaussian graphical models with hidden clustering structure [8.258451067861932]
  We propose a model to estimate the sparse Gaussian graphical models with hidden clustering structure. We develop a symmetric Gauss-Seidel based alternating direction method of the multipliers. Numerical experiments on both synthetic data and real data demonstrate the good performance of our model.
  arXiv  Detail & Related papers  (2020-04-17T08:43:31Z)
• Nonconvex Matrix Completion with Linearly Parameterized Factors [10.163102766021373]
  Parametric Factorization holds for important examples including subspace and completion simulations. The effectiveness of our unified nonconstrained matrix optimization method is also illustrated.
  arXiv  Detail & Related papers  (2020-03-29T22:40:47Z)

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.