The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices

• URL: http://arxiv.org/abs/2311.06632v1
• Date: Sat, 11 Nov 2023 18:31:25 GMT
• Title: The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices
• Authors: Mehdi Molkaraie
• Abstract summary: For a specific class of sparse Gaussian graphical models, we provide a closed-form solution for the determinant of the covariance matrix. In our framework, the graphical interaction model is equal to replacement product of $mathcalK_n$ and $mathcalK_n-1$.
• Score: 5.330240017302621
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For a specific class of sparse Gaussian graphical models, we provide a closed-form solution for the determinant of the covariance matrix. In our framework, the graphical interaction model (i.e., the covariance selection model) is equal to replacement product of $\mathcal{K}_{n}$ and $\mathcal{K}_{n-1}$, where $\mathcal{K}_n$ is the complete graph with $n$ vertices. Our analysis is based on taking the Fourier transform of the local factors of the model, which can be viewed as an application of the Normal Factor Graph Duality Theorem and holographic algorithms. The closed-form expression is obtained by applying the Matrix Determinant Lemma on the transformed graphical model. In this context, we will also define a notion of equivalence between two Gaussian graphical models.

Related papers

• Graph Generation via Spectral Diffusion [51.60814773299899]
  We present GRASP, a novel graph generative model based on 1) the spectral decomposition of the graph Laplacian matrix and 2) a diffusion process. Specifically, we propose to use a denoising model to sample eigenvectors and eigenvalues from which we can reconstruct the graph Laplacian and adjacency matrix. Our permutation invariant model can also handle node features by concatenating them to the eigenvectors of each node.
  arXiv  Detail & Related papers  (2024-02-29T09:26:46Z)
• Learning High-Dimensional Differential Graphs From Multi-Attribute Data [12.94486861344922]
  We consider the problem of estimating differences in two Gaussian graphical models (GGMs) which are known to have similar structure. Existing methods for differential graph estimation are based on single-attribute (SA) models. In this paper, we analyze a group lasso penalized D-trace loss function approach for differential graph learning from multi-attribute data.
  arXiv  Detail & Related papers  (2023-12-05T18:54:46Z)
• 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)
• On the Equivalence of Causal Models: A Category-Theoretic Approach [0.0]
  We develop a criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors. It is shown that when one model is a $Phi$-abstraction of another, the intervention of the former can be consistently translated into that of the latter.
  arXiv  Detail & Related papers  (2022-01-18T13:43:06Z)
• Duality for Continuous Graphical Models [4.56877715768796]
  The dual normal factor graph and the factor graph duality theorem have been considered for discrete graphical models. We show an application of the factor graph duality theorem to continuous graphical models.
  arXiv  Detail & Related papers  (2021-11-02T23:11:34Z)
• Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields [51.07460861448716]
  This paper presents a convex-analytic framework to learn from data. We show that a triangular convexity decomposition is guaranteed by a transform of the corresponding to its upper part.
  arXiv  Detail & Related papers  (2021-09-17T17:46:12Z)
• Random Geometric Graphs on Euclidean Balls [2.28438857884398]
  We consider a latent space model for random graphs where a node $i$ is associated to a random latent point $X_i$ on the Euclidean unit ball. For certain link functions, the model considered here generates graphs with degree distribution that have tails with a power-law-type distribution.
  arXiv  Detail & Related papers  (2020-10-26T17:21:57Z)
• Semiparametric Nonlinear Bipartite Graph Representation Learning with Provable Guarantees [106.91654068632882]
  We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution. We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.
  arXiv  Detail & Related papers  (2020-03-02T16:40:36Z)
• Permutation Invariant Graph Generation via Score-Based Generative Modeling [114.12935776726606]
  We propose a permutation invariant approach to modeling graphs, using the recent framework of score-based generative modeling. In particular, we design a permutation equivariant, multi-channel graph neural network to model the gradient of the data distribution at the input graph. For graph generation, we find that our learning approach achieves better or comparable results to existing models on benchmark datasets.
  arXiv  Detail & Related papers  (2020-03-02T03:06:14Z)

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.