Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold

Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava

doi:10.1214/19-ba1176

Cited by 6 publications

(5 citation statements)

References 51 publications

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“…The model considered here, as most of the literature on statistics for Stiefel manifolds, is estimated with MLE or MAP. Recently, [46] proposed a Bayesian framework for von Mises-Fisher distributions which allows computing the posterior distribution of F given observations of X. An interesting question would be to analyze the behavior of this posterior distribution in a hierarchical model where X is a latent variable, in a direction similar to the works of [41] and [20].…”

Section: Discussionmentioning

confidence: 99%

“…The distribution considered for X is the von Mises-Fisher (vMF) distribution, also called Matrix Langevin distribution in the literature. It was first introduced by [32], who derived basic properties of the distribution and its Maximum Likelihood Estimator (MLE), and was further studied for both theoretical and algorithmic purposes [12,30,35,46]. The von Mises-Fisher distribution over V np is defined by its probability density function (p.d.f.)…”

Section: Eigenvectors Distributionmentioning

confidence: 99%

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Asymptotic Analysis of a Matrix Latent Decomposition Model

2022

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Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model.

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Section: Discussionmentioning

confidence: 99%

Section: Eigenvectors Distributionmentioning

confidence: 99%

Asymptotic Analysis of a Matrix Latent Decomposition Model

2022

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“…Correlation can be introduced between the patterns by using Fisher–Bingham distributions on the Stiefel manifold [ 38 ] and between pattern weights with full Gaussian covariance matrices. Another direction to develop is the quantification of the uncertainty: by adding prior distributions on F and

, a Bayesian analysis would naturally provide posterior confidence regions for the model parameters [ 47 ]. Finally, our framework could be adapted to model graph Laplacian matrices instead of adjacency matrices.…”

Section: Discussionmentioning

confidence: 99%

“…The main obstacle to retrieving the parameter F given samples

is the normalizing constant of the distribution: though analytically known, it is hard to compute in practice (see Pal et al [ 47 ] for a computation procedure when

). Jupp and Mardia [ 48 ] proved that the MLE exists and is unique as long as

and

, or

and

.…”

Section: A Maximum Likelihood Estimation Algorithmmentioning

confidence: 99%

Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold

Mantoux

Couvy‐Duchesne

Cacciamani

et al. 2021

Entropy

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Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.

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“…Parameter estimation in models of directional data have largely focused on large sample asymptotics, maximum likelihood methods and Bayesian methods [12,37].…”

Section: Introductionmentioning

confidence: 99%

Equivariant Estimation of Fréchet Means

McCormack¹,

Hoff²

2021

Preprint

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The Fréchet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fréchet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space is encoded by its isometry group. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models there exists an optimal equivariant estimator, which necessarily will perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fréchet mean. We derive the general form of this minimum risk equivariant estimator and in a few cases provide explicit expressions for it. In other models the isometry group is not large enough relative to the parametric family of distributions for there to exist a minimum risk equivariant estimator. In such cases, we introduce an adaptive equivariant estimator that uses the data to select a submodel for which there is an MRE. Simulations results show that the adaptive equivariant estimator performs favorably relative to alternative estimators.

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Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold

Cited by 6 publications

References 51 publications

Asymptotic Analysis of a Matrix Latent Decomposition Model

Asymptotic Analysis of a Matrix Latent Decomposition Model

Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold

Equivariant Estimation of Fréchet Means

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