Bayesian Factorizations of Big Sparse Tensors

Zhou, Jing; Bhattacharya, Anirban; Herring, Amy H.; Dunson, David B.

doi:10.1080/01621459.2014.983233

Cited by 50 publications

(47 citation statements)

References 45 publications

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“…Matrix and array observations are becoming increasingly available in the big data era thanks to the rapid advance in the information technology and the need to store data in structured forms; see, for example, Li, Kim and Altman [14], Hoff [11], Leng and Tang [13], Zhou, Li and Zhu [24], and Zhou et al [25]. Consider independent and identically distributed matrixvariates X 1 , .…”

Section: Introductionmentioning

confidence: 99%

Covariance estimation via sparse Kronecker structures

Leng¹,

Pan²

2018

Bernoulli

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The problem of estimating covariance matrices is central to statistical analysis and is extensively addressed when data are vectors. This paper studies a novel Kronecker-structured approach for estimating such matrices when data are matrices and arrays. Focusing on matrix-variate data, we present simple approaches to estimate the row and the column correlation matrices, formulated separately via convex optimization. We also discuss simple thresholding estimators motivated by the recent development in the literature. Nonasymptotic results show that the proposed method greatly outperforms methods that ignore the matrix structure of the data. In particular, our framework allows the dimensionality of data to be arbitrary order even for fixed sample size, and works for flexible distributions beyond normality. Simulations and data analysis further confirm the competitiveness of the method. An extension to general array-data is also outlined.

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Section: Introductionmentioning

confidence: 99%

Covariance estimation via sparse Kronecker structures

Leng¹,

Pan²

2018

Bernoulli

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“…This is a generalization of the sparse PARAFAC (sp-PARAFAC) model of [44] to the case of more than two groups, and gives much stronger control over parameter growth than PARAFAC when the truth consists of marginally independent groups of variables. This is shown formally for the special case of graphical models with empty separators in Theorem 4.2.…”

Section: Collapsed Tucker Decompositionsmentioning

confidence: 99%

“…Dunson and Xing [15] showed that a single latent class model is equivalent to a reduced-rank nonnegative PARAFAC decomposition of the joint probability tensor π , while the multiple latent class model in [3] implied a Tucker decomposition. See also [44] and [30] for extensions of these models to more complex settings.…”

Section: Introductionmentioning

confidence: 99%

Tensor decompositions and sparse log-linear models

2017

Self Cite

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Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. We derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

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“…More specifically, the S = 2 solution (total number of components = 6) should not be preferred to the T3 (2, 2, 1) one (approximately same fit, 1 component less). A similar reasoning allowed us to discard the CP solutions with S = 3 and S = 4 components, taking into account the T3 ones in cases (3,3,1) and (4,4,2). In the latter case, the more parsimonious T3 model (10 components) had a higher fit than the CP one with S = 4 (12 components).…”

Section: Three-way Analysismentioning

confidence: 99%

“…[1][2][3][4] However, research on tensor-based methods has a long history. In fact, the first works on tensor decompositions were due to Hitchcock 5,6 and Cattell.…”

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confidence: 99%

A review of tensor‐based methods and their application to hospital care data

Giordani

Kiers

2017

Statistics in Medicine

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In many situations, a researcher is interested in the analysis of the scores of a set of observation units on a set of variables. However, in medicine, it is very frequent that the information is replicated at different occasions. The occasions can be time‐varying or refer to different conditions. In such cases, the data can be stored in a 3‐way array or tensor. The Candecomp/Parafac and Tucker3 methods represent the most common methods for analyzing 3‐way tensors. In this work, a review of these methods is provided, and then this class of methods is applied to a 3‐way data set concerning hospital care data for a hospital in Rome (Italy) during 15 years distinguished in 3 groups of consecutive years (1892–1896, 1940–1944, 1968–1972). The analysis reveals some peculiar aspects about the use of health services and its evolution along the time.

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Bayesian Factorizations of Big Sparse Tensors

Cited by 50 publications

References 45 publications

Covariance estimation via sparse Kronecker structures

Covariance estimation via sparse Kronecker structures

Tensor decompositions and sparse log-linear models

A review of tensor‐based methods and their application to hospital care data

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