On Estimation of Covariance Matrices With Kronecker Product Structure

Werner, Karl; Jansson, Magnus; Stoica, Petre

doi:10.1109/tsp.2007.907834

Cited by 191 publications

(210 citation statements)

References 13 publications

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“…As compared to the standard saturated (unstructured) model, the number of independent parameters in (1) is reduced from Θ(p 2 f 2 ) to Θ(p 2 ) + Θ(f 2 ). Furthermore, as shown [1], factorization (1) results in a significant reduction in estimation mean squared error and in estimator computational complexity. In this paper…”

Section: Introductionmentioning

confidence: 87%

“…This motivates the flip-flop algorithm [1]. Adapting the notation from [1], define the mappingsÂ(·),B(·):…”

Section: Graphical Lasso Frameworkmentioning

confidence: 99%

“…Model (1) arises in channel modeling for MIMO wireless communications, where A 0 is a transmit covariance matrix and B 0 is a receive covariance matrix, and in other applications, see [1]. Let Θ 0 := Σ −1 0 denote the inverse covariance, or precision matrix.…”

Section: Introductionmentioning

confidence: 99%

“…we propose estimation of a sparse version of the Kronecker product model (1) resulting in even more significant performance improvements than for the saturated model studied in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Penalized likelihood estimators for Gaussian graphical models, such as the graphical lasso (Glasso) have been proposed [3,4,5]. The maximum-likelihood (ML) estimator of the Kronecker product (1) has been studied in [1,6]. While the ML estimator has no known closed-form solution, an approximation to the solution can be iteratively computed via an alternating algorithm: the flip-flop (FF) algorithm [1,6].…”

Section: Introductionmentioning

confidence: 99%

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Kronecker graphical lasso

Tsiligkaridis

Hero

Zhou

2012

2012 IEEE Statistical Signal Processing Workshop (SSP)

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We consider high-dimensional estimation of a (possibly sparse) Kronecker-decomposable covariance matrix given i.i.d. Gaussian samples. We propose a sparse covariance estimation algorithm, Kronecker Graphical Lasso (KGlasso), for the high dimensional setting that takes advantage of structure and sparsity. Convergence and limit point characterization of this iterative algorithm is established. Compared to standard Glasso, KGlasso has low computational complexity as the dimension of the covariance matrix increases. We derive a tight MSE convergence rate for KGlasso and show it strictly outperforms standard Glasso and FF. Simulations validate these results and shows that KGlasso outperforms the maximum-likelihood solution (FF), in the high-dimensional small-sample regime.

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Section: Introductionmentioning

confidence: 87%

“…This motivates the flip-flop algorithm [1]. Adapting the notation from [1], define the mappingsÂ(·),B(·):…”

Section: Graphical Lasso Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…we propose estimation of a sparse version of the Kronecker product model (1) resulting in even more significant performance improvements than for the saturated model studied in [1].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Kronecker graphical lasso

Tsiligkaridis

Hero

Zhou

2012

2012 IEEE Statistical Signal Processing Workshop (SSP)

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Separable covariance models for health care quality measures across years and topics

Hatfield

Zaslavsky

2018

Statistics in Medicine

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Public quality reports for Medicare Advantage health plans include 11 measures of patient experiences reported in the annual Consumer Assessment of Healthcare Providers and Systems surveys. Computing summaries at the health plan level (of multiple measures in multiple years) yields an array-structured random variable. To summarize associations among measures and years, we model the variance-covariance matrix governing the plan-level vectors of yearly quality measures as a Kronecker product of an across-measure matrix and an across-year matrix, or a sum of such Kronecker products. This approach extends separable covariance structure to Fay-Herriot models. In addition, we develop linear combinations of Kronecker products similar to principal components for array random variables. To each Kronecker-product term, we apply post hoc analyses suited to the corresponding dimension of the cross-classification: 1-way factor analysis for the across-measure factor and time-series analysis to the across-year factor. These methods draw out key patterns of variation in the quality measures over time and suggest new strategies for reporting quality information to consumers.

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Estimation and testing for separable variance–covariance structures

Dutilleul

2018

WIREs Computational Stats

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The statistical analysis of data for a p‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq(pq + 1)/2 to the Kronecker product of two matrices with p(p + 1)/2 and q(q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed. Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives. This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms

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On Estimation of Covariance Matrices With Kronecker Product Structure

Cited by 191 publications

References 13 publications

Kronecker graphical lasso

Kronecker graphical lasso

Separable covariance models for health care quality measures across years and topics

Estimation and testing for separable variance–covariance structures

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