Kronecker graphical lasso

Tsiligkaridis, Theodoros; Hero, Alfred O.; Zhou, Shengxi

doi:10.1109/ssp.2012.6319849

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…of unknown parameters. This result extends the recent high-dimensional results obtained in [2], [3], [43] for the single Kronecker product model (i.e., r = 1).…”

Section: A High Dimensional Operator Norm Bound For the Permuted Samsupporting

confidence: 89%

“…This asymptotic MSE convergence rate of the estimated covariance to the true covariance reflects the number of degrees of freedom of the model, which is on the order of the total number r(p 2 + q 2 ) of unknown parameters. This result extends the recent high-dimensional results obtained in [2], [3], [43] for the single Kronecker product model (i.e., r = 1).…”

Section: B High Dimensional Mse Convergence Rate For Prlssupporting

confidence: 88%

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Covariance Estimation in High Dimensions Via Kronecker Product Expansions

Tsiligkaridis

Hero

2013

IEEE Trans. Signal Process.

Self Cite

126

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This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices [1]. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm [2], [3] for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank r that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements. Index TermsStructured covariance estimation, penalized least squares, Kronecker product decompositions, high dimensional convergence rates, mean-square error, multivariate prediction.

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“…of unknown parameters. This result extends the recent high-dimensional results obtained in [2], [3], [43] for the single Kronecker product model (i.e., r = 1).…”

Section: A High Dimensional Operator Norm Bound For the Permuted Samsupporting

confidence: 89%

Section: B High Dimensional Mse Convergence Rate For Prlssupporting

confidence: 88%

Covariance Estimation in High Dimensions Via Kronecker Product Expansions

Tsiligkaridis

Hero

2013

IEEE Trans. Signal Process.

Self Cite

126

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show abstract

“…Several authors have proposed and studied penalized likelihood estimators of Φ * and ∆ * when J = 1 (Allen and Tibshirani, 2010;Zhang and Schneider, 2010;Tsiligkaridis et al, 2012;Leng and Tang, 2012;Zhou, 2014).…”

Section: Introductionmentioning

confidence: 99%

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

Molstad

Rothman

2018

Journal of Computational and Graphical Statistics

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We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrices to have equal entries and also encourage zeros in the precision matrix. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze an EEG dataset to demonstrate our method's interpretability and classification accuracy.

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Graphical model selection and estimation for high dimensional tensor data

Yin

et al. 2014

Journal of Multivariate Analysis

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Kronecker graphical lasso

Cited by 3 publications

References 10 publications

Covariance Estimation in High Dimensions Via Kronecker Product Expansions

Covariance Estimation in High Dimensions Via Kronecker Product Expansions

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

Graphical model selection and estimation for high dimensional tensor data

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