Applications of Mathematics, Vol. 65, No. 6, pp. 829-844, 2020

Block matrix approximation via entropy loss function

Malwina Janiszewska, Augustyn Markiewicz, Monika Mokrzycka

Received January 29, 2020.   Published online November 2, 2020.

Abstract:  The aim of the paper is to present a procedure for the approximation of a symmetric positive definite matrix by symmetric block partitioned matrices with structured off-diagonal blocks. The entropy loss function is chosen as approximation criterion. This procedure is applied in a simulation study of the statistical problem of covariance structure identification.
Keywords:  approximation; block covariance structure; entropy loss function
Classification MSC:  15A30, 15B99, 62H20, 65F99
DOI:  10.21136/AM.2020.0023-20

[1] X. Cui, C. Li, J. Zhao, L. Zeng, D. Zhang, J. Pan: Covariance structure regularization via Frobenius-norm discrepancy. Linear Algebra Appl. 510 (2016), 124-145. DOI 10.1016/j.laa.2016.08.013 | MR 3551623 | Zbl 1352.62090
[2] D. K. Dey, C. Srinivasan: Estimation of a covariance matrix under Stein's loss. Ann. Stat. 13 (1985), 1581-1591. DOI 10.1214/aos/1176349756 | MR 0811511 | Zbl 0582.62042
[3] P. L. Fackler: Notes on matrix calculus. Available at (2005), 14 pages.
[4] K. Filipiak, D. Klein: Approximation with a Kronecker product structure with one component as compound symmetry or autoregression. Linear Algebra Appl. 559 (2018), 11-33. DOI 10.1016/j.laa.2018.08.031 | MR 3857535 | Zbl 1402.62117
[5] K. Filipiak, D. Klein, A. Markiewicz, M. Mokrzycka: Approximation with a Kronecker product structure with one component as compound symmetry or autoregression via entropy loss function. To appear in Linear Algebra Appl. (2020). DOI 10.1016/j.laa.2020.10.013
[6] K. Filipiak, D. Klein, M. Mokrzycka: Estimators comparison of separable covariance structure with one component as compound symmetry matrix. Electron. J. Linear Algebra 33 (2018), 83-98. DOI 10.13001/1081-3810.3740 | MR 3962256 | Zbl 1406.15020
[7] K. Filipiak, A. Markiewicz, A. Mieldzioc, A. Sawikowska: On projection of a positive definite matrix on a cone of nonnegative definite Toeplitz matrices. Electron. J. Linear Algebra 33 (2018), 74-82. DOI 10.13001/1081-3810.3750 | MR 3962255 | Zbl 1406.15028
[8] M. Gilson, D. Dahmen, R. Moreno-Bote, A. Insabato, M. Helias: The covariance perceptron: A new paradigm for classication and processing of time series in recurrent neuronal networks. Available at (2020), 39 pages. DOI 10.1101/562546
[9] W. James, C. Stein: Estimation with quadratic loss. Proc. 4th Berkeley Symp. Math. Stat. Probab. Vol. 1. University of California Press, Berkeley, 1961, 361-379. MR 0133191 | Zbl 1281.62026
[10] M. John, A. Mieldzioc: The comparison of the estimators of banded Toeplitz covariance structure under the high-dimensional multivariate model. Commun. Stat., Simulation Comput. 49 (2020), 734-752. DOI 10.1080/03610918.2019.1614622 | MR 4068485
[11] T. Kollo, D. von Rosen: Advanced Multivariate Statistics with Matrices. Mathematics and Its Applications 579. Springer, Dordrecht (2005). DOI 10.1007/1-4020-3419-9 | MR 2162145 | Zbl 1079.62059
[12] L. Lin, N. J. Higham, J. Pan: Covariance structure regularization via entropy loss function. Comput. Stat. Data Anal. 72 (2014), 315-327. DOI 10.1016/j.csda.2013.10.004 | MR 3139365 | Zbl 06983908
[13] N. Lu, D. L. Zimmerman: The likelihood ratio test for a separable covariance matrix. Stat. Probab. Lett. 73 (2005), 449-457. DOI 10.1016/j.spl.2005.04.020 | MR 2187860 | Zbl 1071.62052
[14] J. R. Magnus, H. Neudecker: Symmetry, 0-1 matrices and Jacobians: A review. Econom. Theory 2 (1986), 157-190. DOI 10.1017/S0266466600011476
[15] J. R. Magnus, H. Neudecker: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley Series in Probability and Statistics. Wiley, Chichester (1999). DOI 10.1002/9781119541219 | MR 1698873 | Zbl 0912.15003
[16] A. Mieldzioc, M. Mokrzycka, A. Sawikowska: Covariance regularization for metabolomic data on the drought resistance of barley. Biometrical Lett. 56 (2019), 165-181. DOI 10.2478/bile-2019-0010
[17] J.-X. Pan, K.-T. Fang: Growth Curve Models and Statistical Diagnostics. Springer Series in Statistics. Springer, New York (2002). DOI 10.1007/978-0-387-21812-0 | MR 1937691 | Zbl 1024.62025
[18] M. S. Srivastava, T. von Rosen, D. von Rosen: Models with a Kronecker product covariance structure: Estimation and testing. Math. Methods Stat. 17 (2008), 357-370. DOI 10.3103/S1066530708040066 | MR 2483463 | Zbl 1231.62101
[19] T. H. Szatrowski: Estimation and testing for block compound symmetry and other patterned covariance matrices with linear and non-linear structure. Technical Report OLK NSF 107, Stanford University, Stanford (1976), 187 pages.
[20] T. H. Szatrowski: Testing and estimation in the block compound symmetry problem. J. Educ. Behavioral Stat. 7 (1982), 3-18. DOI 10.3102/10769986007001003
[21] A. Szczepańska-Álvarez, C. Hao, Y. Liang, D. von Rosen: Estimation equations for multivariate linear models with Kronecker structured covariance matrices. Commun. Stat., Theory Methods 46 (2017), 7902-7915. DOI 10.1080/03610926.2016.1165852 | MR 3660028 | Zbl 1373.62262

Affiliations:   Malwina Janiszewska, Augustyn Markiewicz (corresponding author), Department of Mathematical and Statistical Methods, Poznań University of Life Sciences, Wojska Polskiego 28, 60-637 Poznań, Poland, e-mail:; Monika Mokrzycka (, Institute of Plant Genetics, Polish Academy of Sciences, Strzeszyńska 34, 60-479 Poznań, Poland.

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