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Keywords:
matrix decomposition; factor analysis; covariance matrices; low rank matrices; projections
matrix decomposition; factor analysis; covariance matrices; low rank matrices; projections
Summary:
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem. This algorithm appears to perform extremely well and is extremely fast even when the given covariance matrix has a very large dimension. The effectiveness of the algorithm is assessed through simulation studies and by applications to three real benchmark datasets that are considered. A local convergence analysis of the algorithm is also presented.
References:
[1] Agarwal, A., Negahban, S., Wainwright, M. J.: Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. Ann. Statist. 40 (2012), 2, 1171-1197. DOI 10.1214/12-aos1000 | MR 2985947
[2] Bai, J., Li, K., al., et: Statistical analysis of factor models of high dimension. Ann. Statist. 40 (2012), 1, 436-465. DOI 10.1214/11-AOS966 | MR 3014313
[3] Bai, J., Ng, S.: Determining the number of factors in approximate factor models. Econometrica 70 (2002), 1, 191-221. DOI 10.1111/1468-0262.00273 | MR 1926259
[4] Bai, J., Ng, S.: Large dimensional factor analysis. Found. Trends Econometr. 3 (2008), 2, 89-163. DOI 10.1561/0800000002
[5] Bertsimas, D., Copenhaver, M. S., Mazumder, R.: Certifiably optimal low rank factor analysis. J. Machine Learning Res. 18 (2017), 29, 1-53. MR 3634896
[6] Ciccone, V., Ferrante, A., Zorzi, M.: Factor analysis with finite data. In: Proc. 56th Annual Conference on Decision and Control (CDC), VIC, Melbourne 2017, pp. 4046-4051. DOI 10.1109/cdc.2017.8264253
[7] Ciccone, V., Ferrante, A., Zorzi, M.: Robust identification of "Sparse Plus Low-rank" graphical models: An optimization approach. In: Proc. 2018 IEEE Conference on Decision and Control (CDC), Miami Beach 2018, pp. 2241-2246. DOI 10.1109/cdc.2018.8619796
[8] Ciccone, V., Ferrante, A., Zorzi, M.: Factor models with real data: A robust estimation of the number of factors. IEEE Trans. Automat. Control 64 (2019), 6, 2412-2425. DOI 10.1109/tac.2018.2867372 | MR 3960088
[9] Deistler, M., Zinner, C.: Modelling high-dimensional time series by generalized linear dynamic factor models: An introductory survey. Comm. Inform. Systems 7 (2007), 2, 153-166. DOI 10.4310/cis.2007.v7.n2.a3 | MR 2344194
[10] Riccia, G. Della, Shapiro, A.: Minimum rank and minimum trace of covariance matrices. Psychometrika 47 (1982), 443-448. DOI 10.1007/bf02293708 | MR 0691830
[11] Fazel, M.: Matrix rank minimization with applications. Elec. Eng. Dept. Stanford University 54 (2002), 1-130.
[12] Fazel, M., Hindi, H., Boyd, S.: Rank minimization and applications in system theory. In: Proc. American Control Conference 4 (2004), pp. 3273-3278. DOI 10.23919/acc.2004.1384521
[13] Geweke, J.: The dynamic factor analysis of economic time series models. In: Latent Variables in Socio-Economic Models, SSRI workshop series, North-Holland 1977, pp. 365-383.
[14] Guttman, L.: Some necessary conditions for common-factor analysis. Psychometrika 19 (1954), 2, 149-161. DOI 10.1007/bf02289162 | MR 0091235
[15] Harman, H. H., Jones, W. H.: Factor analysis by minimizing residuals (minres). Psychometrika 31 (1066), 3, 351-368. DOI 10.1007/bf02289468 | MR 0201008
[16] Heij, C., Scherrer, W., Deistler, M.: System identification by dynamic factor models. SIAM J. Control Optim. 35 (1997), 6, 1924-1951. DOI 10.1137/s0363012995282127 | MR 1478648
[17] Helmke, U., Shayman, M. A.: Critical points of matrix least squares distance functions. Linear Algebra Appl. 215 (1995), 1-19. DOI 10.1016/0024-3795(93)00070-g | MR 1317470
[18] Lax, P. D.: Linear Algebra and Its Applications. Second edition. Wiley-Interscience, 2007. MR 2356919
[19] Lewis, A. S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33 (2008), 1, 216-234. DOI 10.1287/moor.1070.0291 | MR 2393548
[20] Ning, L., Georgiou, T. T., Tannenbaum, A., Boyd, S. P.: Linear models based on noisy data and the Frisch scheme. SIAM Rev. 57 (2015), 2, 167-197. DOI 10.1137/130921179 | MR 3345338
[21] Reiersøl, O.: Identifiability of a linear relation between variables which are subject to error. Econometrica: J. Economet. Soc. 18 (1950), 4, 375-389. DOI 10.2307/1907835 | MR 0038054
[22] Scherrer, W., Deistler, M.: A structure theory for linear dynamic errors-in-variables models. SIAM J. Control Optim. 36 (1998), 6, 2148-2175. DOI 10.1137/s0363012994262464 | MR 1638960
[23] Shapiro, A.: Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis. Psychometrika 47 (1982), 2, 187-199. DOI 10.1007/bf02296274 | MR 0667012
[24] Shapiro, A., Berge, J. M. F. Ten: Statistical inference of minimum rank factor analysis. Psychometrika 67 (2002), 1, 79-94. DOI 10.1007/bf02294710 | MR 1960829
[25] Zorzi, M., Sepulchre, R.: AR identification of latent-variable graphical models. IEEE Trans. Automat. Control 61 (2016), 9, 2327-2340. DOI 10.1109/tac.2015.2491678 | MR 3545056
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