Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

Positive semi-definite matrices commonly occur as normal of least squares problems in statistics or kernel machine learning and approximation theory. They are typically large dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea reduce computational complexity is approximate the by one low rank. The optimal well understood choice based on eigenvalue decomposition matrix. Unfortunately, this computationally expensive. Cheaper methods Gaussian elimination but they require pivoting. We show how invariant theory provides explicit error formulas for an averaged volume sampling. formula leads ratios elementary symmetric polynomials eigenvalues. discuss several bounds expected norm include examples where computed exactly. References A. Dax. “On extremum properties orthogonal quotients matrices”. In: Lin. Alg. Appl. 432.5 (2010), pp. 1234–1257. doi: 10.1016/j.laa.2009.10.034. M. Dereziński W. Mahoney. Determinantal Point Processes Randomized Numerical Linear Algebra. 2020. url: https://arxiv.org/abs/2005.03185. Deshpande, L. Rademacher, S. Vempala, G. Wang. “Matrix projective clustering via sampling”. Proceedings Seventeenth Annual ACM-SIAM Symposium Discrete Algorithm. SODA ’06. Miami, Florida: Society Industrial Applied Mathematics, 2006, 1117–1126. https://dl.acm.org/doi/abs/10.5555/1109557.1109681. Goreinov, E. Tyrtyshnikov, N. Zamarashkin. “A pseudoskeleton approximations”. 261.1 (1997), 1–21. 10.1016/S0024-3795(96)00301-1. Mahoney P. Drineas. “CUR decompositions improved data analysis”. Proc. Nat. Acad. Sci. 106.3 (Jan. 20, 2009), 697–702. 10.1073/pnas.0803205106. Marcus Lopes. “Inequalities functions Hermitian Can. J. Math. 9 (1957), 305–312. 10.4153/CJM-1957-037-9.