Abstract
Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N 0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N 0, δ n ), any N × n random matrix A = (a ij ) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value s n of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Adamczak, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, Journal of the American Mathematical Society 23 2010, 535–561.
R. Adamczak, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Sharp bounds on the rate of convergence of the empirical covariance matrix, Comptes Rendus Mathématique. Académie des Sciences. Paris 349 2011, 195–200.
Z. D. Bai and Y. Q. Yin, Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix, Annals of Probability 21 1993, 1275–1294.
O. Guédon, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Restricted isometry property for random matrices with heavy-tailed columns, Comptes Rendus Mathématique. Académie des Sciences. Paris 352 2014, 431–434.
V. Koltchinskii and S. Mendelson, Bounding the smallest singular value of a random matrix without concentration, International Mathematics Research Notices, (2015), doi: 10.1093/imrn/rnv096.
P. Lévy, Théorie de l’addition des variables aléatoires, second edition, Gauthier-Villars, Paris, 1954.
A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Advances in Mathematics 195 2005, 491–523.
A. E. Litvak S. Spektor, Quantitative version of a Silverstein’s result, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2116 2014, 335–340.
S. Mendelson and G. Paouris, On the singular values of random matrices, Journal of the European Mathematical Society 16 2014, 823–834.
R. I. Oliveira, The lower tail of random quadratic forms, with applications to ordinary least squares and restricted eigenvalue properties, arXiv:1312.2903.
B. A. Rogozin, On the increase of dispersion of sums of independent random variables, (Russian) Teorija Verojatnostĭ i ee Primenenija 6 1961, 106–108.
M. Rudelson and R. Vershynin, Non-asymptotic theory of random matrices: extreme singular values, in Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1576–102.
M. Rudelson and R. Vershynin, Small ball probabilities for linear images of high dimensional distributions, International Mathematics Research Notices, (2014), doi: 10.1093/imrn/rnu243.
M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular matrix, Communications on Pure and Applied Mathematics 62 2009, 1707–1739.
M. Rudelson and R. Vershynin, The Littlewood–Offord problem and invertibility of random matrices, Advances in Mathematics 218 2008, 600–633.
A. Sankar, D. A. Spielman and S.-H. Teng, Smoothed analysis of the condition numbers and growth factors of matrices, SIAM Journal on Matrix Analysis and Applications 28 2006, 446–476.
D. A. Spielman and S.-H. Teng, Smoothed analysis of algorithms, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 597–606.
N. Srivastava and R. Vershynin, Covariance estimation for distributions with 2 + e moments, Annals of Probability 41 2013, 3081–3111.
T. Tao and V. Vu, Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Annals of of Mathematics 169 2009, 595–632.
T. Tao and V. Vu, Smooth analysis of the condition number and the least singular value, Mathematics of Computation 79 2010, 2333–2352.
T. Tao and V. Vu, The condition number of a randomly perturbed matrix, in STOC’07–Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, 2007, pp. 248–255.
R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing: Theory and Applications, Cambridge University Press, Cambridge, 2012, pp. 210–268.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tikhomirov, K.E. The smallest singular value of random rectangular matrices with no moment assumptions on entries. Isr. J. Math. 212, 289–314 (2016). https://doi.org/10.1007/s11856-016-1287-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1287-8