×

Controlling the least eigenvalue of a random Gram matrix. (English) Zbl 1381.60024

Summary: Consider a \(n\times p\) random matrix \(X\) with i.i.d. rows. We show that the least eigenvalue of \(n^{-1} X^\top X\) is bounded away from zero with high probability when \(p/n \leqslant y\) for some fixed \(y\) in \((0,1)\) and normalized orthogonal projections of rows are not too close to zero. The principal difference from the previous results is that \(y\) can be arbitrarily close to one. Our results cover many cases of interest in high-dimensional statistics and random matrix theory.

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
15B48 Positive matrices and their generalizations; cones of matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adamczak, R., A note on the Hanson-Wright inequality for random vectors with dependencies, Electron. Commun. Probab., 20, 1-3 (2015), Article 72
[2] Batson, J. D.; Spielman, D. A.; Srivastava, N., Twice-Ramanujan sparsifiers, SIAM J. Comput., 41, 6, 225-262 (2009) · Zbl 1304.05130
[3] Caron, R. J.; Traynor, T., The zero set of a polynomial (May 2005), Department of Mathematics and Statistics, University of Windsor: Department of Mathematics and Statistics, University of Windsor Windsor, ON, Canada, Available:
[4] Chafai, D.; Tikhomirov, K., On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence · Zbl 1384.15008
[5] van de Geer, S.; Muro, A., On higher order isotropy conditions and lower bounds for sparse quadratic forms, Electron. J. Stat., 8, 3031-3061 (2014) · Zbl 1308.62148
[6] El Karoui, N., Concentration of measure and spectra of random matrices: applications to correlation matrices, elliptical distributions and beyond, Ann. Appl. Probab., 19, 2362-2405 (2009) · Zbl 1255.62156
[7] Koltchinskii, V.; Mendelson, S., Bounding the smallest singular value of a random matrix without concentration, Int. Math. Res. Not. IMRN (2015) · Zbl 1331.15027
[8] Lecué, G.; Mendelson, S., Sparse recovery under weak moment assumptions, J. Eur. Math. Soc. (JEMS) (2016), forthcoming · Zbl 1414.62135
[9] Marshall, A. W.; Olkin, I.; Arnold, B. C., Inequalities: Theory of Majorization and Its Applications (2010), Springer
[10] Maurer, A., A bound on the deviation probability for sums of non-negative random variables, JIPAM. J. Inequal. Pure Appl. Math., 4, 1-6 (2003) · Zbl 1021.60036
[11] Oliveira, R. I., The lower tail of random quadratic forms, with applications to ordinary least squares and restricted eigenvalue properties
[12] Pajor, A.; Pastur, L., On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution, Studia Math., 195, 11-29 (2009) · Zbl 1178.15023
[13] Pastur, L.; Shcherbina, M., Eigenvalue Distribution of Large Random Matrices, Math. Surveys Monogr., vol. 171 (2011), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1244.15002
[14] Srivastava, N.; Vershynin, R., Covariance estimation for distributions with \(2 + \varepsilon\) moments, Ann. Probab., 41, 3081-3111 (2013) · Zbl 1293.62121
[15] Tikhomirov, K., The smallest singular value of random rectangular matrices with no moment assumptions on entries, Israel J. Math., 1-26 (2016)
[16] Vu, V.; Wang, K., Random weighted projections, random quadratic forms and random eigenvectors, Random Structures Algorithms, 47, 4, 792-821 (2015) · Zbl 1384.60029
[17] Yaskov, P. A., Lower bounds on the smallest eigenvalue of a sample covariance matrix, Electron. Commun. Probab., 19, 1-10 (2014), Article 83 · Zbl 1320.60023
[18] Yaskov, P. A., Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition, Electron. Commun. Probab., 20, 1-9 (2015), Article 44 · Zbl 1318.60009
[19] Yaskov, P. A., The universality principle for spectral distributions of sample covariance matrices · Zbl 1396.62219
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.