Abstract
We study the structure of \(n \times n\) random matrices with centered i.i.d. entries having only two finite moments. In the recent joint work with R. Vershynin, we have shown that the operator norm of such matrix A can be reduced to the optimal order \(O(\sqrt{n})\) with high probability by zeroing out a small submatrix of A, but did not describe the structure of this “bad” submatrix nor provide a constructive way to find it. In the current paper, we give a very simple description of a small “bad” subset of entries. We show that it is enough to zero out a small fraction of the rows and columns of A with largest \(L_2\) norms to bring the operator norm of A to the almost optimal order \(O(\sqrt{n \log \log n})\), under additional assumption that the matrix entries are symmetrically distributed. As a corollary, we also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same norm regularization. The main component of the proof is the development of techniques extending constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.
Similar content being viewed by others
References
Auffinger, A., Tang, S.: Extreme eigenvalues of sparse, heavy tailed random matrices. Stoch. Process. Appl. 126(11), 3310–3330 (2016)
Bandeira, A.S., van Handel, R.: Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Prob. 44(4), 2479–2506 (2016)
Chin, P., Rao, A., Vu, V.: Stochastic block model and community detection in sparse graphs: a spectral algorithm with optimal rate of recovery. In: Conference on Learning Theory, pp. 391–423 (2015)
David, H.A., Nagaraja, H.N.: Order statistics. Encycl. Stat. Sci. 9, 22 (2004)
Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Random Struct. Algorithms 27(2), 251–275 (2005)
Friedman, J., Kahn, J., Szemeredi, E.: On the second eigenvalue of random regular graphs. In: Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, pp. 587–598 (1989)
Krivelevich, M., Sudakov, B.: The largest eigenvalue of sparse random graphs. Comb. Prob. Comput. 12(1), 61–72 (2003)
Latała, R.: Some estimates of norms of random matrices. Am. Math. Soc. 133(5), 1273–1282 (2005)
Le, C.M., Levina, E., Vershynin, R.: Concentration and regularization of random graphs. Random Struct. Algorithms 51(3), 538–561 (2017)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (2013)
Litvak, A.E., Spektor, S.: Quantitative version of a Silverstein’s result. In: Klartag, B., Milman, E. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2116, pp. 335–340. Springer, Cham (2014)
Rebrova, E., Tikhomirov, K.: Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries. Israel J. Math. 227(2), 507–544 (2018)
Rebrova, E., Vershynin, R.: Norms of random matrices: local and global problems. Adv. Math. 324, 40–83 (2018)
Seginer, Y.: The expected norm of random matrices. Comb. Prob. Comput. 9(2), 149–166 (2000)
Silverstein, J.W.: On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivar. Anal. 30(2), 307–311 (1989)
Tao, T.: Topics in Random Matrix Theory, vol. 132. American Mathematical Society, London (2012)
Tropp, J.A.: An introduction to matrix concentration inequalities. Found. Trends Mach. Learn. 8(1–2), 1–230 (2015)
van Handel, R.: On the spectral norm of Gaussian random matrices. Trans. Am. Math. Soc. 369(11), 8161–8178 (2017)
Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Compressed Sensing Theory and Applications, pp. 210–268 (2012)
Vershynin, R.: High-Dimensional Probability: An Introduction with Applications in Data Science, vol. 47. Cambridge University Press, Cambridge (2018)
Yin, Y.-Q., Bai, Z.-D., Krishnaiah, P.R.: On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Prob. Theory Related Fields 78(4), 509–521 (1988)
Zieliński, R., Zieliński, W.: Best exact nonparametric confidence intervals for quantiles. Statistics 39(1), 67–71 (2005)
Acknowledgements
The author is grateful to Roman Vershynin for the suggestion to look at the work of Feige and Ofek, helpful and encouraging discussions, as well as comments related to the presentation of the paper. The author is also grateful to Konstantin Tikhomirov for mentioning the idea to estimate quantiles of the entries distribution from their order statistics, which made Algorithm 1 more elegant.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Work supported in part by Allen Shields Memorial Fellowship.
Rights and permissions
About this article
Cite this article
Rebrova, E. Constructive Regularization of the Random Matrix Norm. J Theor Probab 33, 1768–1790 (2020). https://doi.org/10.1007/s10959-019-00929-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-019-00929-6