## The Annals of Probability

### Lower bounds for the smallest singular value of structured random matrices

Nicholas Cook

#### Abstract

We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form

$M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),$ where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.

#### Article information

Source
Ann. Probab., Volume 46, Number 6 (2018), 3442-3500.

Dates
Revised: November 2017
First available in Project Euclid: 25 September 2018

https://projecteuclid.org/euclid.aop/1537862438

Digital Object Identifier
doi:10.1214/17-AOP1251

Mathematical Reviews number (MathSciNet)
MR3857860

Zentralblatt MATH identifier
06975491

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 15B52: Random matrices

#### Citation

Cook, Nicholas. Lower bounds for the smallest singular value of structured random matrices. Ann. Probab. 46 (2018), no. 6, 3442--3500. doi:10.1214/17-AOP1251. https://projecteuclid.org/euclid.aop/1537862438

#### References

• [1] Aljadeff, J., Renfrew, D. and Stern, M. (2015). Eigenvalues of block structured asymmetric random matrices. J. Math. Phys. 56 103502.
• [2] Alon, N. and Shapira, A. (2004). Testing subgraphs in directed graphs. J. Comput. System Sci. 69 353–382.
• [3] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
• [4] Bai, Z. D., Silverstein, J. W. and Yin, Y. Q. (1988). A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivariate Anal. 26 166–168.
• [5] Bandeira, A. S. and van Handel, R. (2016). Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Probab. 44 2479–2506.
• [6] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
• [7] Bourgade, P., Erdos, L., Yau, H.-T. and Yin, J. Universality for a class of random band matrices. Preprint. Available at arXiv:1602.02312.
• [8] Bourgain, J. and Tzafriri, L. (1987). Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math. 57 137–224.
• [9] Cook, N. (2018). Supplement to “Lower bounds for the smallest singular value of structured random matrices.” DOI:10.1214/17-AOP1251SUPP.
• [10] Cook, N. A. (2016). Spectral properties of non-Hermitian random matrices. Ph.D. thesis, University of California, Los Angeles.
• [11] Cook, N. A., Hachem, W., Najim, J. and Renfrew, D. Limiting spectral distribution for non-Hermitian random matrices with a variance profile. Preprint. Available at arXiv:1612.04428.
• [12] Edelman, A. (1988). Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 543–560.
• [13] Gowers, W. T. (1997). Lower bounds of tower type for Szemerédi’s uniformity lemma. Geom. Funct. Anal. 7 322–337.
• [14] Hachem, W., Loubaton, P. and Najim, J. (2007). Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 875–930.
• [15] Komlós, J. Circulated manuscript, 1977. Edited version available online at http://www.math.rutgers.edu/~komlos/01short.pdf.
• [16] Komlós, J. (1967). On the determinant of $(0,1)$ matrices. Studia Sci. Math. Hungar. 2 7–21.
• [17] Komlós, J. (1968). On the determinant of random matrices. Studia Sci. Math. Hungar. 3 387–399.
• [18] Komlós, J. and Simonovits, M. (1996). Szemerédi’s regularity lemma and its applications in graph theory. In Combinatorics, Paul Erdős Is Eighty, Vol. 2 (Keszthely, 1993). Bolyai Soc. Math. Stud. 2 295–352. János Bolyai Math. Soc., Budapest.
• [19] Latała, R. (2005). Some estimates of norms of random matrices. Proc. Amer. Math. Soc. 133 1273–1282.
• [20] Litvak, A. E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N. and Youssef, P. (2017). Adjacency matrices of random digraphs: Singularity and anti-concentration. J. Math. Anal. Appl. 445 1447–1491.
• [21] Litvak, A. E., Pajor, A., Rudelson, M. and Tomczak-Jaegermann, N. (2005). Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 491–523.
• [22] Litvak, A. E., Pajor, A., Rudelson, M., Tomczak-Jaegermann, N. and Vershynin, R. (2005). Euclidean embeddings in spaces of finite volume ratio via random matrices. J. Reine Angew. Math. 589 1–19.
• [23] Litvak, A. E. and Rivasplata, O. (2012). Smallest singular value of sparse random matrices. Studia Math. 212 195–218.
• [24] Marcus, A. W., Spielman, D. A. and Srivastava, N. (2014). Ramanujan graphs and the solution of the Kadison–Singer problem. In Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III 363–386. Kyung Moon Sa, Seoul.
• [25] Nguyen, H. H. and Vu, V. H. (2018). Normal vector of a random hyperplane. International Mathematics Research Notices 2018 1754-1778.
• [26] Rajan, K. and Abbott, L. F. (2006). Eigenvalue spectra of random matrices for neural networks. Phys. Rev. Lett. 97 188104.
• [27] Rebrova, E. and Tikhomirov, K. Covering of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries. Preprint. Available at arXiv:1508.06690.
• [28] Rudelson, M. (2008). Invertibility of random matrices: Norm of the inverse. Ann. of Math. (2) 168 575–600.
• [29] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
• [30] Rudelson, M. and Vershynin, R. (2008). The least singular value of a random square matrix is $O(n^{-1/2})$. C. R. Math. Acad. Sci. Paris 346 893–896.
• [31] Rudelson, M. and Zeitouni, O. (2016). Singular values of Gaussian matrices and permanent estimators. Random Structures Algorithms 48 183–212.
• [32] Sankar, A., Spielman, D. A. and Teng, S.-H. (2006). Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Anal. Appl. 28 446–476.
• [33] Spielman, D. A. and Srivastava, N. (2012). An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 83–91.
• [34] Szemerédi, E. (1978). Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloq. Internat. CNRS 260 399–401. CNRS, Paris.
• [35] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34.
• [36] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
• [37] Tao, T. and Vu, V. (2010). Random matrices: The distribution of the smallest singular values. Geom. Funct. Anal. 20 260–297.
• [38] Tao, T. and Vu, V. (2010). Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 2023–2065. With an appendix by Manjunath Krishnapur.
• [39] Tao, T. and Vu, V. (2010). Smooth analysis of the condition number and the least singular value. Math. Comp. 79 2333–2352.
• [40] Tao, T. and Vu, V. H. (2009). Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. of Math. (2) 169 595–632.
• [41] Tulino, A. M. and Verdú, S. (2004). Random Matrix Theory and Wireless Communications 1. Now Publishers Inc.
• [42] van Handel, R. (2017). On the spectral norm of Gaussian random matrices. Trans. Amer. Math. Soc. 369 8161–8178.
• [43] Vershynin, R. (2011). Spectral norm of products of random and deterministic matrices. Probab. Theory Related Fields 150 471–509.
• [44] von Neumann, J. and Goldstine, H. H. (1947). Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 53 1021–1099.
• [45] Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 509–521.

#### Supplemental materials

• Supplement to “Lower bounds for the smallest singular value of structured random matrices”. This supplement contains the proofs of Corollary 1.16 and Lemmas 2.5, 2.7 and 2.8.