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Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1301–1347 | Cite as

The smallest singular value of a shifted d-regular random square matrix

  • Alexander E. LitvakEmail author
  • Anna Lytova
  • Konstantin Tikhomirov
  • Nicole Tomczak-Jaegermann
  • Pierre Youssef
Article

Abstract

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let \(C_1<d< c n/\log ^2 n\) and let \(\mathcal {M}_{n,d}\) be the set of all \(n\times n\) square matrices with 0 / 1 entries, such that each row and each column of every matrix in \(\mathcal {M}_{n,d}\) has exactly d ones. Let M be a random matrix uniformly distributed on \(\mathcal {M}_{n,d}\). Then the smallest singular value \(s_{n} (M)\) of M is greater than \(n^{-6}\) with probability at least \(1-C_2\log ^2 d/\sqrt{d}\), where c, \(C_1\), and \(C_2\) are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form \(M-z\,\mathrm{Id}\), where \(\mathrm{Id}\) is the identity matrix and z is a fixed complex number.

Keywords

Adjacency matrices Anti-concentration Condition number Invertibility Littlewood–Offord theory Random graphs Random matrices Regular graphs Singular probability Singularity Sparse matrices Smallest singular value 

Mathematics Subject Classification

Primary: 60B20 15B52 46B06 05C80 Secondary: 46B09 60C05 

Notes

Acknowledgements

We are grateful to an anonymous referee for careful reading the first draft of the manuscript and many valuable suggestions, which helped us to improve presentation. The second and the third named authors would like to thank University of Alberta for excellent working conditions in January–August 2016, when a significant part of this work was done.

References

  1. 1.
    Adamczak, R., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Quantitative estimates of the convergence of the empirical covariance matrix in log-concave Ensembles. J. Am. Math. Soc. 23, 535–561 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adamczak, R., Guedon, O., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Condition number of a square matrix with i.i.d. columns drawn from a convex body. Proc. Am. Math. Soc. 140, 987–998 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bai, Z., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices, Springer Series in Statistics, 2nd edn. Springer, New York (2010)zbMATHGoogle Scholar
  4. 4.
    Basak, A., Cook, N., Zeitouni, O.: Circular law for the sum of random permutation matrices. Electron. J. Probab. 23(33), 1–51 (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Basak, A., Rudelson, M.: Invertibility of sparse non-hermitian matrices. Adv. Math. 310, 426–483 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bau, D., Trefethen, L.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997)zbMATHGoogle Scholar
  7. 7.
    Bordenave, C., Chafaï, D.: Around the circular law. Probab. Surv. 9, 1–89 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.: Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs, vol. 196. American Mathematical Society, Providence (2014)zbMATHGoogle Scholar
  9. 9.
    Cook, N.A.: Discrepancy properties for random regular digraphs. Random Struct. Algorithms 50(1), 23–58 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cook, N.A.: On the singularity of adjacency matrices for random regular digraphs. Probab. Theory Relat. Fields 167(1–2), 143–200 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cook, N.: The circular law for random regular digraphs. arXiv:1703.05839
  12. 12.
    Costello, K.P., Vu, V.: The rank of random graphs. Random Struct. Algorithms 33(3), 269–285 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davidson, K.R., Szarek, S.J.: Local Operator Theory, Random Matrices and Banach Spaces. Handbook of the Geometry of Banach Spaces, vol. 1, p. 317366. North-Holland, Amsterdam (2001)Google Scholar
  14. 14.
    Erdős, P.: On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51, 898–902 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Erdös, L., Yau, H.-T.: A Dynamical Approach to Random Matrix Theory, Courant Lecture Notes in Mathematics, 28. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2017)Google Scholar
  16. 16.
    Frieze, A.: Random structures and algorithms. Proc. ICM 1, 311–340 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Guedon, O., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: On the interval of fluctuation of the singular values of random matrices. J. Eur. Math. Soc. (JEMS) 19(5), 1469–1505 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kashin, B.S.: Diameters of some finite-dimensional sets and classes of smooth functions. Izv. Akad. Nauk SSSR, Ser. Mat. 41, 334–351 (1977)MathSciNetGoogle Scholar
  19. 19.
    Kleitman, D.J.: On a Lemma of Littlewood and Offord on the distributions of linear combinations of vectors. Adv. Math. 5, 155–157 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koltchinskii, V., Mendelson, S.: Bounding the smallest singular value of a random matrix without concentration. Int. Math. Res. Not. 23, 12991–13008 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Litvak, A.E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., Youssef, P.: Adjacency matrices of random digraphs: singularity and anti-concentration. J. Math. Anal. Appl. 445, 1447–1491 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Litvak, A.E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., Youssef, P.: Anti-concentration property for random digraphs and invertibility of their adjacency matrices. C.R. Math. Acad. Sci. Paris 354, 121–124 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Litvak, A.E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., Youssef, P.: Circular law for sparse random regular digraphs. SubmittedGoogle Scholar
  25. 25.
    Litvak, A.E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., Youssef, P.: Structure of eigenvectors of random regular digraphs. SubmittedGoogle Scholar
  26. 26.
    Litvak, A.E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., Youssef, P.: The rank of random regular digraphs of constant degree. J. Complex. (2018).  https://doi.org/10.1016/j.jco.2018.05.004 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Litvak, A.E., Pajor, A., Rudelson, M., Tomczak-Jaegermann, N.: Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195, 491–523 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Litvak, A.E., Pajor, A., Rudelson, M., Tomczak-Jaegermann, N., Vershynin, R.: Random Euclidean embeddings in spaces of bounded volume ratio. C.R. Acad. Sci. Paris, Ser 1, Math. 339, 33–38 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mendelson, S., Paouris, G.: On singular values of matrices. J. EMS 16, 823–834 (2014)zbMATHGoogle Scholar
  30. 30.
    Oliveira, R.I.: The lower tail of random quadratic forms, with applications to ordinary least squares and restricted eigenvalue properties. PTRF 166, 1175–1194 (2016)zbMATHGoogle Scholar
  31. 31.
    Rebrova, E., Tikhomirov, K.: Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries. Israel J. Math. To appear. arXiv:1508.06690
  32. 32.
    Rudelson, M., Vershynin, R.: The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218, 600–633 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rudelson, M., Vershynin, R.: Smallest singular value of a random rectangular matrix. Commun. Pure Appl. Math. 62(12), 1707–1739 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rudelson, M., Vershynin, R.: Non-asymptotic theory of random matrices: extreme singular values. In: Proceedings ICM, Vol. III, 1576–1602, Hindustan Book Agency, New Delhi (2010)Google Scholar
  35. 35.
    Sankar, A., Spielman, D.A., Teng, S.-H.: Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Anal. Appl. 28(2), 446–476 (2006). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schechtman, G.: Special orthogonal splittings of \(L_1^{2k}\). Isr. J. Math. 139, 337–347 (2004)CrossRefzbMATHGoogle Scholar
  37. 37.
    Smale, S.: On the efficiency of algorithms of analysis. Bull. Am. Math. Soc. (N.S.) 13, 87–121 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms. In: Proceedings ICM, vol. I, pp. 597–606. Higher Ed. Press, Beijing (2002)Google Scholar
  39. 39.
    Srivastava, N., Vershynin, R.: Covariance estimation for distributions with \(2+\varepsilon \) moments. Ann. Probab. 41(5), 3081–3111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Tao, T., Vu, V.: The condition number of a randomly perturbed matrix. In: STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 248–255, ACM, New York (2007)Google Scholar
  41. 41.
    Tao, T., Vu, V.: Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. Math. 169, 595–632 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Tao, T., Vu, V.: Smooth analysis of the condition number and the least singular value. Math. Comput. 79(272), 2333–2352 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Tao, T., Vu, V.: Random matrices: universality of ESDs and the circular law. Ann. Probab. 38(5), 2023–2065 (2010). With an appendix by Manjunath KrishnapurMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tikhomirov, K.: Sample covariance matrices of heavy-tailed distributions. Int. Math. Res. Not. (2017).  https://doi.org/10.1093/imrn/rnx067 zbMATHGoogle Scholar
  45. 45.
    Tomczak-Jaegermann, N.: Banach–Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1989)Google Scholar
  46. 46.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    von Neumann, J.: Collected works. Vol. V: Design of Computers, Theory of Automata and Numerical Analysis. General editor: A. H. Taub. A Pergamon Press Book The Macmillan Co., New York (1963)Google Scholar
  48. 48.
    von Neumann, J., Goldstine, H.H.: Numerical inverting of matrices of high order. Bull. Am. Math. Soc. 53, 1021–1099 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Vu, V.: Random discrete matrices, Horizons of combinatorics, Bolyai Soc. Math. Stud., 17, 257–280, Springer, Berlin (2008)Google Scholar
  50. 50.
    Vu, V.H.: Combinatorial problems in random matrix theory. In: Proceedings ICM, Vol. IV, pp. 489–508, Kyung Moon Sa, Seoul (2014)Google Scholar
  51. 51.
    Yaskov, P.: Lower bounds on the smallest eigenvalue of a sample covariance matrix. Electron. Commun. Probab. 19, 1–10 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yaskov, P.: Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition. Electron. Commun. Probab. 20(44), 9 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alexander E. Litvak
    • 1
    Email author
  • Anna Lytova
    • 2
  • Konstantin Tikhomirov
    • 3
  • Nicole Tomczak-Jaegermann
    • 1
  • Pierre Youssef
    • 4
  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Faculty of Mathematics, Physics, and Computer ScienceUniversity of OpoleOpolePoland
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA
  4. 4.Laboratoire de Probabilités, Statistique et ModélisationUniversité Paris DiderotParisFrance

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