Restricted Invertibility Revisited



Suppose that \(m,n \in \mathbb{N}\) and that \(A: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is a linear operator. It is shown here that if \(k,r \in \mathbb{N}\) satisfy \(k <r\leqslant \mathbf{rank}(A)\) then there exists a subset σ ⊆ {1, , m} with | σ | = k such that the restriction of A to \(\mathbb{R}^{\sigma } \subseteq \mathbb{R}^{m}\) is invertible, and moreover the operator norm of the inverse \(A^{-1}: A(\mathbb{R}^{\sigma }) \rightarrow \mathbb{R}^{m}\) is at most a constant multiple of the quantity \(\sqrt{mr/((r - k)\sum _{i=r }^{m }\mathsf{s } _{i } (A)^{2 } )}\), where \(\mathsf{s}_{1}(A)\geqslant \ldots \geqslant \mathsf{s}_{m}(A)\) are the singular values of A. This improves over a series of works, starting from the seminal Bourgain–Tzafriri Restricted Invertibility Principle, through the works of Vershynin, Spielman–Srivastava and Marcus–Spielman–Srivastava. In particular, this directly implies an improved restricted invertibility principle in terms of Schatten–von Neumann norms.



We thank Bill Johnson for helpful discussions. This work was initiated while we were participating in the workshop Beyond Kadison–Singer: paving and consequences at the American Institute of Mathematics. We thank the organizers for the excellent working conditions.


  1. 1.
    R. P. Anstee, L. Rónyai, A. Sali, Shattering news. Graphs Combin. 18(1), 59–73 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    J. Batson, D. A. Spielman, N. Srivastava, Twice-Ramanujan sparsifiers. SIAM J. Comput. 41(6), 1704–1721 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    K. Berman, H. Halpern, V. Kaftal, G. Weiss, Matrix norm inequalities and the relative Dixmier property. Integr. Equ. Oper. Theory 11(1), 28–48 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    R. Bhatia, in Matrix Analysis. Graduate Texts in Mathematics, vol. 169 (Springer, New York, 1997). ISBN:0-387-94846-5. doi:10.1007/ 978-1-4612-0653-8Google Scholar
  5. 5.
    J. Bourgain, L. Tzafriri, Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis. Isr. J. Math. 57(2), 137–224 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    J. Bourgain, L. Tzafriri, On a problem of Kadison and Singer. J. Reine Angew. Math. 420, 1–43 (1991)MathSciNetMATHGoogle Scholar
  7. 7.
    J. Bourgain, S. J. Szarek, The Banach-Mazur distance to the cube and the Dvoretzky–Rogers factorization. Isr. J. Math. 62(2), 169–180 (1988)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    J. Bourgain, L. Tzafriri, Restricted invertibility of matrices and applications, in Analysis at Urbana, Volume II, Urbana, 1986–1987. London Mathematical Society Lecture Note Series, vol. 138 (Cambridge University Press, Cambridge, 1989), pp. 61–107Google Scholar
  9. 9.
    P. J. Davis, in Circulant Matrices. Pure and Applied Mathematics (Wiley, New York/Chichester/Brisbane, 1979). ISBN:0-471-05771-1. A Wiley-Interscience PublicationGoogle Scholar
  10. 10.
    J. Diestel, H. Jarchow, A. Tonge, in Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43 (Cambridge University Press, Cambridge, 1995). ISBN:0-521-43168-9. doi:10.1017/ CBO9780511526138Google Scholar
  11. 11.
    K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. I. Proc. Natl. Acad. Sci. U. S. A. 35, 652–655 (1949)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.A. Giannopoulos, A note on the Banach–Mazur distance to the cube, in Geometric Aspects of Functional Analysis, Israel, 1992–1994. Operator Theory: Advances and Applications, vol. 77 (Birkhäuser, Basel, 1995), pp. 67–73Google Scholar
  13. 13.
    A. A. Giannopoulos, A proportional Dvoretzky-Rogers factorization result. Proc. Am. Math. Soc. 124(1), 233–241 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)MathSciNetGoogle Scholar
  15. 15.
    I. Kra, S. R. Simanca, On circulant matrices. Not. Am. Math. Soc. 59(3), 368–377 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    A. W. Marcus, D. A. Spielman, N. Srivastava, Ramanujan graphs and the solution of the Kadison–Singer problem, in Proceedings of the 2014 International Congress of Mathematicians, Volume III (2014), pp. 363–386. Available at
  17. 17.
    A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. Math. (2) 182(1), 307–325 (2015)Google Scholar
  18. 18.
    A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families II: mixed characteristic polynomials and the Kadison-Singer problem. Ann. Math. (2) 182(1), 327–350 (2015)Google Scholar
  19. 19.
    A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families III: improved restricted invertibility estimates (2016, in preparation)Google Scholar
  20. 20.
    A. Naor, Sparse quadratic forms and their geometric applications [following Batson, Spielman and Srivastava]. Astérisque, (348): Exp. No. 1033, viii, 189–217 (2012). Séminaire Bourbaki: vol. 2010/2011. Exposés 1027–1042Google Scholar
  21. 21.
    A. Pajor, Sous-espacesl 1 n des espaces de Banach. Travaux en Cours [Works in Progress], vol. 16 (Hermann, Paris, 1985). ISBN:2–7056-6021–6. With an introduction by Gilles PisierGoogle Scholar
  22. 22.
    A. Pietsch, Absolut p-summierende Abbildungen in normierten Räumen. Stud. Math. 28, 333–353 (1966/1967)Google Scholar
  23. 23.
    G. Pisier, in Factorization of Linear Operators and Geometry of Banach Spaces. CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, vol. 60 (American Mathematical Society, Providence, 1986). ISBN:0-8218-0710-2Google Scholar
  24. 24.
    N. Sauer, On the density of families of sets. J. Comb. Theory Ser. A 13, 145–147 (1972)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages. Pac. J. Math. 41, 247–261 (1972)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    D. A. Spielman, N. Srivastava, An elementary proof of the restricted invertibility theorem. Isr. J. Math. 190, 83–91 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    N. Srivastava, R. Vershynin, Covariance estimation for distributions with 2 + ɛ moments. Ann. Probab. 41(5), 3081–3111 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    E. Størmer, in Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics (Springer, Heidelberg, 2013). ISBN:978-3-642-34368-1; 978-3-642-34369-8. doi:10.1007/978-3-642-34369-8Google Scholar
  29. 29.
    S. J. Szarek, On the geometry of the Banach-Mazur compactum, in Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Mathematics, vol. 1470 (Springer, Berlin, 1991), pp. 48–59. doi:10.1007/ BFb0090211Google Scholar
  30. 30.
    S.J. Szarek, M. Talagrand, An “isomorphic” version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, in Geometric Aspects of Functional Analysis (1987–1988). Lecture Notes in Mathematics, vol. 1376 (Springer, Berlin, 1989), pp. 105–112. doi:10. 1007/BFb0090050Google Scholar
  31. 31.
    N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38 (Longman Scientific & Technical, Harlow; co-published in the United States with John Wiley & Sons, Inc., New York, 1989). ISBN:0-582-01374-7Google Scholar
  32. 32.
    J.A. Tropp, Column subset selection, matrix factorization, and eigenvalue optimization, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia, 2009), pp. 978–986Google Scholar
  33. 33.
    R. Vershynin, John’s decompositions: selecting a large part. Isr. J. Math. 122, 253–277 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    P. Youssef, Restricted invertibility and the Banach-Mazur distance to the cube. Mathematika 60(1), 201–218 (2014)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton University Fine HallPrincetonUSA
  2. 2.Laboratoire de Probabilités et de Modèles AléatoiresUniversité Paris-DiderotParisFrance

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