Journal of Fourier Analysis and Applications

, Volume 19, Issue 6, pp 1123–1149 | Cite as

The Road to Deterministic Matrices with the Restricted Isometry Property

  • Afonso S. Bandeira
  • Matthew Fickus
  • Dustin G. Mixon
  • Percy Wong


The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.


Restricted isometry property Compressed sensing Equiangular tight frames 

Mathematics Subject Classification

15A42 05E30 15B52 60F10 94A12 


  1. 1.
    Alexeev, B., Cahill, J., Mixon, D.G.: Full spark frames. J. Fourier Anal. Appl. 18(6), 1167–1194 (2012) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput. 35, 787–803 (2006) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Applebaum, L., Howard, S.D., Searle, S., Calderbank, R.: Chirp sensing codes: deterministic compressed sensing measurements for fast recovery. Appl. Comput. Harmon. Anal. 26, 283–290 (2009) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bandeira, A.S., Dobriban, E., Mixon, D.G., Sawin, W.F.: Certifying the restricted isometry property is hard. IEEE Trans. Inf. Theory 59, 3448–3450 (2013) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baraniuk, R., Davenport, M., DeVore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28, 253–263 (2008) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bernstein, S.N.: Theory of Probability, 4th edn. Gostechizdat, Moscow-Leningrad (1946) Google Scholar
  7. 7.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge Univ. Press, Cambridge (2001) CrossRefMATHGoogle Scholar
  8. 8.
    Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159, 145–185 (2011) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris, Ser. I 346, 589–592 (2008) CrossRefMATHGoogle Scholar
  10. 10.
    Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 44, 4203–4215 (2005) CrossRefGoogle Scholar
  11. 11.
    Candès, E.J., Tao, T.: The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35, 2313–2351 (2007) CrossRefMATHGoogle Scholar
  12. 12.
    Chung, F.R.K., Graham, R.L., Wilson, R.M.: Quasi-random graphs. Combinatorica 9, 345–362 (1989) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cohen, S.D.: Clique numbers of Paley graphs. Quaest. Math. 11, 225–231 (1988) CrossRefMATHGoogle Scholar
  14. 14.
    Davidson, K.R., Szarek, S.J.: Local operator theory, random matrices and Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook in Banach Spaces, vol. I, pp. 317–366. Elsevier, Amsterdam (2001) CrossRefGoogle Scholar
  15. 15.
    DeVore, R.A.: Deterministic constructions of compressed sensing matrices. J. Complex. 23, 918–925 (2007) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fickus, M., Mixon, D.G., Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436, 1014–1027 (2012) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gerschgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. 7, 749–754 (1931) Google Scholar
  19. 19.
    Graham, S.W., Ringrose, C.J.: Lower bounds for least quadratic non-residues. Prog. Math. 85, 269–309 (1990) MathSciNetGoogle Scholar
  20. 20.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43, 439–561 (2006) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Laurent, B., Massart, P.: Adaptive estimation of a quadratic functional by model selection. Ann. Stat. 28, 1302–1338 (2000) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Peralta, R.: On the distribution of quadratic residues and nonresidues modulo a prime number. Math. Comput. 58, 433–440 (1992) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rauhut, H.: Stability results for random sampling of sparse trigonometric polynomials. IEEE Trans. Inf. Theory 54, 5661–5670 (2008) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Renes, J.M.: Equiangular tight frames from Paley tournaments. Linear Algebra Appl. 426, 497–501 (2007) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rudelson, M., Vershynin, R.: On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61, 1025–1045 (2008) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sachs, H.: Über selbstkomplementäre Graphen. Publ. Math. (Debr.) 9, 270–288 (1962) MathSciNetGoogle Scholar
  28. 28.
    Seidel, J.J.: A survey of two-graphs. In: Proc. Intern. Coll. Teorie Combinatorie, pp. 481–511 (1973) Google Scholar
  29. 29.
    Stevenhagen, P., Lenstra, H.W.: Chebotarëv and his density theorem. Math. Intell. 18, 26–37 (1996) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Strohmer, T., Heath, R.W.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Tao, T.: Open question: deterministic UUP matrices.
  32. 32.
    Temlyakov, V.: Greedy Approximations. Cambridge University Press, Cambridge (2011) CrossRefGoogle Scholar
  33. 33.
    van Lint, J.H., Seidel, J.J.: Equilateral point sets in elliptic geometry. Nederl. Akad. Wetensch. Proc. Ser. A 69, 335–348 (1966). Indag. Math. 28 MATHGoogle Scholar
  34. 34.
    Waldron, S.: On the construction of equiangular frames from graphs. Linear Algebra Appl. 431, 2228–2242 (2009) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20, 397–399 (1974) MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51, 1900–1907 (2005) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Yurinskii, V.V.: Exponential inequalities for sums of random vectors. J. Multivar. Anal. 6, 473–499 (1976) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Afonso S. Bandeira
    • 1
  • Matthew Fickus
    • 2
  • Dustin G. Mixon
    • 1
  • Percy Wong
    • 1
  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson Air Force BaseUSA

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