Abstract
Designing computational experiments involving ℓ 1 minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number k of nonzero entries is, in general, difficult. Several conditions were introduced which guarantee that, for example for small k or for certain matrices, simply placing entries with desired characteristics on a randomly chosen support will produce vectors which can be recovered by ℓ 1 minimization. In this work, we consider the case of large k and introduce a method which constructs vectors which support has the cardinality k and which can be recovered via ℓ 1 minimization. Especially, such vectors with largest possible support can be constructed. Further, we propose a methodology to quickly check whether a given vector is recoverable. This method can be cast as a linear program and we compare it with solving ℓ 1 minimization directly. Moreover, we gain new insights in the recoverability in a non-asymptotic regime. Our proposal for quickly checking vectors bases on optimality conditions for exact solutions of the ℓ 1 minimization. These conditions can be used to establish equivalence classes of recoverable vectors which have a support of the same cardinality. Further, by these conditions we deduce a geometrical interpretation which identifies an equivalence class with a face of an hypercube which is cut by a certain affine subspace. Due to the new geometrical interpretation we derive new results on the number of equivalence classes which are illustrated by computational experiments.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barnette, D.W.: The minimum number of vertices of a simple polytope. Israel J. Math. 10(1), 121–125 (1971)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)
Donoho, D.L.: Neighborly polytopes and sparse solutions of underdetermined linear equations. Statistics Departement Stanford University (2004)
Donoho, D.L.: High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimension. Disc. Comp. Geom. 35(4), 617–652 (2006)
Tsaig, Y., Donoho, D.L.: Breakdown of local equivalence between sparse solution and ℓ 1 minimization. Signal Process. 86(3), 533–548 (2006)
Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of system of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2002)
Dossal, C., Peyré, G., Fadili, J.: A numerical exploration of compressed sensing recovery. Linear Algebra and its Applications 482(7), 1663–1679 (2010)
Foucart, S., Rauhut, H.: A mathematical introduction to compressive sensing. Birkhäuser, Boston (2013)
Fuchs, J.-J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Th. 50(6), 1341–1344 (2004)
Grünbaum, B.: Convex polytopes, volume 221 of Graduate Texts in Mathematics, 2nd. Springer-Verlag, New York (2003)
Dohono, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inform. Theory 47(7), 2845–2862 (2001)
Strohmer, T., Heath, R.W. Jr.: Grassmannian frames with applications to coding and communnication. App. Compu. Harm. Anal. 14(3), 257–275 (2003)
Kruschel, C.: Geometrical Interpretations and Algorithmic Verification of Exact Solutions in Compressed Sensing. PhD thesis, TU Braunschweig (2014)
Lawrence, J.: Cutting the d-cube. J. Res. Natl. Bur. Stand., 84 (1979)
Lonke, Y.: On Random Sections of the Cube. Discret. Comput. Geom. 23, 157–169 (2000)
Lorenz, D.A.: Constructing test instances for Basis Pursuit Denoising. IEEE Trans. Signal Process. 61(5) (2013)
Barany, I., Lovasz, L.: Borsuks Theorem and the number of facets of centrally symmetric polytopes. Acta Math. Sci. Hungar. 40, 323–389 (1982)
Istomina, M.N., Pevnyi, A.B.: The Mercedes–Benz frame in the n-dimensional space [in Russian]. Vestnik Syktyvkar. Gos. Univ. Ser. 1, 219–222 (2006)
Malozemov, V.N., Borisovich, P.A.: Equiangular tight frames. J. Math. Sci. 157, 789–815 (2009)
McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970)
Gribonval, R., Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inform. Theory 49(12), 3320–3325 (2003)
O’Neil, P.E.: Hyperplane cuts of an n-cube. Discret. Math. 1(2), 193–195 (1971)
Plumbley, M.D.: On polar polytopes and the recovery of sparse representations. IEEE Trans. Inf. Theory 53(9), 3188–3195 (2007)
Poincaré, H: Sur la genéralisation d’un théorème d’Euler relatif aux polyèdreś. C. R. Acad. Sci. Paris 177, 144–145 (1893)
Poincaré, H.: Complement à l’Analysis Situś. Rend. Circ. Mat. Palermo. 13, 285–343 (1899)
Chen, S.S., Donoho, D.L., Saunders, A.: Atomic Decomposition by Basis Pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)
Affentranger, F., Schneider, R.: Random projections of regular simplices. Discret. Comput. Geom. 7, 219–226 (1992)
Shepard, P.M., Geoffrey, C.: Diagrams for centrally symmetric d-polytopes. Mathematika 15, 123–138 (1968)
Steinitz, E.: Über die Eulersche Polyederrelationen. Arch. Math. Phys. 11, 86–88 (1906)
Donoho, D.L., Tanner, J.: Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. Royal Soc. A: Math., Phys. Eng. Sci. 367(1906), 4273–4293 (2009)
Tropp, J.A.: Recovery of short, complex, linear combinations va ℓ 1-minimization. IEEE Trans. Inform. Theory 51, 1568–1570 (2006)
Ziegler, G.M.: Lectures on polytopes, (152). Graduate Texts in Mathematics. Springer-Verlag, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Raymond H. Chan
Rights and permissions
About this article
Cite this article
Kruschel, C., Lorenz, D.A. Computing and analyzing recoverable supports for sparse reconstruction. Adv Comput Math 41, 1119–1144 (2015). https://doi.org/10.1007/s10444-015-9403-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-015-9403-6