𝓁p-Spread Properties of Sparse Matrices

Random subspaces of R= of dimension proportional to = are, with high probability, wellspread with respect to the l?-norm (for ? ∈ [1, 2]). Namely, every nonzero G ∈ is “robustly non-sparse” in the following sense: G is ‖G‖?-far in l?-distance from all =-sparse vectors, for positive constants , bounded away from 0. This “l?-spread” property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and, for ? = 2, corresponds to being a Euclidean section of the l1 unit ball. Explicit l?spread subspaces of dimensionΩ(=), however, are not known except for ? = 1. The construction for ? = 1, as well as the best known constructions for ? ∈ (1, 2] (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of certain sparse matrices. Motivated by this, we study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors G that are >(1) · ‖G‖2-close to >(=)-sparse with respect to the l2-norm, and in particular are not l2spread. This is strikingly different from the case of random LDPC codes, whose distance is asymptotically almost as good as that of (dense) random linear codes. On the other hand, for ? < 2 we prove that such subspaces are l?-spread with high probability. The spread property of sparse random matrices thus exhibits a threshold behavior at ? = 2. Our proof for ? < 2 moreover shows that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the l? norm. In fact, we show that RIP follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the l1 norm [BGI +08]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of l?-spread subspaces and l?-RIP matrices for 1 ≤ ? < ?0, where 1 < ?0 < 2 is an absolute constant.

[1]  Piotr Indyk,et al.  Combining geometry and combinatorics: A unified approach to sparse signal recovery , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[2]  Zeyuan Allen Zhu,et al.  Restricted Isometry Property for General p-Norms , 2014, IEEE Transactions on Information Theory.

[3]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[4]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[5]  Ryan O'Donnell,et al.  Explicit near-fully X-Ramanujan graphs , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[6]  Yuan Zhou,et al.  Hypercontractivity, sum-of-squares proofs, and their applications , 2012, STOC '12.

[7]  S. Szarek,et al.  Almost-Euclidean Subspaces of 1 N via Tensor Products : A Simple Approach to Randomness Reduction , 2010 .

[8]  Sidhanth Mohanty,et al.  The SDP value for random two-eigenvalue CSPs , 2019, STACS.

[9]  Piotr Indyk,et al.  Uncertainty principles, extractors, and explicit embeddings of l2 into l1 , 2007, STOC '07.

[10]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[11]  Avi Wigderson,et al.  Randomness conductors and constant-degree lossless expanders , 2002, STOC '02.

[12]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[13]  Mark Rudelson,et al.  Invertibility of Sparse non-Hermitian matrices , 2015, 1507.03525.

[14]  Venkatesan Guruswami,et al.  Euclidean Sections of with Sublinear Randomness and Error-Correction over the Reals , 2008, APPROX-RANDOM.

[15]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[16]  Charles Bordenave,et al.  A new proof of Friedman's second eigenvalue theorem and its extension to random lifts , 2015, Annales scientifiques de l'École normale supérieure.

[17]  T. Tao Topics in Random Matrix Theory , 2012 .

[18]  Sidhanth Mohanty,et al.  Explicit near-Ramanujan graphs of every degree , 2020, STOC.

[19]  B. Collins,et al.  Eigenvalues of random lifts and polynomials of random permutation matrices , 2018, Annals of Mathematics.

[20]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[21]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[22]  Piotr Indyk,et al.  Stable distributions, pseudorandom generators, embeddings, and data stream computation , 2006, JACM.

[23]  A. Bandeira,et al.  Sharp nonasymptotic bounds on the norm of random matrices with independent entries , 2014, 1408.6185.

[24]  Daniel A. Spielman,et al.  Expander codes , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[25]  V. Temlyakov,et al.  A remark on Compressed Sensing , 2007 .

[26]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .