Derandomizing Restricted Isometries via the Legendre Symbol

Abstract

Abstract. The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special-our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.

Discrete Uncertainty Principles and Sparse Signal Processing

“…This is perhaps not surprising, considering various choices of structured random matrices are known to form restricted isometries with high probability [12,37,35,28,34,8,3]. To prove Theorem 8, we show that the structured matrix enjoys restricted orthogonality with high probability, and then appeal to Lemma 6.…”

Book Review: A mathematical introduction to compressive sensing

“…There is also a long-standing connection between sparsity and the analytic principle of the large sieve [DL92]. More recently, authors have used properties of the Legendre symbol to construct nonrandom measurement matrices that mimic the properties of random measurements [BFMM16].…”

Derandomized compressed sensing with nonuniform guarantees for $\ell_1$ recovery

“…In 2007, Tao [39] posed the problem of finding explicit s-restricted isometries A ∈ C m×N with N ǫ ≤ m ≤ (1 − ǫ)N and m = s polylog N. One may view this as an instance of Avi Wigderson's hay in a haystack problem [4]. To be clear, we say a sequence {A N } of m(N) × N matrices with N → ∞ is explicit if there exists an algorithm that on input N produces A N in time that is polynomial in N. For example, we currently know of several explicit sequences of matrices A with unit-norm columns {a i } i∈[N ] and minimum coherence:…”