On the restricted isometry property of the Paley matrix

Geometric lifting of the integrable cellular automata with periodic boundary conditions

“…In order to explain the definition of the matrix g * (x, s; λ), here we introduce a useful notion. For any n × n matrix A and 1 r n, let C r (A) be the rth contravariant alternating tensor representation of A, which is an n r × n r matrix that consists of all the order r minor determinants of A (see, for example [22]). That is, we define…”

“…(This formula is derived by using calculus of a determinant. See, for example [22]. An alternative derivation for the case of A ∈ M n (C) will be given as a consequence of corollary 29.)…”

On Compressed Sensing Matrices Breaking the Square-Root Bottleneck

“…On the other hand, in [3], Bandeira, Fickus, Mixon and Wong conjectured that the Paley matrix, a (p + 1)/2 × (p + 1) matrix defined by quadratic residues modulo an odd prime p, satisfies the (K, δ)-RIP with K ≥ C 1 • p/ log C 2 p and some δ < √ 2 − 1, where C 1 , C 2 > 0 are universal constants. Under a number-theoretic conjecture in [10], Bandeira, Mixon and Moreira [4] proved that when p ≡ 1 (mod 4), the Paley matrix has the (K, o(1))-RIP with K = Ω(p γ ) for some γ > 1/2, which provides a conditional solution to Problem 2; recently, assuming that the Paley graph conjecture (see Remark 9) holds, Satake [21] extended the result in [4] for general odd primes p, and also gave some implications to a Ramsey-theoretic problem proposed by Erdős and Moser [11]. Also, under another type of number-theoretic conjecture, Arian and Yilmaz [1] proved that for a sufficiently large prime p ≡ 3 (mod 4), a (p + 1)/2 × p matrix obtained by deleting the last column from the Paley matrix (see Remark 14) has the (K, δ)-RIP for any K < p 5/7 /2 and δ < 1/ √ 2.…”

“…p + 1 Ω(p γ ) some 1 2 < γ < 1, p ≡ 1 (mod 4) [21] In this paper, we aim to investigate RIP of certain matrices defined by higher power residues modulo primes. In particular, under the widely-believed generalized Paley graph conjecture formulated in Section 2, we prove that these matrices are new solutions to Problem 2, which forms our main theorem as described below.…”

“…Let p be an odd prime. The Paley matrix in [4] and [21] is a (p + 1)/2 × (p + 1) matrix obtained from the matrix Φ…”