On the restricted isometry property of the Paley matrix

Explicit Matrices with the Restricted Isometry Property: Breaking the Square-Root Bottleneck

Matrices with the restricted isometry property (RIP) are of particular interest in compressed sensing. To date, the best known RIP matrices are constructed using random processes, while explicit constructions are notorious for performing at the "square-root bottleneck," i.e., they only accept sparsity levels on the order of the square root of the number of measurements. The only known explicit matrix which surpasses this bottleneck was constructed by Bourgain, Dilworth, Ford, Konyagin and Kutzarova. This chapter provides three contributions to further the groundbreaking work of Bourgain et al.…

高斯矩阵受限等距常数的改进界

Improved Bounds on Restricted Isometry Constants for Gaussian Matrices

本文通过利用共享大量列的矩阵奇异值相似性，改进了高斯矩阵的受限等距常数（RIC）上界，这是目前已知的大规模矩形矩阵的最小RIC界。

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-ε) with list size L=O(1/ε^2) and rate R=Ω_q(ε^2/(log^3(1/ε))). Up to the polylogarithmic factor in (1/ε) and constant factors depending on q, this matches the lower bound L=Ω_q(1/ε^2) for the list size and upper bound R=O_q(ε^2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q=2 (Guruswami, Håstad, Sudan and Zuckerman, 2002). In order to obtain our result, we employ a relaxed version of the well known Johnson bound on…

Derandomizing restricted isometries via the Legendre symbol

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.

重新审视 Paley 图团数的块对角 SDP 松弛

Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs

本文通过计算 821 至 997 之间素数的 Paley 图块对角 SDP（L2）松弛值，为团数上界及 SOS-4 松弛可能以小于平方根数量级增长提供了新的数值证据，同时指出更大规模计算仍是一个开放问题。

不包含$K_{2,3}$诱导极小图的图可加误差地准等距于树宽至多为二的图

$K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two

本文提出了一种在亚二次时间内计算普遍可签名图直径的算法，通过证明$K_{2,3}$-诱导极小图可加误差地准等距于树宽至多为二的图。