Numerically erasure-robust frames

Given a channel with additive noise and adversarial erasures, the task is to design a frame that allows for stable signal reconstruction from transmitted frame coefficients. To meet these specifications, we introduce numerically erasure-robust frames. We first consider a variety of constructions, including random frames, equiangular tight frames and group frames. Later, we show that arbitrarily large erasure rates necessarily induce numerical instability in signal reconstruction. We conclude with a few observations, including some implications for maximal equiangular tight frames and sparse frames.Comment: 15 page

Computational Complexity of Certifying Restricted Isometry Property

Given a matrix $A$ with $n$ rows, a number $k<n$, and $0<\delta < 1$, $A$ is $(k,\delta)$-RIP (Restricted Isometry Property) if, for any vector $x \in \mathbb{R}^n$, with at most $k$ non-zero co-ordinates, $$(1-\delta) \|x\|_2 \leq \|A x\|_2 \leq (1+\delta)\|x\|_2$$ In many applications, such as compressed sensing and sparse recovery, it is desirable to construct RIP matrices with a large $k$ and a small $\delta$. Given the efficacy of random constructions in generating useful RIP matrices, the problem of certifying the RIP parameters of a matrix has become important. In this paper, we prove that it is hard to approximate the RIP parameters of a matrix assuming the Small-Set-Expansion-Hypothesis. Specifically, we prove that for any arbitrarily large constant $C>0$ and any arbitrarily small constant $0<\delta<1$, there exists some $k$ such that given a matrix $M$, it is SSE-Hard to distinguish the following two cases: - (Highly RIP) $M$ is $(k,\delta)$-RIP. - (Far away from RIP) $M$ is not $(k/C, 1-\delta)$-RIP. Most of the previous results on the topic of hardness of RIP certification only hold for certification when $\delta=o(1)$. In practice, it is of interest to understand the complexity of certifying a matrix with $\delta$ being close to $\sqrt{2}-1$, as it suffices for many real applications to have matrices with $\delta = \sqrt{2}-1$. Our hardness result holds for any constant $\delta$. Specifically, our result proves that even if $\delta$ is indeed very small, i.e. the matrix is in fact \emph{strongly RIP}, certifying that the matrix exhibits \emph{weak RIP} itself is SSE-Hard. In order to prove the hardness result, we prove a variant of the Cheeger's Inequality for sparse vectors

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

This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue modulo certain prime powers. Improving the upper bound for the Paley graph clique number for odd prime powers is an open problem in combinatorics. Moreover, since quadratic residues exhibit pseudorandom properties, Paley graphs are related to the construction of deterministic restricted isometries, an open problem in compressed sensing and sparse recovery. Recent work provides evidence that the current upper bounds can be improved by the sum-of-squares (SOS) relaxations. In particular the bounds given by the SOS relaxations of degree 4 (SOS-4) are asymptotically growing at an order smaller than square root of the prime. However computations of SOS-4 become intractable with respect to large graphs. Gvozdenovic et al. introduced a more computationally efficient block-diagonal hierarchy of SDPs that refines the SOS hierarchy. They computed the values of these SDPs of degrees 2 and 3 (L2 and L3 respectively) for the Paley graph clique numbers associated with primes p less or equal to 809. These values bound from the above the values of the corresponding SOS-4 and SOS-6 relaxations respectively. We revisit these computations and determine the values of the L2 relaxation for larger p's. Our results provide additional numerical evidence that the L2 relaxations, and therefore also the SOS-4 relaxations, are asymptotically growing at an order smaller than the square root of p