The problem of sparse support recovery in Multiple Measurement Vector (MMV) models is considered, where the support size (K) can be larger than the dimension (M) of each measurement vector. Most results in literature address the case where K < M. We propose a sequential detector for the MMV problem, which despite its simplicity, it can recover supports of size K = O(M2), for suitable measurement matrices, with a probability of error decaying to zero exponentially fast as the number of independent measurement vectors (L) goes to infinity. By exactly characterizing the distribution of the detection statistic, we derive explicit forms for the error exponent. We show that the required conditions on the measurement matrix can be met for equiangular tight frames. Although certain constructive methods show existence of equiangular tight frames of size N > M (N being the number of columns), the question of existence of equiangular tight frames of size N = O(M2), for arbitrarily large M, is still an open problem. We review some of the well-known results on this topic, and make connections to the support recovery problem when K > M.
Mapping the topology of a complex network in a resource-efficient manner is a challenging problem with applications in internet mapping, social network inference, and so forth. We propose a new entropy driven algorithm leveraging ideas from matrix completion, to map the network using monitors (or sensors) which, when placed on judiciously selected nodes, are capable of discovering their immediate neighbors. The main challenge is to maximize the portion of discovered network using only a limited number of available monitors. To this end, (i) a new measure of entropy or uncertainty is associated with each node, in terms of the currently discovered edges incident on that node, and (ii) a greedy algorithm is developed to select a candidate node for monitor placement based on its entropy. Utilizing the fact that many complex networks of interest (such as social networks), have a low-rank adjacency matrix, a matrix completion algorithm, namely 1-bit matrix completion, is combined with the greedy algorithm to further boost its performance. The low rank property of the network adjacency matrix can be used to extrapolate a portion of missing edges, and consequently update the node entropies, so as to efficiently guide the network discovery algorithm towards placing monitors on the nodes that can turn out to be more informative. Simulations performed on a variety of real world networks such as social networks and peer networks demonstrate the superior performance of the matrix-completion guided approach in discovering the network topology.