Abstract—Compressive phase retrieval algorithms attempt to reconstruct a “sparse high-dimensional vector ” from its “low-dimensional intensity measurements”. Suppose x is any length-n input vector over C with exactly k non-zero entries, and A is an m × n (k < m n) phase measurement matrix over C. The decoder is handed m “intensity measurements” (|A1x|,..., |Amx|) (corresponding to component-wise absolute values of the linear measurement Ax) – here Ai’s correspond to the rows of the measurement matrix A. In this work, we present a class of measurement matrices A, and a corresponding decoding algorithm that we call SUPER, which can reconstruct x up to a global phase from intensity measurements. The SUPER algorithm is the first to simultaneously have the following properties: (a) it requires only O(k) (order-optimal) measurements, (b) the computational complexity of decoding is O(k log k) (near order-optimal) arithmetic operations, (c) it succeeds with high probability over the design of A. Our results hold for all k ∈ {1, 2,..., n}. I.

We address the problem of robustly recovering the support of high-dimensional sparse signals1 from linear measurements in a low-dimensional subspace. We introduce a new compressed sensing framework through carefully designed sparse measurement matrices associated with low measurement costs and low-complexity recovery algorithms. The measurement system in our framework captures observations of the signal through well-designed measurement matrices sparsified by capacity-approaching sparse-graph codes, and then recovers the signal by using a simple peeling decoder. As a result, we can simultaneously reduce both the measurement cost and the computational complexity. In this paper, we formally connect general sparse recovery problems in compressed sensing with sparse-graph decoding in packet-communication systems, and analyze our design in terms of the measurement cost, computational complexity and recovery performance. Specifically, by structuring the measurements through sparse-graph codes, we propose two families of mea-surement matrices, the Fourier family and the binary family respectively, which lead to different measurement and computational costs. In the noiseless setting, our framework recovers the sparse support of any K-sparse signal in time2 O(K) with 2K measurements obtained by the Fourier family, or in time O(K logN) using K log2N + K measurements obtained by the binary family. In the presence of noise, both measurement and

The fundamental problem considered in this paper is, basically, “what is the energy consumed for the implementation of compressive sensing algorithm on a circuit? " We use the proposed “bit-meters1 " measure as a proportional measurement for energy, i.e., the product of number bits transmitted and the distance of information transport. Be Analogous to the friction imposed on relative motion between two surfaces, this model is so-called “information-friction " model. By using this “information-friction " model, a fundamental lower bound for the implementation of compressive sensing algorithms on a circuit is provided. Further, we explore and compare a series of decoding algorithms based on different implementation-circuits. As our second main result, an order-tight asymptotic result on a fixed precision bits for the regime m = O(k) (m is the number of measurements and k is the number of non-zero entries in the compressible vector) is provided. We thus attest in this paper that the asymptotic lower bound is order-optimal with a sub-linear sparsity k such that k = n1−β (β ∈ (0, 1), n is the total number of input entries) since a proposed algorithm with corresponding construction of implementation-circuit can achieve an upper bound with the same order.

Abstract—We study the problem of link and node delay estimation in undirected networks when at most k out of n links or nodes in the network are congested. Our approach relies on end-to-end measurements of path delays across pre-specified paths in the network. We present a class of algorithms that we call FRANTIC. The FRANTIC algorithms are motivated by compressive sensing; however, unlike traditional compressive sensing, the measurement design here is constrained by the network topology and the matrix entries are constrained to be positive integers. A key component of our design is a new compressive sensing algorithm SHO-FA-INT that is related to the SHO-FA algorithm [1] for compressive sensing, but unlike SHO-FA, the matrix entries here are drawn from the set of integers {0, 1,...,M}. We show that O(k logn / logM) measure-ments suffice both for SHO-FA-INT and FRANTIC. Further, we show that the computational complexity of decoding is also O(k logn / logM) for each of these algorithms. Finally, we look at efficient constructions of the measurement operations through Steiner Trees. I.