Abstract. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix Φ and then uses linear programming to decode information about x from Φx. The combinatorial approach constructs Φ and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of high-quality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the ℓp norm for p ≈ 1, and then show that unbalanced expanders are essentially equivalent to RIP-p matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance. 1.

Abstract—Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, sub-Nyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform asymptotically optimal Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with sparse encoding matrices. To decode a length- signal containing large coefficients, our CS-BP decoding algorithm uses ( log ()) measurements and ( log 2 ()) computation. Finally, although we focus on a two-state mixture Gaussian model, CS-BP is easily adapted to other signal models. Index Terms—Bayesian inference, belief propagation, compressive sensing, fast algorithms, sparse matrices. I.

We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector x from its lower-dimensional sketch Ax. A popular way of performing this recovery is by finding x # such that Ax = Ax # , and �x # �1 is minimal. It is known that this approach “works” if A is a random dense matrix, chosen from a proper distribution. In this paper, we investigate this procedure for the case where A is binary and very sparse. We show that, both in theory and in practice, sparse matrices are essentially as “good” as the dense ones. At the same time, sparse binary matrices provide additional benefits, such as reduced encoding and decoding time.

Abstract — Compressed sensing is a new framework for acquiring sparse signals based on the revelation that a small number of linear projections (measurements) of the signal contain enough information for its reconstruction. The foundation of Compressed sensing is built on the availability of noise-free measurements. However, measurement noise is unavoidable in analog systems and must be accounted for. We demonstrate that measurement noise is the crucial factor that dictates the number of measurements needed for reconstruction. To establish this result, we evaluate the information contained in the measurements by viewing the measurement system as an information theoretic channel. Combining the capacity of this channel with the ratedistortion function of the sparse signal, we lower bound the rate-distortion performance of a compressed sensing system. Our approach concisely captures the effect of measurement noise on the performance limits of signal reconstruction, thus enabling to benchmark the performance of specific reconstruction algorithms. I.

Abstract—We survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to several areas, including compressive sensing, data stream computing and group testing. I.

A Compressive Sensing Microarray (CSM) is a new device for DNA-based identification of target organisms that leverages the nascent theory of Compressive Sensing (CS). In contrast to a conventional DNA microarray, in which each genetic sensor spot is designed to respond to a single target organism, in a CSM each sensor spot responds to a group of targets. As a result, significantly fewer total sensor spots are required. In this paper, we study how to design group identifier probes that simultaneously account for both the constraints from the CS theory and the biochemistry of probe-target DNA hybridization. We employ Belief Propagation as a CS recovery method to estimate target concentrations from the microarray intensities.

Abstract—Compressed sensing is a novel technique to acquire sparse signals with few measurements. Normally, compressed sensing uses random projections as measurements. Here we design deterministic measurements and an algorithm to accomplish signal recovery with computational efficiently. A measurement matrix is designed with chirp sequences forming the columns. Chirps are used since an efficient method using FFTs can recover the parameters of a small superposition. We show empirically that this type of matrix is valid as compressed sensing measurements. This is done by a comparison with random projections and a modified reduced isometry property. Further, by implementing our algorithm, simulations show successful recovery of signals with sparsity levels similar to those possible by Matching Pursuit with random measurements. For sufficiently sparse signals, our algorithm recovers the signal with computational complexity O(K log K) for K measurements. This is a significant improvement over existing algorithms. I.

We propose a signal recovery method using Belief Propagation (BP) for nonlinear Compressed Sensing (CS) and demonstrate its utility in DNA array decoding. In a CS DNA microarray, the array spots identify DNA sequences that are shared between multiple organisms, thereby reducing the number of spots required. The sparsity in DNA sequence commonality between different organisms translates to conditions that render Belief Propagation (BP) efficient for signal reconstruction. However, an excessively high concentration of target DNA molecules has a nonlinear effect on the measurements — it causes saturation in the measurement intensities at the array spots. We use a modified BP to estimate the target signal coefficients since it is flexible to handle the nonlinearity unlike ℓ1 decoding or other greedy algorithms and show that the original signal coefficients can be recovered from saturated measurements of their linear combinations.

sensors that operate using group testing and compressive sensing (CS) principles. In contrast to conventional DNA microarrays, in which each genetic sensor is designed to respond to a single target, in a CSM each sensor responds to a set of targets. We study the problem of designing CSMs that simultaneously account for both the constraints from compressive sensing theory and the biochemistry of probe-target DNA hybridization. An appropriate cross-hybridization model is proposed for CSMs, and several methods are developed for probe design and CS signal recovery based on the new model. Our lab experiments suggest that, in order to achieve accurate hybridization profiling, consensus probe sequences are required to have sequence homology of at least 80 % with all targets to be detected. Furthermore, outof-equilibrium datasets are usually as accurate as those obtained from equilibrium conditions. Consequently, one can use CSMs in applications for which only short hybridization times are allowed. Index Terms—Compressive sensing, DNA microarray, group testing, hybridization affinity, probe design I.

presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder. I.