This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. We support our results with experimental demonstrations of reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process.
This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of t…
We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency matrices of lossless expander graphs. We present tail bounds on the probability that the cardinality of the set of neighbors for these graphs will be less than the expected value. The bounds are derived through the analysis of collisions in unions of sets using a {\em dyadic splitting} technique. This analysis led to the derivation of better constants that allow fo…
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any $n$-dimensional vector that is $k$-sparse (with $k\ll n$) can be fully recovered using $O(k\log\frac{n}{k})$ measurements and only $O(k\log n)$ simple recovery iterations. In this paper we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only $O(k)$ recovery iterations are required, which is a significant improvement when $n$ is large. In fact, full recovery c…
Is it possible to obliviously construct a set of hyperplanes H such that you can approximate a unit vector x when you are given the side on which the vector lies with respect to every h in H? In the sparse recovery literature, where x is approximately k-sparse, this problem is called one-bit compressed sensing and has received a fair amount of attention the last decade. In this paper we obtain the first scheme that achieves almost optimal measurements and sublinear decoding time for one-bit compressed sensing in the non-uniform case. For a large range of parameters, we improve the state of the…
This paper describes performance bounds for compressed sensing in the presence of Poisson noise when the underlying signal, a vector of Poisson intensities, is sparse or compressible (admits a sparse approximation). The signal-independent and bounded noise models used in the literature to analyze the performance of compressed sensing do not accurately model the effects of Poisson noise. However, Poisson noise is an appropriate noise model for a variety of applications, including low-light imaging, where sensing hardware is large or expensive, and limiting the number of measurements collected i…
One of the purposes of network tomography is to infer the status of parameters (e.g., delay) for the links inside a network through end-to-end probing between (external) boundary nodes along predetermined routes. In this work, we apply concepts from compressed sensing and expander graphs to the delay estimation problem. We first show that a relative majority of network topologies are not expanders for existing expansion criteria. Motivated by this challenge, we then relax such criteria, enabling us to acquire simulation evidence that link delays can be estimated for 30% more networks. That is,…
This paper describes a low-complexity approach for reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process. Assuming that the rate vector of this Poisson process is sparse or approximately sparse, the goal is to maintain a compressed summary of the process sample paths using a small number of counters, such that at any time it is possible to reconstruct both the total number of packets in each flow and the underlying rate vector. We show that these tasks can be accomp…
The expicit restricted isometry property (RIP) measurement matrices are needed in practical application of compressed sensing in signal processing. RIP matrices from Reed-Solomon codes, BCH codes, orthogonal codes, expander graphs have been proposed and analysised. On the other hand binary constant weight codes have been studied for many years and many optimal or near-optimal small weight and ditance constant weight codes have been determined. In this paper we propose a new deterministic construction of RIP measurement matrices in compressed sensing from binary and ternary contant weight codes…
Most existing bounds for signal reconstruction from compressive measurements make the assumption of additive signal-independent noise. However in many compressive imaging systems, the noise statistics are more accurately represented by Poisson or Poisson-Gaussian noise models. In this paper, we derive upper bounds for signal reconstruction error from compressive measurements which are corrupted by Poisson or Poisson-Gaussian noise. The features of our bounds are as follows: (1) The bounds are derived for a probabilistically motivated, computationally tractable convex estimator with principled …
Reconstruction error bounds in compressed sensing under Gaussian or uniform bounded noise do not translate easily to the case of Poisson noise. Reasons for this include the signal dependent nature of Poisson noise, and also the fact that the negative log likelihood (NLL) in case of a Poisson distribution (which is related to the generalized Kullback-Leibler divergence (GKLD)) is not a metric and does not obey the triangle inequality. There exist prior theoretical results in the form of provable error bounds for computationally tractable estimators for compressed sensing problems under Poisson …
This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, s…
In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for given linear sketches of the true signal vector which is known to be group-sparse in the sense that its support is contained in the union of a small number of these groups, we study algorithms which successfully recover the true signal just by the knowledge of its linear sketches. We derive model projection complexity results and algorithms for more general group models than the state-of-the-art. We consider two versions of the c…
Coronavirus disease 2019 (COVID-19) is an ongoing pandemic infectious disease outbreak that has significantly harmed and threatened the health and lives of millions or even billions of people. COVID-19 has also negatively impacted the social and economic activities of many countries significantly. With no approved vaccine available at this moment, extensive testing of COVID-19 viruses in people are essential for disease diagnosis, virus spread confinement, contact tracing, and determining right conditions for people to return to normal economic activities. Identifying people who have antibodie…
This paper introduces the Reed Muller Sieve, a deterministic measurement matrix for compressed sensing. The columns of this matrix are obtained by exponentiating codewords in the quaternary second order Reed Muller code of length $N$. For $k=O(N)$, the Reed Muller Sieve improves upon prior methods for identifying the support of a $k$-sparse vector by removing the requirement that the signal entries be independent. The Sieve also enables local detection; an algorithm is presented with complexity $N^2 \log N$ that detects the presence or absence of a signal at any given position in the data doma…
In another related work, U-statistics were used for non-asymptotic "average-case" analysis of random compressed sensing matrices. In this companion paper the same analytical tool is adopted differently - here we perform non-asymptotic "worst-case" analysis. Simple union bounds are a natural choice for "worst-case" analyses, however their tightness is an issue (and questioned in previous works). Here we focus on a theoretical U-statistical result, which potentially allows us to prove that these union bounds are tight. To our knowledge, this kind of (powerful) result is completely new in the con…
A novel solution is proposed to undertake a frequent task in wireless networks, which is to let all nodes broadcast information to and receive information from their respective one-hop neighboring nodes. The contribution is two-fold. First, as each neighbor selects one message-bearing codeword from its unique codebook for transmission, it is shown that decoding their messages based on a superposition of those codewords through the multiaccess channel is fundamentally a problem of compressed sensing. In the case where each message consists of a small number of bits, an iterative algorithm based…
