The Pros and Cons of Compressive Sensing for Wideband Signal Acquisition: Noise Folding vs. Dynamic Range

Compressive sensing (CS) exploits the sparsity present in many signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors and any associated communication links. This paper examines the use of CS in the design of a wideband radio receiver in a noisy environment. We formulate the problem statement for such a receiver and establish a reasonable set of requirements that a receiver should meet to be practically useful. We then evaluate the performance of a CS-based receiver in two ways: via a theoretical analysis of its expected performance, with a particular emphasis on noise and dynamic range, and via simulations that compare the CS receiver against the performance expected from a conventional implementation. On the one hand, we show that CS-based systems that aim to reduce the number of acquired measurements are somewhat sensitive to signal noise, exhibiting a 3dB SNR loss per octave of subsampling, which parallels the classic noise-folding phenomenon. On the other hand, we demonstrate that since they sample at a lower rate, CS-based systems can potentially attain a significantly larger dynamic range. Hence, we conclude that while a CS-based system has inherent limitations that do impose some restrictions on its potential applications, it also has attributes that make it highly desirable in a number of important practical settings

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds $$m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$$ where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples

Measurements design and phenomena discrimination

The construction of measurements suitable for discriminating signal components produced by phenomena of different types is considered. The required measurements should be capable of cancelling out those signal components which are to be ignored when focusing on a phenomenon of interest. Under the hypothesis that the subspaces hosting the signal components produced by each phenomenon are complementary, their discrimination is accomplished by measurements giving rise to the appropriate oblique projector operator. The subspace onto which the operator should project is selected by nonlinear techniques in line with adaptive pursuit strategies

Perceptual Compressive Sensing

Compressive sensing (CS) works to acquire measurements at sub-Nyquist rate and recover the scene images. Existing CS methods always recover the scene images in pixel level. This causes the smoothness of recovered images and lack of structure information, especially at a low measurement rate. To overcome this drawback, in this paper, we propose perceptual CS to obtain high-level structured recovery. Our task no longer focuses on pixel level. Instead, we work to make a better visual effect. In detail, we employ perceptual loss, defined on feature level, to enhance the structure information of the recovered images. Experiments show that our method achieves better visual results with stronger structure information than existing CS methods at the same measurement rate.Comment: Accepted by The First Chinese Conference on Pattern Recognition and Computer Vision (PRCV 2018). This is a pre-print version (not final version

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the latter also being known as l1-heavy hitters), norm estimation, and approximate inner product. We focus on devising a fixed matrix A in R^{m x n} and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. Our results improve upon existing work, the following being our main contributions: * A proof that linf/l1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. * A new lower bound for the number of linear measurements required to solve l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude. * A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover |x|_2 +/- eps|x|_1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of l1/l1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

Effects of beta-alanine supplementation on brain homocarnosine/carnosine signal and cognitive function: an exploratory study

Objectives: Two independent studies were conducted to examine the effects of 28 d of beta-alanine supplementation at 6.4 g d-1 on brain homocarnosine/carnosine signal in omnivores and vegetarians (Study 1) and on cognitive function before and after exercise in trained cyclists (Study 2). Methods: In Study 1, seven healthy vegetarians (3 women and 4 men) and seven age- and sex-matched omnivores undertook a brain 1H-MRS exam at baseline and after beta-alanine supplementation. In study 2, nineteen trained male cyclists completed four 20-Km cycling time trials (two pre supplementation and two post supplementation), with a battery of cognitive function tests (Stroop test, Sternberg paradigm, Rapid Visual Information Processing task) being performed before and after exercise on each occasion. Results: In Study 1, there were no within-group effects of beta-alanine supplementation on brain homocarnosine/carnosine signal in either vegetarians (p = 0.99) or omnivores (p = 0.27); nor was there any effect when data from both groups were pooled (p = 0.19). Similarly, there was no group by time interaction for brain homocarnosine/carnosine signal (p = 0.27). In study 2, exercise improved cognitive function across all tests (P0.05) of beta-alanine supplementation on response times or accuracy for the Stroop test, Sternberg paradigm or RVIP task at rest or after exercise. Conclusion: 28 d of beta-alanine supplementation at 6.4g d-1 appeared not to influence brain homocarnosine/ carnosine signal in either omnivores or vegetarians; nor did it influence cognitive function before or after exercise in trained cyclists

A chronic fatigue syndrome – related proteome in human cerebrospinal fluid

BACKGROUND: Chronic Fatigue Syndrome (CFS), Persian Gulf War Illness (PGI), and fibromyalgia are overlapping symptom complexes without objective markers or known pathophysiology. Neurological dysfunction is common. We assessed cerebrospinal fluid to find proteins that were differentially expressed in this CFS-spectrum of illnesses compared to control subjects. METHODS: Cerebrospinal fluid specimens from 10 CFS, 10 PGI, and 10 control subjects (50 μl/subject) were pooled into one sample per group (cohort 1). Cohort 2 of 12 control and 9 CFS subjects had their fluids (200 μl/subject) assessed individually. After trypsin digestion, peptides were analyzed by capillary chromatography, quadrupole-time-of-flight mass spectrometry, peptide sequencing, bioinformatic protein identification, and statistical analysis. RESULTS: Pooled CFS and PGI samples shared 20 proteins that were not detectable in the pooled control sample (cohort 1 CFS-related proteome). Multilogistic regression analysis (GLM) of cohort 2 detected 10 proteins that were shared by CFS individuals and the cohort 1 CFS-related proteome, but were not detected in control samples. Detection of ≥1 of a select set of 5 CFS-related proteins predicted CFS status with 80% concordance (logistic model). The proteins were α-1-macroglobulin, amyloid precursor-like protein 1, keratin 16, orosomucoid 2 and pigment epithelium-derived factor. Overall, 62 of 115 proteins were newly described. CONCLUSION: This pilot study detected an identical set of central nervous system, innate immune and amyloidogenic proteins in cerebrospinal fluids from two independent cohorts of subjects with overlapping CFS, PGI and fibromyalgia. Although syndrome names and definitions were different, the proteome and presumed pathological mechanism(s) may be shared