DOCSLIB.ORG

Structured Compressed Sensing - Using Patterns in Sparsity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Structured Compressed Sensing - Using Patterns in Sparsity Structured Compressed Sensing - Using Patterns in Sparsity Johannes Maly Technische Universit¨atM¨unchen, Department of Mathematics, Chair of Applied Numerical Analysis [email protected] CoSIP Workshop, Berlin, Dezember 9, 2016 Overview Classical Compressed Sensing Structures in Sparsity I - Joint Sparsity Structures in Sparsity II - Union of Subspaces Conclusion Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 2 of 44 Classical Compressed Sensing Overview Classical Compressed Sensing Structures in Sparsity I - Joint Sparsity Structures in Sparsity II - Union of Subspaces Conclusion Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 3 of 44 Classical Compressed Sensing Compressed Sensing N Let x 2 R be some unknown k-sparse signal. Then, x can be recovered from few linear measurements y = A · x m×N m where A 2 R is a (random) matrix, y 2 R is the vector of measurements and m N. Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 4 of 44 Classical Compressed Sensing Compressed Sensing N Let x 2 R be some unknown k-sparse signal. Then, x can be recovered from few linear measurements y = A · x m×N m where A 2 R is a (random) matrix, y 2 R is the vector of measurements and m N. It is sufficient to have N m Ck log & k measurements to recover x (with high probability) by greedy strategies, e.g. Orthogonal Matching Pursuit, or convex optimization, e.g. `1-minimization. Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 5 of 44 OMP INPUT: matrix A; measurement vector y: INIT: T0 = ;; x0 = 0: ITERATION: until stopping criterion is met T jn+1 arg maxj2[N] (A (y − Axn))j ; Tn+1 Tn [ fjn+1g; xn+1 arg minz2RN fky − Azk2; supp(z) ⊂ Tn+1g : OUTPUT: then ~-sparse approximationx ^ := xn~ Classical Compressed Sensing Orthogonal Matching Pursuit OMP is a simple algorithm that tries to find the true support of x by k greedy steps. It selects stepwise those columns of A that have the highest correlation with the measurements to build a support estimate T . Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 6 of 44 Classical Compressed Sensing Orthogonal Matching Pursuit OMP is a simple algorithm that tries to find the true support of x by k greedy steps. It selects stepwise those columns of A that have the highest correlation with the measurements to build a support estimate T . OMP INPUT: matrix A; measurement vector y: INIT: T0 = ;; x0 = 0: ITERATION: until stopping criterion is met T jn+1 arg maxj2[N] (A (y − Axn))j ; Tn+1 Tn [ fjn+1g; xn+1 arg minz2RN fky − Azk2; supp(z) ⊂ Tn+1g : OUTPUT: then ~-sparse approximationx ^ := xn~ Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 6 of 44 0 1 0 1 ha1; yi ha1; a`i T B . C B . C )A · y = @ . A = @ . A ) j1 = `: haN ; yi haN ; a`i Classical Compressed Sensing Orthogonal Matching Pursuit OMP is a simple algorithm that tries to find the true support of x by k greedy steps. It selects stepwise those columns of A that have the highest correlation with the measurements to build a support estimate T . 001 0 1 B.C j j B.C B C A · e` = @a1 ··· aN A · B1C = a` =: y B C j j B.C @.A 0 Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 7 of 44 ) j1 = `: Classical Compressed Sensing Orthogonal Matching Pursuit OMP is a simple algorithm that tries to find the true support of x by k greedy steps. It selects stepwise those columns of A that have the highest correlation with the measurements to build a support estimate T . 001 0 1 B.C j j B.C B C A · e` = @a1 ··· aN A · B1C = a` =: y B C j j B.C @.A 0 0 1 0 1 ha1; yi ha1; a`i T B . C B . C )A · y = @ . A = @ . A haN ; yi haN ; a`i Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 7 of 44 Classical Compressed Sensing Orthogonal Matching Pursuit OMP is a simple algorithm that tries to find the true support of x by k greedy steps. It selects stepwise those columns of A that have the highest correlation with the measurements to build a support estimate T . 001 0 1 B.C j j B.C B C A · e` = @a1 ··· aN A · B1C = a` =: y B C j j B.C @.A 0 0 1 0 1 ha1; yi ha1; a`i T B . C B . C )A · y = @ . A = @ . A ) j1 = `: haN ; yi haN ; a`i Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 7 of 44 Structures in Sparsity I - Joint Sparsity Overview Classical Compressed Sensing Structures in Sparsity I - Joint Sparsity Structures in Sparsity II - Union of Subspaces Conclusion Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 8 of 44 but L different measurements y1;:::; yL given by L different signals x1;:::; xL sharing a common support TX ⊂ [N]; jTX j ≤ k 0 j j 1 0 j j 1 A · @x1 ··· xLA = @y1 ··· yLA , A · X = Y ; j j j j N×L which can be written into matrices X 2 R , k-row-sparse, and m×L Y 2 R . Structures in Sparsity I - Joint Sparsity Joint Sparsity with Multiple Measurement Vectors We want now to improve on classical CS by using additional structure in sparsity. Possibly we not only have one measurement vector y from one sparse signal A · x = y; Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 9 of 44 Structures in Sparsity I - Joint Sparsity Joint Sparsity with Multiple Measurement Vectors We want now to improve on classical CS by using additional structure in sparsity. Possibly we not only have one measurement vector y from one sparse signal A · x = y; but L different measurements y1;:::; yL given by L different signals x1;:::; xL sharing a common support TX ⊂ [N]; jTX j ≤ k 0 j j 1 0 j j 1 A · @x1 ··· xLA = @y1 ··· yLA , A · X = Y ; j j j j N×L which can be written into matrices X 2 R , k-row-sparse, and m×L Y 2 R . Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 9 of 44 Based on the spark a necessary and sufficient condition for the measurements y = Ax to uniquely determine each k-sparse vector x is given by spark(A) k < ; 2 which leads to the requirement m ≥ 2k. Structures in Sparsity I - Joint Sparsity MMV in Theory Definition (spark(A)) m×N The spark of a matrix A 2 R is the smallest number of linearly dependent columns of A. It fulfills spark(A) ≤ m + 1. Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 10 of 44 Structures in Sparsity I - Joint Sparsity MMV in Theory Definition (spark(A)) m×N The spark of a matrix A 2 R is the smallest number of linearly dependent columns of A. It fulfills spark(A) ≤ m + 1. Based on the spark a necessary and sufficient condition for the measurements y = Ax to uniquely determine each k-sparse vector x is given by spark(A) k < ; 2 which leads to the requirement m ≥ 2k. Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 10 of 44 Structures in Sparsity I - Joint Sparsity MMV in Theory Definition (spark(A)) m×N The spark of a matrix A 2 R is the smallest number of linearly dependent columns of A. It fulfills spark(A) ≤ m + 1. In the MMV case a sufficient condition for the measurements Y = AX to uniquely determine the jointly k-sparse matrix X is spark(A)−1 + rank(X ) k < ; 2 which leads to the requirement m ≥ k + 1 if spark(A) and rank(X ) are optimal, i.e. spark(A) = m + 1 and rank(X ) = k. (see [1]) Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 11 of 44 Structures in Sparsity I - Joint Sparsity Simultaneous OMP SOMP uses a small modification of OMP to benefit from several different measurement vectors. Choosing the support index by the residual's largest row norm shall improve the support recovery. OMP INPUT: matrix A; measurement vector y: INIT: T0 = ;; x0 = 0: ITERATION: until stopping criterion is met T jn+1 arg maxj2[N] (A (y − Axn))j ; Tn+1 Tn [ fjn+1g; xn+1 arg minz2RN fky − Azk2; supp(z) ⊂ Tn+1g : OUTPUT: then ~-sparse approximationx ^ := xn~ Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 12 of 44 Structures in Sparsity I - Joint Sparsity Simultaneous OMP SOMP uses a small modification of OMP to benefit from several different measurement vectors. Choosing the support index by the residual's largest row norm shall improve the support recovery. SOMP INPUT: matrix A; measurement vectors Y = y1;:::; yL: INIT: T0 = ;; X0 = 0: ITERATION: until stopping criterion is met T jn+1 arg maxj2[N] (A (Y − AXn))j p; Tn+1 Tn [ fjn+1g; Xn+1 arg minZ2RN×L fkY − AZk2; supp(Z) ⊂ Tn+1g : OUTPUT: then ~ row-sparse approximation X^ := Xn~ Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 13 of 44 Structures in Sparsity I - Joint Sparsity SOMP - Numerics SOMP comparison with N = 256, m = 32 and L = 1; 2; 4; 8; 16; 32 (from left to right); Source: [2] Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 14 of 44 Structures in Sparsity II - Union of Subspaces Overview Classical Compressed Sensing Structures in Sparsity I - Joint Sparsity Structures in Sparsity II - Union of Subspaces Conclusion Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 15 of 44 N m ≈ k log k Structures in Sparsity II - Union of Subspaces Why Structure? Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 16 of 44 Structures in Sparsity II - Union of Subspaces Why Structure? N m ≈ k log k Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 16 of 44 m ≈ ? Structures in Sparsity II - Union of Subspaces Why Structure? Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity 17 of 44 Structures in Sparsity II - Union of Subspaces Why Structure? m ≈ ? Johannes Maly Structured Compressed Sensing - Using Patterns in Sparsity
Load more

Recommended publications
  • Compressive Phase Retrieval Lei Tian
    Compressive Phase Retrieval by Lei Tian B.S., Tsinghua University (2008) S.M., Massachusetts Institute of Technology (2010) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2013 c Massachusetts Institute of Technology 2013. All rights reserved. Author.............................................................. Department of Mechanical Engineering May 18, 2013 Certified by. George Barbastathis Professor Thesis Supervisor Accepted by . David E. Hardt Chairman, Department Committee on Graduate Students 2 Compressive Phase Retrieval by Lei Tian Submitted to the Department of Mechanical Engineering on May 18, 2013, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Recovering a full description of a wave from limited intensity measurements remains a central problem in optics. Optical waves oscillate too fast for detectors to measure anything but time{averaged intensities. This is unfortunate since the phase can reveal important information about the object. When the light is partially coherent, a complete description of the phase requires knowledge about the statistical correlations for each pair of points in space. Recovery of the correlation function is a much more challenging problem since the number of pairs grows much more rapidly than the number of points. In this thesis, quantitative phase imaging techniques that works for partially co- herent illuminations are investigated. In order to recover the phase information with few measurements, the sparsity in each underly problem and efficient inversion meth- ods are explored under the framework of compressed sensing. In each phase retrieval technique under study, diffraction during spatial propagation is exploited as an ef- fective and convenient mechanism to uniformly distribute the information about the unknown signal into the measurement space.
    [Show full text]
  • Compressed Digital Holography: from Micro Towards Macro
    Compressed digital holography: from micro towards macro Colas Schretter, Stijn Bettens, David Blinder, Beatrice Pesquet-Popescu, Marco Cagnazzo, Frederic Dufaux, Peter Schelkens To cite this version: Colas Schretter, Stijn Bettens, David Blinder, Beatrice Pesquet-Popescu, Marco Cagnazzo, et al.. Compressed digital holography: from micro towards macro. Applications of Digital Image Processing XXXIX, SPIE, Aug 2016, San Diego, United States. hal-01436102 HAL Id: hal-01436102 https://hal.archives-ouvertes.fr/hal-01436102 Submitted on 10 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Compressed digital holography: from micro towards macro Colas Schrettera,c, Stijn Bettensa,c, David Blindera,c, Béatrice Pesquet-Popescub, Marco Cagnazzob, Frédéric Dufauxb, and Peter Schelkensa,c aDept. of Electronics and Informatics (ETRO), Vrije Universiteit Brussel, Brussels, Belgium bLTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, Paris, France ciMinds, Technologiepark 19, Zwijnaarde, Belgium ABSTRACT The age of computational imaging is merging the physical hardware-driven approach of photonics with advanced signal processing methods from software-driven computer engineering and applied mathematics. The compressed sensing theory in particular established a practical framework for reconstructing the scene content using few linear combinations of complex measurements and a sparse prior for regularizing the solution.
    [Show full text]
  • Compressive Sensing Off the Grid
    Compressive sensing off the grid Abstract— We consider the problem of estimating the fre- Indeed, one significant drawback of the discretization quency components of a mixture of s complex sinusoids approach is the performance degradation when the true signal from a random subset of n regularly spaced samples. Unlike is not exactly supported on the grid points, the so called basis previous work in compressive sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the mismatch problem [13], [15], [16]. When basis mismatch normalized frequency domain [0; 1]. We propose an atomic occurs, the true signal cannot be sparsely represented by the norm minimization approach to exactly recover the unobserved assumed dictionary determined by the grid points. One might samples, which is then followed by any linear prediction method attempt to remedy this issue by using a finer discretization. to identify the frequency components. We reformulate the However, increasing the discretization level will also increase atomic norm minimization as an exact semidefinite program. By constructing a dual certificate polynomial using random the coherence of the dictionary. Common wisdom in com- kernels, we show that roughly s log s log n random samples pressive sensing suggests that high coherence would also are sufficient to guarantee the exact frequency estimation with degrade the performance. It remains unclear whether over- high probability, provided the frequencies are well separated. discretization is beneficial to solving the problems. Finer Extensive numerical experiments are performed to illustrate gridding also results in higher computational complexity and the effectiveness of the proposed method.
    [Show full text]
  • Sparse Penalties in Dynamical System Estimation
    1 Sparsity Penalties in Dynamical System Estimation Adam Charles, M. Salman Asif, Justin Romberg, Christopher Rozell School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, Georgia 30332-0250 Email: facharles6,sasif,jrom,[email protected] Abstract—In this work we address the problem of state by the following equations: estimation in dynamical systems using recent developments x = f (x ) + ν in compressive sensing and sparse approximation. We n n n−1 n (1) formulate the traditional Kalman filter as a one-step update yn = Gnxn + n optimization procedure which leads us to a more unified N framework, useful for incorporating sparsity constraints. where xn 2 R represents the signal of interest, N N We introduce three combinations of two sparsity conditions fn(·)jR ! R represents the (assumed known) evo- (sparsity in the state and sparsity in the innovations) M lution of the signal from time n − 1 to n, yn 2 R and write recursive optimization programs to estimate the M is a set of linear measurements of xn, n 2 R state for each model. This paper is meant as an overview N of different methods for incorporating sparsity into the is the associated measurement noise, and νn 2 R dynamic model, a presentation of algorithms that unify the is our modeling error for fn(·) (commonly known as support and coefficient estimation, and a demonstration the innovations). In the case where Gn is invertible that these suboptimal schemes can actually show some (N = M and the matrix has full rank), state estimation performance improvements (either in estimation error or at each iteration reduces to a least squares problem.
    [Show full text]
  • Compressed Sensing Applied to Modeshapes Reconstruction Joseph Morlier, Dimitri Bettebghor
    Compressed sensing applied to modeshapes reconstruction Joseph Morlier, Dimitri Bettebghor To cite this version: Joseph Morlier, Dimitri Bettebghor. Compressed sensing applied to modeshapes reconstruction. XXX Conference and exposition on structural dynamics (IMAC 2012), Jan 2012, Jacksonville, FL, United States. pp.1-8, 10.1007/978-1-4614-2425-3_1. hal-01852318 HAL Id: hal-01852318 https://hal.archives-ouvertes.fr/hal-01852318 Submitted on 1 Aug 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 5163 To cite this document: Morlier, Joseph and Bettebghor, Dimitri Compressed sensing applied to modeshapes reconstruction. (2011) In: IMAC XXX Conference and exposition on structural dynamics, 30 Jan – 02 Feb 2012, Jacksonville, USA. Any correspondence concerning this service should be sent to the repository administrator: [email protected] Compressed sensing applied to modeshapes reconstruction Joseph Morlier 1*, Dimitri Bettebghor 2 1 Université de Toulouse, ICA, ISAE DMSM ,10 avenue edouard Belin 31005 Toulouse cedex 4, France 2 Onera DTIM MS2M, 10 avenue Edouard Belin, Toulouse, France * Corresponding author, Email: [email protected], Phone no: + (33) 5 61 33 81 31, Fax no: + (33) 5 61 33 83 30 ABSTRACT Modal analysis classicaly used signals that respect the Shannon/Nyquist theory.
    [Show full text]
  • Accelerating Matching Pursuit for Multiple Time-Frequency Dictionaries
    Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020 ACCELERATING MATCHING PURSUIT FOR MULTIPLE TIME-FREQUENCY DICTIONARIES ZdenˇekPr˚uša,Nicki Holighaus and Peter Balazs ∗ Acoustics Research Institute Austrian Academy of Sciences Vienna, Austria [email protected],[email protected],[email protected] ABSTRACT An overview of greedy algorithms, a class of algorithms MP Matching pursuit (MP) algorithms are widely used greedy meth- falls under, can be found in [10, 11] and in the context of audio ods to find K-sparse signal approximations in redundant dictionar- and music processing in [12, 13, 14]. Notable applications of MP ies. We present an acceleration technique and an implementation algorithms in the audio domain include analysis [15], [16], coding of the matching pursuit algorithm acting on a multi-Gabor dictio- [17, 18, 19], time scaling/pitch shifting [20] [21], source separation nary, i.e., a concatenation of several Gabor-type time-frequency [22], denoising [23], partial and harmonic detection and tracking dictionaries, consisting of translations and modulations of possi- [24]. bly different windows, time- and frequency-shift parameters. The We present a method for accelerating MP-based algorithms proposed acceleration is based on pre-computing and thresholding acting on a single Gabor-type time-frequency dictionary or on a inner products between atoms and on updating the residual directly concatenation of several Gabor dictionaries with possibly different in the coefficient domain, i.e., without the round-trip to thesig- windows and parameters. The main idea of the present accelera- nal domain.
    [Show full text]
  • Kalman Filtered Compressed Sensing
    KALMAN FILTERED COMPRESSED SENSING Namrata Vaswani Dept. of ECE, Iowa State University, Ames, IA, [email protected] ABSTRACT that we address is: can we do better than performing CS at each time separately, if (a) the sparsity pattern (support set) of the transform We consider the problem of reconstructing time sequences of spa- coefficients’ vector changes slowly, i.e. every time, none or only a tially sparse signals (with unknown and time-varying sparsity pat- few elements of the support change, and (b) a prior model on the terns) from a limited number of linear “incoherent” measurements, temporal dynamics of its current non-zero elements is available. in real-time. The signals are sparse in some transform domain re- Our solution is motivated by reformulating the above problem ferred to as the sparsity basis. For a single spatial signal, the solu- as causal minimum mean squared error (MMSE) estimation with a tion is provided by Compressed Sensing (CS). The question that we slow time-varying set of dominant basis directions (or equivalently address is, for a sequence of sparse signals, can we do better than the support of the transform vector). If the support is known, the CS, if (a) the sparsity pattern of the signal’s transform coefficients’ MMSE solution is given by the Kalman filter (KF) [9] for this sup- vector changes slowly over time, and (b) a simple prior model on port. But what happens if the support is unknown and time-varying? the temporal dynamics of its current non-zero elements is available. The initial support can be estimated using CS [7].
    [Show full text]
  • Robust Compressed Sensing Using Generative Models
    Robust Compressed Sensing using Generative Models Ajil Jalal ∗ Liu Liu ECE, UT Austin ECE, UT Austin [email protected] [email protected] Alexandros G. Dimakis Constantine Caramanis ECE, UT Austin ECE, UT Austin [email protected] [email protected] Abstract The goal of compressed sensing is to estimate a high dimensional vector from an underdetermined system of noisy linear equations. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the vector is represented by a deep generative model G : Rk ! Rn. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is sub-Gaussian. However, when the measurement matrix and measurements are heavy-tailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the Median-of-Means (MOM). Our algorithm guarantees recovery for heavy-tailed data, even in the presence of outliers. Theoretically, our results show our novel MOM-based algorithm enjoys the same sample complexity guarantees as ERM under sub-Gaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavy-tailed and/or corrupted data, while our approach exhibits the predicted robustness. 1 Introduction Compressive or compressed sensing is the problem of reconstructing an unknown vector x∗ 2 Rn after observing m < n linear measurements of its entries, possibly with added noise: y = Ax∗ + η; where A 2 Rm×n is called the measurement matrix and η 2 Rm is noise. Even without noise, this is an underdetermined system of linear equations, so recovery is impossible without a structural assumption on the unknown vector x∗.
    [Show full text]
  • An Overview of Compressed Sensing
    What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered An Overview of Compressed sensing M. Vidyasagar FRS Cecil & Ida Green Chair, The University of Texas at Dallas [email protected], www.utdallas.edu/∼m.vidyasagar Distinguished Professor, IIT Hyderabad [email protected] Ba;a;ga;va;tua;l .=+a;ma mUa; a;tRa and Ba;a;ga;va;tua;l Za;a:=+d;a;}ba Memorial Lecture University of Hyderabad, 16 March 2015 M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Compressed Sensing: Basic Problem Formulation n Suppose x 2 R is known to be k-sparse, where k n. That is, jsupp(x)j ≤ k n, where supp denotes the support of a vector. However, it is not known which k components are nonzero. Can we recover x exactly by taking m n linear measurements of x? M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Precise Problem Statement: Basic Define the set of k-sparse vectors n Σk = fx 2 R : jsupp(x)j ≤ kg: m×n Do there exist an integer m, a \measurement matrix" A 2 R , m n and a \demodulation map" ∆ : R ! R , such that ∆(Ax) = x; 8x 2 Σk? Note: Measurements are linear but demodulation map could be nonlinear.
    [Show full text]
  • Paper, We Present Data Fusion Across Multiple Signal Sources
    68 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 51, NO. 1, JANUARY 2016 A Configurable 12–237 kS/s 12.8 mW Sparse-Approximation Engine for Mobile Data Aggregation of Compressively Sampled Physiological Signals Fengbo Ren, Member, IEEE, and Dejan Markovic,´ Member, IEEE Abstract—Compressive sensing (CS) is a promising technology framework, the CS framework has several intrinsic advantages. for realizing low-power and cost-effective wireless sensor nodes First, random encoding is a universal compression method that (WSNs) in pervasive health systems for 24/7 health monitoring. can effectively apply to all compressible signals regardless of Due to the high computational complexity (CC) of the recon- struction algorithms, software solutions cannot fulfill the energy what their sparse domain is. This is a desirable merit for the efficiency needs for real-time processing. In this paper, we present data fusion across multiple signal sources. Second, sampling a 12—237 kS/s 12.8 mW sparse-approximation (SA) engine chip and compression can be performed at the same stage in CS, that enables the energy-efficient data aggregation of compressively allowing for a sampling rate that is significantly lower than the sampled physiological signals on mobile platforms. The SA engine Nyquist rate. Therefore, CS has a potential to greatly impact chip integrated in 40 nm CMOS can support the simultaneous reconstruction of over 200 channels of physiological signals while the data acquisition devices that are sensitive to cost, energy consuming <1% of a smartphone’s power budget. Such energy- consumption, and portability, such as wireless sensor nodes efficient reconstruction enables two-to-three times energy saving (WSNs) in mobile and wearable applications [5].
    [Show full text]
  • Credit Spread Prediction Using Compressed Sensing
    CREDIT SPREAD PREDICTION USING COMPRESSED SENSING Graham L. Giller Data Science Research, Equity Division April 2017 What is Compressed Sensing? Parameter estimation technology developed by Tao, Candès, Donoho. When estimating coefficients of sparse linear systems, minimizing 퐿1 norms dominates traditional “least squares” analysis in accuracy. Only requires 표 푘푆 data points for perfect reconstruction of a noiseless 푆-sparse system of 푁 ≫ 푆 data points. 푘 ≈ 4, 12, ln 푁 … (depending on proof) Nyquist-Shannon sampling theorem says 2푁 ≫ 푘푆 Results are extendable to noisy systems. Applications: MRI Images, Radar Images, Factor Models! Why does it Work? (This due to Tao, who is very much smarter than me!) 2 2 Minimize 푥1 + 푥2 Minimize 푥1 + 푥2 Such that 푎푥1 + 푏푥2 = 푐 (푎, 푏, 푐 are random) Such that 푎푥1 + 푏푥2 = 푐 푥 푥 And 1 is sparse (lies on axis). And 1 is sparse. 푥2 푥2 Probability of sparse solution: Probability of sparse solution: 4 ∞ i.e. almost never. 4 4 i.e. almost always. Interpolating Polynomials Joseph Louis Lagrange, 1795 푁 10 푖−1 푝푁 푥 = 휃푖푥 푖=1 8 6 4 2 0 푁 -2 For any set of 푁 pairs 푦푖, 푥푖 푖=1 I can create a polynomial of order 푁 − 1, 푝푁 푥 -4 such that 푦푖 ≡ 푝푁 푥푖 ∀ 푖 i.e. The polynomial reproduces the observed data with zero error! -6 0 1 2 3 4 5 6 7 Sparse Interpolating Polynomials 10 5 0 -5 -10 If 푦푖 = 푝푆 푥푖 ∀ 푖 and 휽 0 = 푆 ≪ 푁 we have a sparse interpolating polynomial. -15 From an estimation point-of-view, this is a linear system since the polynomial is linear in the 2 푁−1 푇 -20 parameters vector: 푝푁 푥 = 1 푥 푥 ⋯ 푥 휽 Thus this problem is amenable to compressed sensing.
    [Show full text]
  • Can Compressed Sensing Beat the Nyquist Sampling Rate?
    Can compressed sensing beat the Nyquist sampling rate? Leonid P. Yaroslavsky, Dept. of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 699789, Israel Data saving capability of “Compressed sensing (sampling)” in signal discretization is disputed and found to be far below the theoretical upper bound defined by the signal sparsity. On a simple and intuitive example, it is demonstrated that, in a realistic scenario for signals that are believed to be sparse, one can achieve a substantially larger saving than compressing sensing can. It is also shown that frequent assertions in the literature that “Compressed sensing” can beat the Nyquist sampling approach are misleading substitution of terms and are rooted in misinterpretation of the sampling theory. Keywords: sampling; sampling theorem; compressed sensing. Paper 150246C received Feb. 26, 2015; accepted for publication Jun. 26, 2015. This letter is motivated by recent OPN publications1,2 that Therefore, the inverse of the signal sparsity SPRS=K/N is the advertise wide use in optical sensing of “Compressed sensing” theoretical upper bound of signal dimensionality reduction (CS), a new method of digital image formation that has achievable by means of signal sparse approximation. This obtained considerable attention after publications3-7. This relationship is plotted in Fig. 1 (solid line). attention is driven by such assertions in numerous The upper bound of signal dimensionality reduction publications as “CS theory asserts that one can recover CSDRF of signals with sparsity SSP achievable by CS can be certain signals and images from far fewer samples or evaluated from the relationship: measurements than traditional methods use”6, “Beating 1/CSDRF>C(2SSPlogCSDRF) provided in Ref.
    [Show full text]