The Sparsity Gap. Joel A. Tropp. Computing & Mathematical Sciences California Institute of Technology
|
|
- Aubrie Davis
- 6 years ago
- Views:
Transcription
1 The Sparsity Gap Joel A. Tropp Computing & Mathematical Sciences California Institute of Technology Research supported in part by ONR 1
2 Introduction The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 2
3 Systems of Linear Equations We consider linear systems of the form m Φ }{{} N x = b Assume that Φ has dimensions m N with N > m Φ has full row rank The columns of Φ have unit l 2 norm The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 3
4 The Trichotomy Theorem Theorem 1. For a linear system Φx = b, exactly one of the following situations occurs. 1. No solution exists. 2. The equation has a unique solution. 3. The solutions form a linear subspace of positive dimension. The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 4
5 Regularization via Sparsity A principled approach to underdetermined systems: min x 0 subject to Φx = b (P0) where x 0 = # supp(x) = #{j : x j 0} When x 0 s, then x is called s-sparse If Φx = b and x is s-sparse, then x is an s-sparse representation of b Since Φ has full row-rank, every b has an m-sparse representation Question: What can we say about sparser representations? The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 5
6 Geometry The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 6
7 Key Insight Sparse representations are well behaved when the matrix Φ is sufficiently nice (Column submatrices should not be singular and individual columns should not look like sparse signals) The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 7
8 Quantifying Niceness I We call Φ a tight frame when ΦΦ = N m I Equivalently, the rows of Φ form an orthonormal family (up to scaling) Observe that N/m is the redundancy of the frame Tight frames have minimal spectral norm among conformal matrices The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 8
9 Quantifying Niceness II The coherence of Φ is the quantity µ = max j k ϕ j, ϕ k Measures the angle between columns When N 2m, the coherence satisfies µ 1 m Incoherent matrices appear often in signal processing applications References: [Welch 1974, Mallat Zhang 1993, Donoho Huo 2001, Gribonval Nielsen 2003, Strohmer Heath 2003] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 9
10 Example: Identity + Fourier 1 Impulses Complex Exponentials A very incoherent tight frame The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 10
11 Uniqueness The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 11
12 Uncertainty implies Uniqueness Theorem 2. Suppose that a vector b has two representations: Φx = b = Φy. Then x 0 + y 0 > µ 1. Corollary 3. Suppose that b = Φx where x 0 < 1 2 µ 1. Then x is the unique solution to (P0). Very strict requirement since µ 1 m References: [Donoho Stark 1989, Donoho Huo 2001, Gribonval Nielsen 2003, Donoho Elad 2003] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 12
13 The Square-Root Threshold Sparse representations are not necessarily unique past the m threshold Example: The Dirac Comb Consider the Identity + Fourier matrix with m = p 2 There is a vector b that can be written as either p spikes or p sines By the Poisson summation formula, b(t) = p 1 j=0 δ pj (t) = 1 p 1 m j=0 e 2πipjt/m for t = 0, 1,..., m References: [Donoho Stark 1989] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 13
14 Enter Probability Insight: The bad vectors are atypical The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 14
15 An Uncertainty Principle for Generic Signals Theorem 4. [T 2010] Suppose that b = Φx where the nonzero components of x have a continuous distribution. With probability one, the vector b has no representation b = Φy where supp(x) supp(y) = unless x 0 + y 0 > µ 1 x 1/2 0. Corollary 5. When µ m 1/2, condition becomes [ ] m y 0 > x 0 1 x 0 Even with refinements, this approach does not yield uniqueness! Problem: Some supports could be bad References: [The Sparsity Gap] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 15
16 Enter More Probability Insight: The bad supports are atypical The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 16
17 A Simple Model for Random Sparse Vectors Model (M0) for b = Φx The matrix Φ The support of x The nonzero entries of x is a unit-norm tight frame of size m N with coherence µ c/ log N. has cardinality s cm/ log N and is uniformly random. have a continuous distribution. The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 17
18 Random Submatrices & Sparse Representation Theorem 6. [T 2006, 2008] Assume all parameters satisfy Model (M0). Draw a uniformly random set S of s columns from Φ, and define the random column submatrix A = Φ S. Then Prob { A A I < 1 } % and Prob { max n/ S A ϕ n 2 < 1 } %. References: [Random Subdictionaries, Random Submatrices] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 18
19 The Sparsity Gap Theorem 7. [T 2008, 2010] Let b = Φx be a vector drawn from Model (M0). With probability at least 99.44%, the following statements hold. 1. The vector x is the unique solution to (P0). 2. Furthermore, there is no disjoint representation b = Φy where supp(x) supp(y) = unless ( y 0 > x m ). N References: [Candès Romberg 2006, Random Subdictionaries, Random Submatrices, The Sparsity Gap] The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 19
20 To learn more... Web: Relevant papers: Conditioning of random subdictionaries, ACHA, 2008 Norms of random submatrices, CRAS, 2008 Spikes and sines, JFAA, 2008 The sparsity gap, Proc. CISS, 2010 The Sparsity Gap (Casazza Birthday Conference, College Park, 20 May 2010) 20
Uncertainty principles and sparse approximation
Uncertainty principles and sparse approximation In this lecture, we will consider the special case where the dictionary Ψ is composed of a pair of orthobases. We will see that our ability to find a sparse
More informationLecture: Introduction to Compressed Sensing Sparse Recovery Guarantees
Lecture: Introduction to Compressed Sensing Sparse Recovery Guarantees http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html Acknowledgement: this slides is based on Prof. Emmanuel Candes and Prof. Wotao Yin
More informationRobust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information
1 Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information Emmanuel Candès, California Institute of Technology International Conference on Computational Harmonic
More informationAn Introduction to Sparse Approximation
An Introduction to Sparse Approximation Anna C. Gilbert Department of Mathematics University of Michigan Basic image/signal/data compression: transform coding Approximate signals sparsely Compress images,
More informationThresholds for the Recovery of Sparse Solutions via L1 Minimization
Thresholds for the Recovery of Sparse Solutions via L Minimization David L. Donoho Department of Statistics Stanford University 39 Serra Mall, Sequoia Hall Stanford, CA 9435-465 Email: donoho@stanford.edu
More informationSignal Recovery, Uncertainty Relations, and Minkowski Dimension
Signal Recovery, Uncertainty Relations, and Minkowski Dimension Helmut Bőlcskei ETH Zurich December 2013 Joint work with C. Aubel, P. Kuppinger, G. Pope, E. Riegler, D. Stotz, and C. Studer Aim of this
More informationIEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 9, SEPTEMBER
IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 9, SEPTEMBER 2015 1239 Preconditioning for Underdetermined Linear Systems with Sparse Solutions Evaggelia Tsiligianni, StudentMember,IEEE, Lisimachos P. Kondi,
More informationSparse Optimization Lecture: Sparse Recovery Guarantees
Those who complete this lecture will know Sparse Optimization Lecture: Sparse Recovery Guarantees Sparse Optimization Lecture: Sparse Recovery Guarantees Instructor: Wotao Yin Department of Mathematics,
More informationCoSaMP. Iterative signal recovery from incomplete and inaccurate samples. Joel A. Tropp
CoSaMP Iterative signal recovery from incomplete and inaccurate samples Joel A. Tropp Applied & Computational Mathematics California Institute of Technology jtropp@acm.caltech.edu Joint with D. Needell
More informationGREEDY SIGNAL RECOVERY REVIEW
GREEDY SIGNAL RECOVERY REVIEW DEANNA NEEDELL, JOEL A. TROPP, ROMAN VERSHYNIN Abstract. The two major approaches to sparse recovery are L 1-minimization and greedy methods. Recently, Needell and Vershynin
More informationInverse problems, Dictionary based Signal Models and Compressed Sensing
Inverse problems, Dictionary based Signal Models and Compressed Sensing Rémi Gribonval METISS project-team (audio signal processing, speech recognition, source separation) INRIA, Rennes, France Ecole d
More informationUniqueness and Uncertainty
Chapter 2 Uniqueness and Uncertainty We return to the basic problem (P 0 ), which is at the core of our discussion, (P 0 ) : min x x 0 subject to b = Ax. While we shall refer hereafter to this problem
More informationSparse analysis Lecture III: Dictionary geometry and greedy algorithms
Sparse analysis Lecture III: Dictionary geometry and greedy algorithms Anna C. Gilbert Department of Mathematics University of Michigan Intuition from ONB Key step in algorithm: r, ϕ j = x c i ϕ i, ϕ j
More informationColumn Subset Selection
Column Subset Selection Joel A. Tropp Applied & Computational Mathematics California Institute of Technology jtropp@acm.caltech.edu Thanks to B. Recht (Caltech, IST) Research supported in part by NSF,
More informationSparsity in Underdetermined Systems
Sparsity in Underdetermined Systems Department of Statistics Stanford University August 19, 2005 Classical Linear Regression Problem X n y p n 1 > Given predictors and response, y Xβ ε = + ε N( 0, σ 2
More informationUniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit
Claremont Colleges Scholarship @ Claremont CMC Faculty Publications and Research CMC Faculty Scholarship 6-5-2008 Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit
More informationON THE LINEAR INDEPENDENCE OF SPIKES AND SINES
ON THE LINEAR INDEPENDENCE OF SPIKES AND SINES JOEL A. TROPP Abstract. The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results
More informationError Correction via Linear Programming
Error Correction via Linear Programming Emmanuel Candes and Terence Tao Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 Department of Mathematics, University of California, Los Angeles,
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 11, NOVEMBER On the Performance of Sparse Recovery
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 11, NOVEMBER 2011 7255 On the Performance of Sparse Recovery Via `p-minimization (0 p 1) Meng Wang, Student Member, IEEE, Weiyu Xu, and Ao Tang, Senior
More informationEquivalence Probability and Sparsity of Two Sparse Solutions in Sparse Representation
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 12, DECEMBER 2008 2009 Equivalence Probability and Sparsity of Two Sparse Solutions in Sparse Representation Yuanqing Li, Member, IEEE, Andrzej Cichocki,
More informationRecovering overcomplete sparse representations from structured sensing
Recovering overcomplete sparse representations from structured sensing Deanna Needell Claremont McKenna College Feb. 2015 Support: Alfred P. Sloan Foundation and NSF CAREER #1348721. Joint work with Felix
More informationOn Optimal Frame Conditioners
On Optimal Frame Conditioners Chae A. Clark Department of Mathematics University of Maryland, College Park Email: cclark18@math.umd.edu Kasso A. Okoudjou Department of Mathematics University of Maryland,
More informationUniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit
Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit arxiv:0707.4203v2 [math.na] 14 Aug 2007 Deanna Needell Department of Mathematics University of California,
More informationNear Optimal Signal Recovery from Random Projections
1 Near Optimal Signal Recovery from Random Projections Emmanuel Candès, California Institute of Technology Multiscale Geometric Analysis in High Dimensions: Workshop # 2 IPAM, UCLA, October 2004 Collaborators:
More informationSparse and Low-Rank Matrix Decompositions
Forty-Seventh Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 30 - October 2, 2009 Sparse and Low-Rank Matrix Decompositions Venkat Chandrasekaran, Sujay Sanghavi, Pablo A. Parrilo,
More informationIntroduction to Compressed Sensing
Introduction to Compressed Sensing Alejandro Parada, Gonzalo Arce University of Delaware August 25, 2016 Motivation: Classical Sampling 1 Motivation: Classical Sampling Issues Some applications Radar Spectral
More informationUsing Friendly Tail Bounds for Sums of Random Matrices
Using Friendly Tail Bounds for Sums of Random Matrices Joel A. Tropp Computing + Mathematical Sciences California Institute of Technology jtropp@cms.caltech.edu Research supported in part by NSF, DARPA,
More information5742 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 12, DECEMBER /$ IEEE
5742 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 12, DECEMBER 2009 Uncertainty Relations for Shift-Invariant Analog Signals Yonina C. Eldar, Senior Member, IEEE Abstract The past several years
More informationIntroduction How it works Theory behind Compressed Sensing. Compressed Sensing. Huichao Xue. CS3750 Fall 2011
Compressed Sensing Huichao Xue CS3750 Fall 2011 Table of Contents Introduction From News Reports Abstract Definition How it works A review of L 1 norm The Algorithm Backgrounds for underdetermined linear
More informationSolution-recovery in l 1 -norm for non-square linear systems: deterministic conditions and open questions
Solution-recovery in l 1 -norm for non-square linear systems: deterministic conditions and open questions Yin Zhang Technical Report TR05-06 Department of Computational and Applied Mathematics Rice University,
More informationReconstruction from Anisotropic Random Measurements
Reconstruction from Anisotropic Random Measurements Mark Rudelson and Shuheng Zhou The University of Michigan, Ann Arbor Coding, Complexity, and Sparsity Workshop, 013 Ann Arbor, Michigan August 7, 013
More informationA NEW FRAMEWORK FOR DESIGNING INCOHERENT SPARSIFYING DICTIONARIES
A NEW FRAMEWORK FOR DESIGNING INCOERENT SPARSIFYING DICTIONARIES Gang Li, Zhihui Zhu, 2 uang Bai, 3 and Aihua Yu 3 School of Automation & EE, Zhejiang Univ. of Sci. & Tech., angzhou, Zhejiang, P.R. China
More informationTHEORY OF COMPRESSIVE SENSING VIA l 1 -MINIMIZATION: A NON-RIP ANALYSIS AND EXTENSIONS
THEORY OF COMPRESSIVE SENSING VIA l 1 -MINIMIZATION: A NON-RIP ANALYSIS AND EXTENSIONS YIN ZHANG Abstract. Compressive sensing (CS) is an emerging methodology in computational signal processing that has
More informationFrame Coherence and Sparse Signal Processing
Frame Coherence and Sparse Signal Processing Dustin G ixon Waheed U Bajwa Robert Calderbank Program in Applied and Computational athematics Princeton University Princeton ew Jersey 08544 Department of
More informationLecture 22: More On Compressed Sensing
Lecture 22: More On Compressed Sensing Scribed by Eric Lee, Chengrun Yang, and Sebastian Ament Nov. 2, 207 Recap and Introduction Basis pursuit was the method of recovering the sparsest solution to an
More informationRSP-Based Analysis for Sparsest and Least l 1 -Norm Solutions to Underdetermined Linear Systems
1 RSP-Based Analysis for Sparsest and Least l 1 -Norm Solutions to Underdetermined Linear Systems Yun-Bin Zhao IEEE member Abstract Recently, the worse-case analysis, probabilistic analysis and empirical
More informationCompressed sensing. Or: the equation Ax = b, revisited. Terence Tao. Mahler Lecture Series. University of California, Los Angeles
Or: the equation Ax = b, revisited University of California, Los Angeles Mahler Lecture Series Acquiring signals Many types of real-world signals (e.g. sound, images, video) can be viewed as an n-dimensional
More informationUniqueness Conditions for A Class of l 0 -Minimization Problems
Uniqueness Conditions for A Class of l 0 -Minimization Problems Chunlei Xu and Yun-Bin Zhao October, 03, Revised January 04 Abstract. We consider a class of l 0 -minimization problems, which is to search
More informationHighly sparse representations from dictionaries are unique and independent of the sparseness measure. R. Gribonval and M. Nielsen
AALBORG UNIVERSITY Highly sparse representations from dictionaries are unique and independent of the sparseness measure by R. Gribonval and M. Nielsen October 2003 R-2003-16 Department of Mathematical
More informationConditions for Robust Principal Component Analysis
Rose-Hulman Undergraduate Mathematics Journal Volume 12 Issue 2 Article 9 Conditions for Robust Principal Component Analysis Michael Hornstein Stanford University, mdhornstein@gmail.com Follow this and
More informationCOMPRESSED SENSING IN PYTHON
COMPRESSED SENSING IN PYTHON Sercan Yıldız syildiz@samsi.info February 27, 2017 OUTLINE A BRIEF INTRODUCTION TO COMPRESSED SENSING A BRIEF INTRODUCTION TO CVXOPT EXAMPLES A Brief Introduction to Compressed
More informationCompressed Sensing and Redundant Dictionaries
Compressed Sensing and Redundant Dictionaries Holger Rauhut, Karin Schnass and Pierre Vandergheynst December 2, 2006 Abstract This article extends the concept of compressed sensing to signals that are
More informationA new method on deterministic construction of the measurement matrix in compressed sensing
A new method on deterministic construction of the measurement matrix in compressed sensing Qun Mo 1 arxiv:1503.01250v1 [cs.it] 4 Mar 2015 Abstract Construction on the measurement matrix A is a central
More informationLecture 18: The Rank of a Matrix and Consistency of Linear Systems
Lecture 18: The Rank of a Matrix and Consistency of Linear Systems Winfried Just Department of Mathematics, Ohio University February 28, 218 Review: The linear span Definition Let { v 1, v 2,..., v n }
More informationStable Signal Recovery from Incomplete and Inaccurate Measurements
Stable Signal Recovery from Incomplete and Inaccurate Measurements EMMANUEL J. CANDÈS California Institute of Technology JUSTIN K. ROMBERG California Institute of Technology AND TERENCE TAO University
More informationA Generalized Uncertainty Principle and Sparse Representation in Pairs of Bases
2558 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 48, NO 9, SEPTEMBER 2002 A Generalized Uncertainty Principle Sparse Representation in Pairs of Bases Michael Elad Alfred M Bruckstein Abstract An elementary
More informationInverse problems and sparse models (1/2) Rémi Gribonval INRIA Rennes - Bretagne Atlantique, France
Inverse problems and sparse models (1/2) Rémi Gribonval INRIA Rennes - Bretagne Atlantique, France remi.gribonval@inria.fr Structure of the tutorial Session 1: Introduction to inverse problems & sparse
More informationMATCHING PURSUIT WITH STOCHASTIC SELECTION
2th European Signal Processing Conference (EUSIPCO 22) Bucharest, Romania, August 27-3, 22 MATCHING PURSUIT WITH STOCHASTIC SELECTION Thomas Peel, Valentin Emiya, Liva Ralaivola Aix-Marseille Université
More informationSPARSE NEAR-EQUIANGULAR TIGHT FRAMES WITH APPLICATIONS IN FULL DUPLEX WIRELESS COMMUNICATION
SPARSE NEAR-EQUIANGULAR TIGHT FRAMES WITH APPLICATIONS IN FULL DUPLEX WIRELESS COMMUNICATION A. Thompson Mathematical Institute University of Oxford Oxford, United Kingdom R. Calderbank Department of ECE
More informationAbstract This paper is about the efficient solution of large-scale compressed sensing problems.
Noname manuscript No. (will be inserted by the editor) Optimization for Compressed Sensing: New Insights and Alternatives Robert Vanderbei and Han Liu and Lie Wang Received: date / Accepted: date Abstract
More informationSensing systems limited by constraints: physical size, time, cost, energy
Rebecca Willett Sensing systems limited by constraints: physical size, time, cost, energy Reduce the number of measurements needed for reconstruction Higher accuracy data subject to constraints Original
More informationDesign of Projection Matrix for Compressive Sensing by Nonsmooth Optimization
Design of Proection Matrix for Compressive Sensing by Nonsmooth Optimization W.-S. Lu T. Hinamoto Dept. of Electrical & Computer Engineering Graduate School of Engineering University of Victoria Hiroshima
More informationBeyond incoherence and beyond sparsity: compressed sensing in the real world
Beyond incoherence and beyond sparsity: compressed sensing in the real world Clarice Poon 1st November 2013 University of Cambridge, UK Applied Functional and Harmonic Analysis Group Head of Group Anders
More informationWe will now focus on underdetermined systems of equations: data acquisition system
l 1 minimization We will now focus on underdetermined systems of equations: # samples = resolution/ bandwidth data acquisition system unknown signal/image Suppose we observe y = Φx 0, and given y we attempt
More informationA simple test to check the optimality of sparse signal approximations
A simple test to check the optimality of sparse signal approximations Rémi Gribonval, Rosa Maria Figueras I Ventura, Pierre Vergheynst To cite this version: Rémi Gribonval, Rosa Maria Figueras I Ventura,
More informationNecessary and sufficient conditions of solution uniqueness in l 1 minimization
1 Necessary and sufficient conditions of solution uniqueness in l 1 minimization Hui Zhang, Wotao Yin, and Lizhi Cheng arxiv:1209.0652v2 [cs.it] 18 Sep 2012 Abstract This paper shows that the solutions
More informationNoisy Signal Recovery via Iterative Reweighted L1-Minimization
Noisy Signal Recovery via Iterative Reweighted L1-Minimization Deanna Needell UC Davis / Stanford University Asilomar SSC, November 2009 Problem Background Setup 1 Suppose x is an unknown signal in R d.
More informationThe Kadison-Singer Problem and the Uncertainty Principle Eric Weber joint with Pete Casazza
The Kadison-Singer Problem and the Uncertainty Principle Eric Weber joint with Pete Casazza Illinois-Missouri Applied Harmonic Analysis Seminar, April 28, 2007. Abstract: We endeavor to tell a story which
More informationNew Coherence and RIP Analysis for Weak. Orthogonal Matching Pursuit
New Coherence and RIP Analysis for Wea 1 Orthogonal Matching Pursuit Mingrui Yang, Member, IEEE, and Fran de Hoog arxiv:1405.3354v1 [cs.it] 14 May 2014 Abstract In this paper we define a new coherence
More informationNear Ideal Behavior of a Modified Elastic Net Algorithm in Compressed Sensing
Near Ideal Behavior of a Modified Elastic Net Algorithm in Compressed Sensing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas M.Vidyasagar@utdallas.edu www.utdallas.edu/ m.vidyasagar
More informationSparsity and Incoherence in Compressive Sampling
Sparsity and Incoherence in Compressive Sampling Emmanuel Candès and Justin Romberg Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 Electrical and Computer Engineering, Georgia Tech,
More informationAn algebraic perspective on integer sparse recovery
An algebraic perspective on integer sparse recovery Lenny Fukshansky Claremont McKenna College (joint work with Deanna Needell and Benny Sudakov) Combinatorics Seminar USC October 31, 2018 From Wikipedia:
More informationRandomness-in-Structured Ensembles for Compressed Sensing of Images
Randomness-in-Structured Ensembles for Compressed Sensing of Images Abdolreza Abdolhosseini Moghadam Dep. of Electrical and Computer Engineering Michigan State University Email: abdolhos@msu.edu Hayder
More informationWIRELESS sensor networks (WSNs) have attracted
On the Benefit of using ight Frames for Robust Data ransmission and Compressive Data Gathering in Wireless Sensor Networks Wei Chen, Miguel R. D. Rodrigues and Ian J. Wassell Computer Laboratory, University
More informationStrengthened Sobolev inequalities for a random subspace of functions
Strengthened Sobolev inequalities for a random subspace of functions Rachel Ward University of Texas at Austin April 2013 2 Discrete Sobolev inequalities Proposition (Sobolev inequality for discrete images)
More informationSignal Recovery From Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit
Signal Recovery From Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit Deanna Needell and Roman Vershynin Abstract We demonstrate a simple greedy algorithm that can reliably
More informationDetecting Sparse Structures in Data in Sub-Linear Time: A group testing approach
Detecting Sparse Structures in Data in Sub-Linear Time: A group testing approach Boaz Nadler The Weizmann Institute of Science Israel Joint works with Inbal Horev, Ronen Basri, Meirav Galun and Ery Arias-Castro
More informationAN INTRODUCTION TO COMPRESSIVE SENSING
AN INTRODUCTION TO COMPRESSIVE SENSING Rodrigo B. Platte School of Mathematical and Statistical Sciences APM/EEE598 Reverse Engineering of Complex Dynamical Networks OUTLINE 1 INTRODUCTION 2 INCOHERENCE
More informationIEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 2, FEBRUARY 2006 423 Underdetermined Blind Source Separation Based on Sparse Representation Yuanqing Li, Shun-Ichi Amari, Fellow, IEEE, Andrzej Cichocki,
More informationOn Sparse Solutions of Underdetermined Linear Systems
On Sparse Solutions of Underdetermined Linear Systems Ming-Jun Lai Department of Mathematics the University of Georgia Athens, GA 30602 January 17, 2009 Abstract We first explain the research problem of
More informationLEARNING OVERCOMPLETE SPARSIFYING TRANSFORMS FOR SIGNAL PROCESSING. Saiprasad Ravishankar and Yoram Bresler
LEARNING OVERCOMPLETE SPARSIFYING TRANSFORMS FOR SIGNAL PROCESSING Saiprasad Ravishankar and Yoram Bresler Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University
More informationPre-weighted Matching Pursuit Algorithms for Sparse Recovery
Journal of Information & Computational Science 11:9 (214) 2933 2939 June 1, 214 Available at http://www.joics.com Pre-weighted Matching Pursuit Algorithms for Sparse Recovery Jingfei He, Guiling Sun, Jie
More informationStructured matrix factorizations. Example: Eigenfaces
Structured matrix factorizations Example: Eigenfaces An extremely large variety of interesting and important problems in machine learning can be formulated as: Given a matrix, find a matrix and a matrix
More informationConstructing Explicit RIP Matrices and the Square-Root Bottleneck
Constructing Explicit RIP Matrices and the Square-Root Bottleneck Ryan Cinoman July 18, 2018 Ryan Cinoman Constructing Explicit RIP Matrices July 18, 2018 1 / 36 Outline 1 Introduction 2 Restricted Isometry
More informationNecessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization
Noname manuscript No. (will be inserted by the editor) Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization Hui Zhang Wotao Yin Lizhi Cheng Received: / Accepted: Abstract This
More informationCompressed Sensing and Redundant Dictionaries
Compressed Sensing and Redundant Dictionaries Holger Rauhut, Karin Schnass and Pierre Vandergheynst Abstract This article extends the concept of compressed sensing to signals that are not sparse in an
More informationGreedy Signal Recovery and Uniform Uncertainty Principles
Greedy Signal Recovery and Uniform Uncertainty Principles SPIE - IE 2008 Deanna Needell Joint work with Roman Vershynin UC Davis, January 2008 Greedy Signal Recovery and Uniform Uncertainty Principles
More informationCompressed Sensing and Sparse Recovery
ELE 538B: Sparsity, Structure and Inference Compressed Sensing and Sparse Recovery Yuxin Chen Princeton University, Spring 217 Outline Restricted isometry property (RIP) A RIPless theory Compressed sensing
More informationTHE KADISON-SINGER PROBLEM AND THE UNCERTAINTY PRINCIPLE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 THE KADISON-SINGER PROBLEM AND THE UNCERTAINTY PRINCIPLE PETER G. CASAZZA AND ERIC WEBER Abstract.
More informationSimultaneous Sparsity
Simultaneous Sparsity Joel A. Tropp Anna C. Gilbert Martin J. Strauss {jtropp annacg martinjs}@umich.edu Department of Mathematics The University of Michigan 1 Simple Sparse Approximation Work in the d-dimensional,
More informationFinite Frames for Sparse Signal Processing
Finite Frames for Sparse Signal Processing Waheed U. Bajwa and Ali Pezeshki Abstract Over the last decade, considerable progress has been made towards developing new signal processing methods to manage
More informationOn Rank Awareness, Thresholding, and MUSIC for Joint Sparse Recovery
On Rank Awareness, Thresholding, and MUSIC for Joint Sparse Recovery Jeffrey D. Blanchard a,1,, Caleb Leedy a,2, Yimin Wu a,2 a Department of Mathematics and Statistics, Grinnell College, Grinnell, IA
More informationSparse representations and approximation theory
Journal of Approximation Theory 63 (20) 388 42 wwwelseviercom/locate/jat Sparse representations and approximation theory Allan Pinkus Department of Mathematics, Technion, 32000 Haifa, Israel Received 28
More informationZ Algorithmic Superpower Randomization October 15th, Lecture 12
15.859-Z Algorithmic Superpower Randomization October 15th, 014 Lecture 1 Lecturer: Bernhard Haeupler Scribe: Goran Žužić Today s lecture is about finding sparse solutions to linear systems. The problem
More informationCompressed Sensing: Lecture I. Ronald DeVore
Compressed Sensing: Lecture I Ronald DeVore Motivation Compressed Sensing is a new paradigm for signal/image/function acquisition Motivation Compressed Sensing is a new paradigm for signal/image/function
More informationExact Low-rank Matrix Recovery via Nonconvex M p -Minimization
Exact Low-rank Matrix Recovery via Nonconvex M p -Minimization Lingchen Kong and Naihua Xiu Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, People s Republic of China E-mail:
More informationThe Analysis Cosparse Model for Signals and Images
The Analysis Cosparse Model for Signals and Images Raja Giryes Computer Science Department, Technion. The research leading to these results has received funding from the European Research Council under
More informationEUSIPCO
EUSIPCO 013 1569746769 SUBSET PURSUIT FOR ANALYSIS DICTIONARY LEARNING Ye Zhang 1,, Haolong Wang 1, Tenglong Yu 1, Wenwu Wang 1 Department of Electronic and Information Engineering, Nanchang University,
More informationQuantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions Emmanuel J. Candès and Justin Romberg Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 ovember 2004; Revised
More informationIN a series of recent papers [1] [4], the morphological component
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 2675 Morphological Component Analysis: An Adaptive Thresholding Strategy Jérôme Bobin, Jean-Luc Starck, Jalal M. Fadili, Yassir Moudden,
More informationSparse Solutions of Linear Systems of Equations and Sparse Modeling of Signals and Images!
Sparse Solutions of Linear Systems of Equations and Sparse Modeling of Signals and Images! Alfredo Nava-Tudela John J. Benedetto, advisor 1 Happy birthday Lucía! 2 Outline - Problem: Find sparse solutions
More informationc 2011 International Press Vol. 18, No. 1, pp , March DENNIS TREDE
METHODS AND APPLICATIONS OF ANALYSIS. c 2011 International Press Vol. 18, No. 1, pp. 105 110, March 2011 007 EXACT SUPPORT RECOVERY FOR LINEAR INVERSE PROBLEMS WITH SPARSITY CONSTRAINTS DENNIS TREDE Abstract.
More informationInverse Eigenvalue Problems in Wireless Communications
Inverse Eigenvalue Problems in Wireless Communications Inderjit S. Dhillon Robert W. Heath Jr. Mátyás Sustik Joel A. Tropp The University of Texas at Austin Thomas Strohmer The University of California
More informationQuantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions Emmanuel J. Candès and Justin Romberg Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 June 3, 2005 Abstract
More informationSparse analysis Lecture V: From Sparse Approximation to Sparse Signal Recovery
Sparse analysis Lecture V: From Sparse Approximation to Sparse Signal Recovery Anna C. Gilbert Department of Mathematics University of Michigan Connection between... Sparse Approximation and Compressed
More informationSPARSE signal representations have gained popularity in recent
6958 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 10, OCTOBER 2011 Blind Compressed Sensing Sivan Gleichman and Yonina C. Eldar, Senior Member, IEEE Abstract The fundamental principle underlying
More information6 Compressed Sensing and Sparse Recovery
6 Compressed Sensing and Sparse Recovery Most of us have noticed how saving an image in JPEG dramatically reduces the space it occupies in our hard drives as oppose to file types that save the pixel value
More informationOptimisation Combinatoire et Convexe.
Optimisation Combinatoire et Convexe. Low complexity models, l 1 penalties. A. d Aspremont. M1 ENS. 1/36 Today Sparsity, low complexity models. l 1 -recovery results: three approaches. Extensions: matrix
More informationLinear Systems. Carlo Tomasi. June 12, r = rank(a) b range(a) n r solutions
Linear Systems Carlo Tomasi June, 08 Section characterizes the existence and multiplicity of the solutions of a linear system in terms of the four fundamental spaces associated with the system s matrix
More informationMorphological Diversity and Source Separation
Morphological Diversity and Source Separation J. Bobin, Y. Moudden, J.-L. Starck, and M. Elad Abstract This paper describes a new method for blind source separation, adapted to the case of sources having
More information