Adaptive Compressive Imaging Using Sparse Hierarchical Learned Dictionaries
|
|
- Adelia Spencer
- 6 years ago
- Views:
Transcription
1 Adaptive Compressive Imaging Using Sparse Hierarchical Learned Dictionaries Jarvis Haupt University of Minnesota Department of Electrical and Computer Engineering Supported by
2 Motivation New Agile Sensing Platforms
3 A Host of New Agile Imaging Sensors Original 20% 40% samples samples
4 Overview of This Talk Fusing Adaptive Sensing and Structured Sparsity in theory and in practice
5 Background Sparse Inference and Adaptive Sensing
6 A Model for Sparsity Objects of interest are vectors x 2 R n Signal Support: S, {i : x i =0} Sparse S = k n number of nonzero signal components
7 A Sparse Inference Task Noisy Linear Observation Model: y = x + w R m n w N (0,I m m ) Support Recovery Goal: Obtain an (accurate) estimate b S = b S(y, ) of true support S
8 A Sparse Inference Task Noisy Linear Observation Model: y = x + w R m n w N (0,I m m ) Assume Sensing energy 2 F fixed: 2 F = R x i µ for all i S What conditions are necessary/sufficient for exact support recovery? (eg., such that P (S = b S) 0 as n )
9 Exact Support Recovery? Point sampling y = x + w (Sensing energy R = n) Necessary & Sufficient for Exact Support Recovery: r n µ const. log n R Uncompressed Sensing (Donoho & Jin 2004; JH, Castro, & Nowak 2010) Compressed Sensing (Genovese, Jin, & Wasserman 2009; Aeron, Saligrama & Zhao, 2010)
10 Conditions for Exact Support Recovery Uncompressed / compressed r n µ const. log n R (N + S) Uncompressed Sensing (Donoho & Jin 2004; JH, Castro, & Nowak 2010) Compressed Sensing (Genovese, Jin, & Wasserman 2009; Aeron, Saligrama & Zhao, 2010) Question: Can we do better by exploiting structure, or adaptivity, or both?
11 Conditions for Exact Support Recovery r n µ const. log n R r n µ const. log k R Uncompressed/ Compressed (N + S) Necessity: (Castro 2012) Sufficiency (uncompressed): (Malloy & Nowak, 2010; Malloy & Nowak, 2011) Sufficiency (compressed): (JH, Baraniuk, Castro, & Nowak 2012, Malloy & Nowak 2013) y = x + w R m n w N (0,I m m ) k k 2 F = R
12 Beyond Simple Sparsity The Role of Structure
13 Our Focus: Tree Sparsity Tree T with n nodes and degree d Characteristics of tree structure: Elements of x in one-to-one correspondence with nodes of T Nonzeros of tree-sparse vector form rooted connected subtree of T Question: Does tree structure help in support recovery?
14 Conditions for Exact Support Recovery Detection of simple trail (uncompressed sensing) r n µ const. log n R r n µ const. R (N+S)* r n µ const. log k R Signal Detection Problem: (Arias-Castro, Candes, Helgason, & Zeitouni 2008)
15 Conditions for Exact Support Recovery The intersection of adaptivity and (tree) structure... r n µ const. log n R r n µ const. R r n µ const. log k R s k µ const. log k R (S) (A. Soni & JH, 2011) Akshay Soni University of Minnesota Recent related work: Adaptivity and Structure in finding activated blocks in a matrix (Balakrishnan, Kolar, Rinaldo, and Singh 2012)
16 Adaptive Tree Sensing: An Example If the hypothesis test is correct at each step, then m = dk +1=O(k) y (1) = x 1 + w (1) suppose y (1) > Adaptive Wavelet Tree Sensing in the Literature: Non-Fourier encoded MRI (Panych & Jolesz, 1994) Compressive Imaging (Deutsch, Averbuch, & Dekel, 2009) y (2) = x 2 + w (2) suppose y (2) < (none analyzed the case of noisy measurements...) y (3) = x 5 + w (3) y (4) = x 6 + w (4) y (5) = x 7 + w (5) suppose y (3) > suppose y (4) < suppose y (5) <
17 Orthogonal Dictionaries and Tree Sparsity Consider signals z R p that are sparse in a known dictionary D R p n. That is, z = Dx, where x R n is k-sparse, D satisfies D T D = I n n, and columns of D are d j, j =1, 2,...,n We are interested in the case where x is tree-sparse... Collect (noisy) observations of z by projecting onto (scaled) columns of D. Suppose, for example, that the j-th measurement is obtained by projecting onto columnn d i, then y (j) = d T i z + w (j) where w (j) N (0, 1). Nonnegative scaling factor (equivalently, could consider non-unit noise variance)
18 Support Recovery via Adaptive Tree Sensing Theorem (A. Soni & JH, 2011) Let T n,d be a balanced, rooted connected tree of degree d with n nodes. Suppose that z R p can be expressed as z = Dx, where D is a known dictionary with orthonormal columns and x is k-sparse. Further, suppose the support of x corresponds to a rooted connected subtree of T n,d. Observations of z are of the form of projections of z onto columns of D. Let the index corresponding to the root of T n,d be known, and apply the top-down tree sensing procedure with threshold and scaling parameter. For any c 1 > 0 and c 2 (0, 1), there exists a constant c 3 > 0 such that if µ =min i S x i p c 3 2 log k and = c 2 µ, the tree sensing procedure collects m = dk+1 measurements, and produces a support estimate b S that equals S with probability at least 1 k c 1. q Choose = R (d+1)k, then the theorem guarantees exact support recovery (whp) when µ s c 3 (d + 1) k log k R
19 Conditions for Exact Support Recovery The intersection of adaptivity and (tree) structure... r n µ const. log n R r n µ const. R (*conjecture for support recovery) r n µ const. log k R s k µ const. log k R (S) (A. Soni & JH, 2011) Akshay Soni University of Minnesota
20 LASeR Learning Adaptive Sensing Representations
21 Beyond Wavelet Trees: Learned Representations Given training data Z R p q, want to learn a dictionary D so that Z DX, D R p n, X R n q, and each column of X, x i R n, is tree-sparse in T n,d. Pose this as an optimization: {D, X} = arg min D R p n,d T D=I n n, {x i } qx z i Dx i (x i ) i=1 The regularization term is (x i )= P g G g (x i ) g, where G denotes a set of (overlapping) groups of indices for x, (x i ) g is x i restricted to the indices in the group g G, g are non-negative weights, and the norm can be, eg., 2 or Solve by alternating minimization over D and X (Jenatton, Mairal, Obozinski, & Bach, 2010) Sparse Modeling Software (SPAMS):
22 Group Specifications to Enforce Tree Structure Example: Binary Tree, 15 nodes, 4 levels Number of groups same as number of nodes (but varying sizes)
23 LASeR An Illustrative Example
24 Learning Adaptive Sensing Representations Training' Data' Structured' Sparsity' LASeR' Adap3ve' Sensing' Learn representation for 163 images from Psychological Image Collection at Stirling (PICS) LASeR:'Learning'Adap3ve'Sensing'Representa3ons' Example images ( )
25 Learned Orthogonal Tree-Basis Elements (First four levels of 7 total)
26 Original Image Tree Elements Present in Sparse Representation
27 Qualitative Results Sensing Energy R = Wavelet Tree Sensing m = 20 m = 50 m = 80 PCA CS LASSO CS Tree LASSO LASeR
28 Qualitative Results Sensing Energy R = Wavelet Tree Sensing m = 20 m = 50 m = 80 PCA CS LASSO CS Tree LASSO LASeR
29 Quantitative Results Reconstruction SNR(dB) #projections Reconstruction SNR(dB) #projections Reconstruction SNR(dB) #projections
30 LASeR Imaging via Patch-wise Sensing
31 Patch-wise Sensing Experiment Motivated by EO Imaging Application (Thanks: Bob Lockheed Martin) Training Data: 3 Sample images from the Columbus Large Image Format (CLIF) 2007 Dataset Each image is 1024x1024 Randomly extracted x32 patches (at random locations) and vectorized them into length 1024 vectors Applied PCA and LASeR (7-level 127 node binary tree) to this training data
32 Compare: PCA Basis Elements In the tree-sensing context, can view PCA sensing approach in terms of a tree of degree 1
33 Learned Orthogonal Tree-Basis Elements
34 Example: Approximation by Patch-wise Sensing Test Image (another image from CLIF database) Sense & reconstruct non-overlapping 32x32 patches comparing LASeR, PCA, Wavelets
35 Approximation Results Uniform Sampling Rate Sampling rate: 12.5% LASeR rsnr = 16.5 db PCA rsnr = 17.6 db rsnr, 20 log 10 (kbx xk F /kxk F )
36 Approximation Results Uniform Sampling Rate Sampling rate: 12.5% LASeR rsnr = 16.5 db rsnr, 20 log 10 (kbx xk F /kxk F )
37 Approximation Results Uniform Sampling Rate Sampling rate: 12.5% PCA rsnr = 17.6 db rsnr, 20 log 10 (kbx xk F /kxk F )
38 Approximation Results Uniform Sampling Rate Sampling rate: 12.5% LASeR rsnr = 16.5 db 2D Haar Wavelet rsnr = 13.5 db rsnr, 20 log 10 (kbx xk F /kxk F )
39 Approximation Results Adaptive Sampling Rate Average sampling rate: 7.2% LASeR rsnr = 13.9 db PCA rsnr = 15.0 db rsnr, 20 log 10 (kbx xk F /kxk F )
40 Approximation Results Adaptive Sampling Rate Average sampling rate: 7.2% LASeR rsnr = 13.9 db 2D Haar Wavelet rsnr = 11.9 db rsnr, 20 log 10 (kbx xk F /kxk F )
41 Approximation: Zoomed In A Closer Look... (Average sampling rate: 7.2%) Original LASeR PCA 2D Haar
42 Approximation: Zoomed In A Closer Look... (Average sampling rate: 7.2%) Original LASeR PCA 2D Haar
43 Sampling Rate Adapts to Block Complexity Sampling rate per block (Average sampling rate: 7.2%) LASeR PCA
44 Sampling Rate Adapts to Block Complexity Sampling rate per block (Average sampling rate: 7.2%) LASeR 2D Haar
45 Summary: Adaptivity + Structure In Theory (Support Recovery) r n µ const. log n R r n µ const. R (conj.) r n µ const. log k R s k µ const. log k R Conclusions: Polynomial reduction in SNR required for exact support recovery (for fixed sensing energy )
46 Summary: Adaptivity + Structure In Practice (Learned Representations and Patch-wise Sensing) Original LASeR PCA Conclusions: PCA works very well on small patch sizes (shared, elemental structure) in noise free settings
47 Summary: Adaptivity + Structure m = 20 m = 50 m = 80 Wavelet Tree Sensing PCA Original Image LASeR Conclusions: Potential benefit for learned representations depend on patch size, data regularity, noise. Use LASeR on PCA residuals? (ie, fuse PCA and LASeR) jdhaupt jdhaupt@umn.edu
Efficient Adaptive Compressive Sensing Using Sparse Hierarchical Learned Dictionaries
1 Efficient Adaptive Compressive Sensing Using Sparse Hierarchical Learned Dictionaries Akshay Soni and Jarvis Haupt University of Minnesota, Twin Cities Department of Electrical and Computer Engineering
More informationTopographic Dictionary Learning with Structured Sparsity
Topographic Dictionary Learning with Structured Sparsity Julien Mairal 1 Rodolphe Jenatton 2 Guillaume Obozinski 2 Francis Bach 2 1 UC Berkeley 2 INRIA - SIERRA Project-Team San Diego, Wavelets and Sparsity
More informationRecent Advances in Structured Sparse Models
Recent Advances in Structured Sparse Models Julien Mairal Willow group - INRIA - ENS - Paris 21 September 2010 LEAR seminar At Grenoble, September 21 st, 2010 Julien Mairal Recent Advances in Structured
More information2.3. Clustering or vector quantization 57
Multivariate Statistics non-negative matrix factorisation and sparse dictionary learning The PCA decomposition is by construction optimal solution to argmin A R n q,h R q p X AH 2 2 under constraint :
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 informationConvex relaxation for Combinatorial Penalties
Convex relaxation for Combinatorial Penalties Guillaume Obozinski Equipe Imagine Laboratoire d Informatique Gaspard Monge Ecole des Ponts - ParisTech Joint work with Francis Bach Fête Parisienne in Computation,
More informationBayesian Methods for Sparse Signal Recovery
Bayesian Methods for Sparse Signal Recovery Bhaskar D Rao 1 University of California, San Diego 1 Thanks to David Wipf, Jason Palmer, Zhilin Zhang and Ritwik Giri Motivation Motivation Sparse Signal Recovery
More informationModel-Based Compressive Sensing for Signal Ensembles. Marco F. Duarte Volkan Cevher Richard G. Baraniuk
Model-Based Compressive Sensing for Signal Ensembles Marco F. Duarte Volkan Cevher Richard G. Baraniuk Concise Signal Structure Sparse signal: only K out of N coordinates nonzero model: union of K-dimensional
More informationSparse Estimation and Dictionary Learning
Sparse Estimation and Dictionary Learning (for Biostatistics?) Julien Mairal Biostatistics Seminar, UC Berkeley Julien Mairal Sparse Estimation and Dictionary Learning Methods 1/69 What this talk is about?
More informationCompressed Sensing and Neural Networks
and Jan Vybíral (Charles University & Czech Technical University Prague, Czech Republic) NOMAD Summer Berlin, September 25-29, 2017 1 / 31 Outline Lasso & Introduction Notation Training the network Applications
More informationLow Resolution Adaptive Compressed Sensing for mmwave MIMO receivers
Low Resolution Adaptive Compressed Sensing for mmwave MIMO receivers Cristian Rusu, Nuria González-Prelcic and Robert W. Heath Motivation 2 Power consumption at mmwave in a MIMO receiver Power at 60 GHz
More informationSparsifying Transform Learning for Compressed Sensing MRI
Sparsifying Transform Learning for Compressed Sensing MRI Saiprasad Ravishankar and Yoram Bresler Department of Electrical and Computer Engineering and Coordinated Science Laborarory University of Illinois
More informationGlobal vs. Multiscale Approaches
Harmonic Analysis on Graphs Global vs. Multiscale Approaches Weizmann Institute of Science, Rehovot, Israel July 2011 Joint work with Matan Gavish (WIS/Stanford), Ronald Coifman (Yale), ICML 10' Challenge:
More informationThe Pros and Cons of Compressive Sensing
The Pros and Cons of Compressive Sensing Mark A. Davenport Stanford University Department of Statistics Compressive Sensing Replace samples with general linear measurements measurements sampled signal
More informationEE 381V: Large Scale Optimization Fall Lecture 24 April 11
EE 381V: Large Scale Optimization Fall 2012 Lecture 24 April 11 Lecturer: Caramanis & Sanghavi Scribe: Tao Huang 24.1 Review In past classes, we studied the problem of sparsity. Sparsity problem is that
More informationProximal Methods for Optimization with Spasity-inducing Norms
Proximal Methods for Optimization with Spasity-inducing Norms Group Learning Presentation Xiaowei Zhou Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology
More informationStructured Sparse Estimation with Network Flow Optimization
Structured Sparse Estimation with Network Flow Optimization Julien Mairal University of California, Berkeley Neyman seminar, Berkeley Julien Mairal Neyman seminar, UC Berkeley /48 Purpose of the talk introduce
More informationAlgorithms for sparse analysis Lecture I: Background on sparse approximation
Algorithms for sparse analysis Lecture I: Background on sparse approximation Anna C. Gilbert Department of Mathematics University of Michigan Tutorial on sparse approximations and algorithms Compress data
More informationSparse representation classification and positive L1 minimization
Sparse representation classification and positive L1 minimization Cencheng Shen Joint Work with Li Chen, Carey E. Priebe Applied Mathematics and Statistics Johns Hopkins University, August 5, 2014 Cencheng
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 informationLarge-Scale L1-Related Minimization in Compressive Sensing and Beyond
Large-Scale L1-Related Minimization in Compressive Sensing and Beyond Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, U.S.A. Arizona State University March
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 informationGreedy Dictionary Selection for Sparse Representation
Greedy Dictionary Selection for Sparse Representation Volkan Cevher Rice University volkan@rice.edu Andreas Krause Caltech krausea@caltech.edu Abstract We discuss how to construct a dictionary by selecting
More informationCompressed Sensing Using Reed- Solomon and Q-Ary LDPC Codes
Compressed Sensing Using Reed- Solomon and Q-Ary LDPC Codes Item Type text; Proceedings Authors Jagiello, Kristin M. Publisher International Foundation for Telemetering Journal International Telemetering
More informationCompressed Sensing and Related Learning Problems
Compressed Sensing and Related Learning Problems Yingzhen Li Dept. of Mathematics, Sun Yat-sen University Advisor: Prof. Haizhang Zhang Advisor: Prof. Haizhang Zhang 1 / Overview Overview Background Compressed
More informationTree-Structured Compressive Sensing with Variational Bayesian Analysis
1 Tree-Structured Compressive Sensing with Variational Bayesian Analysis Lihan He, Haojun Chen and Lawrence Carin Department of Electrical and Computer Engineering Duke University Durham, NC 27708-0291
More informationCompressibility of Infinite Sequences and its Interplay with Compressed Sensing Recovery
Compressibility of Infinite Sequences and its Interplay with Compressed Sensing Recovery Jorge F. Silva and Eduardo Pavez Department of Electrical Engineering Information and Decision Systems Group Universidad
More informationSparse & Redundant Signal Representation, and its Role in Image Processing
Sparse & Redundant Signal Representation, and its Role in Michael Elad The CS Department The Technion Israel Institute of technology Haifa 3000, Israel Wave 006 Wavelet and Applications Ecole Polytechnique
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 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 informationOverview. Optimization-Based Data Analysis. Carlos Fernandez-Granda
Overview Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda 1/25/2016 Sparsity Denoising Regression Inverse problems Low-rank models Matrix completion
More informationSignal Recovery from Permuted Observations
EE381V Course Project Signal Recovery from Permuted Observations 1 Problem Shanshan Wu (sw33323) May 8th, 2015 We start with the following problem: let s R n be an unknown n-dimensional real-valued signal,
More information1 Sparsity and l 1 relaxation
6.883 Learning with Combinatorial Structure Note for Lecture 2 Author: Chiyuan Zhang Sparsity and l relaxation Last time we talked about sparsity and characterized when an l relaxation could recover the
More informationOnline Dictionary Learning with Group Structure Inducing Norms
Online Dictionary Learning with Group Structure Inducing Norms Zoltán Szabó 1, Barnabás Póczos 2, András Lőrincz 1 1 Eötvös Loránd University, Budapest, Hungary 2 Carnegie Mellon University, Pittsburgh,
More informationParcimonie en apprentissage statistique
Parcimonie en apprentissage statistique Guillaume Obozinski Ecole des Ponts - ParisTech Journée Parcimonie Fédération Charles Hermite, 23 Juin 2014 Parcimonie en apprentissage 1/44 Classical supervised
More informationSparse linear models
Sparse linear models Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda 2/22/2016 Introduction Linear transforms Frequency representation Short-time
More informationSparse Interactions: Identifying High-Dimensional Multilinear Systems via Compressed Sensing
Sparse Interactions: Identifying High-Dimensional Multilinear Systems via Compressed Sensing Bobak Nazer and Robert D. Nowak University of Wisconsin, Madison Allerton 10/01/10 Motivation: Virus-Host Interaction
More informationFast Hard Thresholding with Nesterov s Gradient Method
Fast Hard Thresholding with Nesterov s Gradient Method Volkan Cevher Idiap Research Institute Ecole Polytechnique Federale de ausanne volkan.cevher@epfl.ch Sina Jafarpour Department of Computer Science
More informationNoise Removal? The Evolution Of Pr(x) Denoising By Energy Minimization. ( x) An Introduction to Sparse Representation and the K-SVD Algorithm
Sparse Representation and the K-SV Algorithm he CS epartment he echnion Israel Institute of technology Haifa 3, Israel University of Erlangen - Nürnberg April 8 Noise Removal? Our story begins with image
More informationLarge-Scale Multiple Testing in Medical Informatics
Large-Scale Multiple Testing in Medical Informatics experiment space model space data IMA Workshop on Large Data Sets in Medical Informatics November 16 2011 Rob Nowak www.ece.wisc.edu/~nowak Joint work
More informationA Generalized Restricted Isometry Property
1 A Generalized Restricted Isometry Property Jarvis Haupt and Robert Nowak Department of Electrical and Computer Engineering, University of Wisconsin Madison University of Wisconsin Technical Report ECE-07-1
More informationarxiv: v1 [cs.it] 21 Feb 2013
q-ary Compressive Sensing arxiv:30.568v [cs.it] Feb 03 Youssef Mroueh,, Lorenzo Rosasco, CBCL, CSAIL, Massachusetts Institute of Technology LCSL, Istituto Italiano di Tecnologia and IIT@MIT lab, Istituto
More informationLinear Methods for Regression. Lijun Zhang
Linear Methods for Regression Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Introduction Linear Regression Models and Least Squares Subset Selection Shrinkage Methods Methods Using Derived
More informationSCRIBERS: SOROOSH SHAFIEEZADEH-ABADEH, MICHAËL DEFFERRARD
EE-731: ADVANCED TOPICS IN DATA SCIENCES LABORATORY FOR INFORMATION AND INFERENCE SYSTEMS SPRING 2016 INSTRUCTOR: VOLKAN CEVHER SCRIBERS: SOROOSH SHAFIEEZADEH-ABADEH, MICHAËL DEFFERRARD STRUCTURED SPARSITY
More informationPerformance Analysis for Sparse Support Recovery
Performance Analysis for Sparse Support Recovery Gongguo Tang and Arye Nehorai ESE, Washington University April 21st 2009 Gongguo Tang and Arye Nehorai (Institute) Performance Analysis for Sparse Support
More informationRandom projection trees and low dimensional manifolds. Sanjoy Dasgupta and Yoav Freund University of California, San Diego
Random projection trees and low dimensional manifolds Sanjoy Dasgupta and Yoav Freund University of California, San Diego I. The new nonparametrics The new nonparametrics The traditional bane of nonparametric
More informationStructured sparsity-inducing norms through submodular functions
Structured sparsity-inducing norms through submodular functions Francis Bach Sierra team, INRIA - Ecole Normale Supérieure - CNRS Thanks to R. Jenatton, J. Mairal, G. Obozinski June 2011 Outline Introduction:
More informationWavelets and Signal Processing
Wavelets and Signal Processing John E. Gilbert Mathematics in Science Lecture April 30, 2002. Publicity Mathematics In Science* A LECTURE SERIES FOR UNDERGRADUATES Wavelets Professor John Gilbert Mathematics
More informationWireless Compressive Sensing for Energy Harvesting Sensor Nodes over Fading Channels
Wireless Compressive Sensing for Energy Harvesting Sensor Nodes over Fading Channels Gang Yang, Vincent Y. F. Tan, Chin Keong Ho, See Ho Ting and Yong Liang Guan School of Electrical and Electronic Engineering,
More informationWavelet Footprints: Theory, Algorithms, and Applications
1306 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 5, MAY 2003 Wavelet Footprints: Theory, Algorithms, and Applications Pier Luigi Dragotti, Member, IEEE, and Martin Vetterli, Fellow, IEEE Abstract
More informationGauge optimization and duality
1 / 54 Gauge optimization and duality Junfeng Yang Department of Mathematics Nanjing University Joint with Shiqian Ma, CUHK September, 2015 2 / 54 Outline Introduction Duality Lagrange duality Fenchel
More informationRecovery of Sparse Signals from Noisy Measurements Using an l p -Regularized Least-Squares Algorithm
Recovery of Sparse Signals from Noisy Measurements Using an l p -Regularized Least-Squares Algorithm J. K. Pant, W.-S. Lu, and A. Antoniou University of Victoria August 25, 2011 Compressive Sensing 1 University
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 informationCSC 576: Variants of Sparse Learning
CSC 576: Variants of Sparse Learning Ji Liu Department of Computer Science, University of Rochester October 27, 205 Introduction Our previous note basically suggests using l norm to enforce sparsity in
More informationSingle-letter Characterization of Signal Estimation from Linear Measurements
Single-letter Characterization of Signal Estimation from Linear Measurements Dongning Guo Dror Baron Shlomo Shamai The work has been supported by the European Commission in the framework of the FP7 Network
More informationMachine Learning for Signal Processing Sparse and Overcomplete Representations. Bhiksha Raj (slides from Sourish Chaudhuri) Oct 22, 2013
Machine Learning for Signal Processing Sparse and Overcomplete Representations Bhiksha Raj (slides from Sourish Chaudhuri) Oct 22, 2013 1 Key Topics in this Lecture Basics Component-based representations
More informationMotivation Sparse Signal Recovery is an interesting area with many potential applications. Methods developed for solving sparse signal recovery proble
Bayesian Methods for Sparse Signal Recovery Bhaskar D Rao 1 University of California, San Diego 1 Thanks to David Wipf, Zhilin Zhang and Ritwik Giri Motivation Sparse Signal Recovery is an interesting
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 informationEnhanced Compressive Sensing and More
Enhanced Compressive Sensing and More Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, U.S.A. Nonlinear Approximation Techniques Using L1 Texas A & M University
More informationPHASE RETRIEVAL OF SPARSE SIGNALS FROM MAGNITUDE INFORMATION. A Thesis MELTEM APAYDIN
PHASE RETRIEVAL OF SPARSE SIGNALS FROM MAGNITUDE INFORMATION A Thesis by MELTEM APAYDIN Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the
More informationLinear Regression. Aarti Singh. Machine Learning / Sept 27, 2010
Linear Regression Aarti Singh Machine Learning 10-701/15-781 Sept 27, 2010 Discrete to Continuous Labels Classification Sports Science News Anemic cell Healthy cell Regression X = Document Y = Topic X
More informationMLCC 2018 Variable Selection and Sparsity. Lorenzo Rosasco UNIGE-MIT-IIT
MLCC 2018 Variable Selection and Sparsity Lorenzo Rosasco UNIGE-MIT-IIT Outline Variable Selection Subset Selection Greedy Methods: (Orthogonal) Matching Pursuit Convex Relaxation: LASSO & Elastic Net
More informationSparse Gaussian Markov Random Field Mixtures for Anomaly Detection
Sparse Gaussian Markov Random Field Mixtures for Anomaly Detection Tsuyoshi Idé ( Ide-san ), Ankush Khandelwal*, Jayant Kalagnanam IBM Research, T. J. Watson Research Center (*Currently with University
More informationarxiv: v1 [cs.it] 26 Oct 2018
Outlier Detection using Generative Models with Theoretical Performance Guarantees arxiv:1810.11335v1 [cs.it] 6 Oct 018 Jirong Yi Anh Duc Le Tianming Wang Xiaodong Wu Weiyu Xu October 9, 018 Abstract This
More informationDATA MINING AND MACHINE LEARNING
DATA MINING AND MACHINE LEARNING Lecture 5: Regularization and loss functions Lecturer: Simone Scardapane Academic Year 2016/2017 Table of contents Loss functions Loss functions for regression problems
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 informationCompressed Sensing: Extending CLEAN and NNLS
Compressed Sensing: Extending CLEAN and NNLS Ludwig Schwardt SKA South Africa (KAT Project) Calibration & Imaging Workshop Socorro, NM, USA 31 March 2009 Outline 1 Compressed Sensing (CS) Introduction
More informationSparse Solutions of an Undetermined Linear System
1 Sparse Solutions of an Undetermined Linear System Maddullah Almerdasy New York University Tandon School of Engineering arxiv:1702.07096v1 [math.oc] 23 Feb 2017 Abstract This work proposes a research
More information25 : Graphical induced structured input/output models
10-708: Probabilistic Graphical Models 10-708, Spring 2013 25 : Graphical induced structured input/output models Lecturer: Eric P. Xing Scribes: Meghana Kshirsagar (mkshirsa), Yiwen Chen (yiwenche) 1 Graph
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 informationHigh-dimensional Statistical Models
High-dimensional Statistical Models Pradeep Ravikumar UT Austin MLSS 2014 1 Curse of Dimensionality Statistical Learning: Given n observations from p(x; θ ), where θ R p, recover signal/parameter θ. For
More informationCompressive Sensing and Beyond
Compressive Sensing and Beyond Sohail Bahmani Gerorgia Tech. Signal Processing Compressed Sensing Signal Models Classics: bandlimited The Sampling Theorem Any signal with bandwidth B can be recovered
More informationExploiting Structure in Wavelet-Based Bayesian Compressive Sensing
Exploiting Structure in Wavelet-Based Bayesian Compressive Sensing 1 Lihan He and Lawrence Carin Department of Electrical and Computer Engineering Duke University, Durham, NC 2778-291 USA {lihan,lcarin}@ece.duke.edu
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 informationLearning-based compression
Nature Learning-based compression Value / Knowledge Volkan Cevher Laboratory for Information and Inference Systems http://lions.epfl.ch A paradigm shift in data generation 1977-5hours A paradigm shift
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 informationA Convex Approach for Designing Good Linear Embeddings. Chinmay Hegde
A Convex Approach for Designing Good Linear Embeddings Chinmay Hegde Redundancy in Images Image Acquisition Sensor Information content
More informationLecture Notes 9: Constrained Optimization
Optimization-based data analysis Fall 017 Lecture Notes 9: Constrained Optimization 1 Compressed sensing 1.1 Underdetermined linear inverse problems Linear inverse problems model measurements of the form
More informationOn the Projection Matrices Influence in the Classification of Compressed Sensed ECG Signals
On the Projection Matrices Influence in the Classification of Compressed Sensed ECG Signals Monica Fira, Liviu Goras Institute of Computer Science Romanian Academy Iasi, Romania Liviu Goras, Nicolae Cleju,
More informationLecture 24: Principal Component Analysis. Aykut Erdem May 2016 Hacettepe University
Lecture 4: Principal Component Analysis Aykut Erdem May 016 Hacettepe University This week Motivation PCA algorithms Applications PCA shortcomings Autoencoders Kernel PCA PCA Applications Data Visualization
More informationA tutorial on sparse modeling. Outline:
A tutorial on sparse modeling. Outline: 1. Why? 2. What? 3. How. 4. no really, why? Sparse modeling is a component in many state of the art signal processing and machine learning tasks. image processing
More informationOn the Relationship Between Compressive Sensing and Random Sensor Arrays
On the Relationship Between Compressive Sensing and Random Sensor Arrays Lawrence Carin Department of Electrical & Computer Engineering Duke University Durham, NC lcarin@ece.duke.edu Abstract Random sensor
More informationCompressive sensing in the analog world
Compressive sensing in the analog world Mark A. Davenport Georgia Institute of Technology School of Electrical and Computer Engineering Compressive Sensing A D -sparse Can we really acquire analog signals
More informationImproving Approximate Message Passing Recovery of Sparse Binary Vectors by Post Processing
10th International ITG Conference on Systems, Communications and Coding (SCC 2015) Improving Approximate Message Passing Recovery of Sparse Binary Vectors by Post Processing Martin Mayer and Norbert Goertz
More informationLEARNING DATA TRIAGE: LINEAR DECODING WORKS FOR COMPRESSIVE MRI. Yen-Huan Li and Volkan Cevher
LARNING DATA TRIAG: LINAR DCODING WORKS FOR COMPRSSIV MRI Yen-Huan Li and Volkan Cevher Laboratory for Information Inference Systems École Polytechnique Fédérale de Lausanne ABSTRACT The standard approach
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationAnalysis of audio intercepts: Can we identify and locate the speaker?
Motivation Analysis of audio intercepts: Can we identify and locate the speaker? K V Vijay Girish, PhD Student Research Advisor: Prof A G Ramakrishnan Research Collaborator: Dr T V Ananthapadmanabha Medical
More informationEstimation Error Guarantees for Poisson Denoising with Sparse and Structured Dictionary Models
Estimation Error Guarantees for Poisson Denoising with Sparse and Structured Dictionary Models Akshay Soni and Jarvis Haupt Department of Electrical and Computer Engineering University of Minnesota Twin
More informationLecture 3: Compressive Classification
Lecture 3: Compressive Classification Richard Baraniuk Rice University dsp.rice.edu/cs Compressive Sampling Signal Sparsity wideband signal samples large Gabor (TF) coefficients Fourier matrix Compressive
More informationCompressive Sensing Theory and L1-Related Optimization Algorithms
Compressive Sensing Theory and L1-Related Optimization Algorithms Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, USA CAAM Colloquium January 26, 2009 Outline:
More informationDetecting Weak but Hierarchically-Structured Patterns in Networks
Aarti Singh Robert Nowak Robert Calderbank Machine Learning Department Carnegie Mellon University aartisingh@cmu.edu Electrical and Computer Engineering University of Wisconsin - Madison nowak@engr.wisc.edu
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 informationHigh-dimensional Statistics
High-dimensional Statistics Pradeep Ravikumar UT Austin Outline 1. High Dimensional Data : Large p, small n 2. Sparsity 3. Group Sparsity 4. Low Rank 1 Curse of Dimensionality Statistical Learning: Given
More informationInvertible Nonlinear Dimensionality Reduction via Joint Dictionary Learning
Invertible Nonlinear Dimensionality Reduction via Joint Dictionary Learning Xian Wei, Martin Kleinsteuber, and Hao Shen Department of Electrical and Computer Engineering Technische Universität München,
More informationThe Dantzig Selector
The Dantzig Selector Emmanuel Candès California Institute of Technology Statistics Colloquium, Stanford University, November 2006 Collaborator: Terence Tao (UCLA) Statistical linear model y = Xβ + z y
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 informationReweighted Laplace Prior Based Hyperspectral Compressive Sensing for Unknown Sparsity: Supplementary Material
Reweighted Laplace Prior Based Hyperspectral Compressive Sensing for Unknown Sparsity: Supplementary Material Lei Zhang, Wei Wei, Yanning Zhang, Chunna Tian, Fei Li School of Computer Science, Northwestern
More informationThe Iteration-Tuned Dictionary for Sparse Representations
The Iteration-Tuned Dictionary for Sparse Representations Joaquin Zepeda #1, Christine Guillemot #2, Ewa Kijak 3 # INRIA Centre Rennes - Bretagne Atlantique Campus de Beaulieu, 35042 Rennes Cedex, FRANCE
More informationDeep Learning: Approximation of Functions by Composition
Deep Learning: Approximation of Functions by Composition Zuowei Shen Department of Mathematics National University of Singapore Outline 1 A brief introduction of approximation theory 2 Deep learning: approximation
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 information