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Sparse Signal Recovery: Theory, Applications
and Algorithms
Bhaskar RaoDepartment of Electrical and Computer Engineering
University of California, San Diego
Collaborators: I. Gorodonitsky, S. Cotter, D. Wipf, J. Palmer, K.Kreutz-DelgadoAcknowledgement: Travel support provided by Office of Naval Research Global under Grant Number: N62909-09-1-1024. The
research was supported by several grants from the National Science Foundation, most recently CCF - 0830612
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 1 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 2 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 3 / 42
Importance of Problem
Organized with Prof. Bresler a Special Session at the 1998 IEEEInternational Conference on Acoustics, Speech and Signal Processing
VOLUME I11
DSPl7: DSP EDUCATION DSP Instead of Circuits? - Transition to Undergraduate DSP Education at Rose-Hulman ........................................... 111-1845 W. Padgett, M. Yoder (Rose-Hulman Institute of Tecnology, USA)
A Java Signal Analysis Tool for Signal Processing Experiments .................................... ~.........,..,....,.,..........,..,...,,..........III-l~9 A. Clausen, A. Spanias, A. Xavier, M. Tampi (Arizona State University, USA) Visualization of Signal Processing Concepts ...................................................................................................................... 111-1853 J. Shaffer, J. Hamaker, J. Picone (Mississippi State University, USA) An Experimental Architecture for Interactive Web-based DSP Education .................................................................... 111-1 857 M. Rahkila, M. Karjalainen (Helsinki University of Technology, Finland)
SPEC-DSP: SIGNAL PROCESSING WITH SPARSENESS CONSTRAINT Signal Processing with the Sparseness Constraint ............................................................................................................. 111-1861 B. Rao (University of California, San Diego, USA) Application of Basis Pursuit in Spectrum Estimation ........................................................................................................ 111-1865 S. Chen (IBM, USA); D. Donoho (Stanford University, USA) Parsimony and Wavelet Method for Denoising ................................................................................................................... 111-1869 H. Krim (MIT, USA); J. Pesquet (University Paris Sud, France); I. Schick (GTE Ynternetworking and Harvard Univ., USA) Parsimonious Side Propagation ........................................................................................................................................... 111-1873 P. Bradley, 0. Mangasarian (University of Wisconsin-Madison, USA) Fast Optimal and Suboptimal Algorithms for Sparse Solutions to Linear Inverse Problems ....................................... 111-1877 G. Harikumar (Tellabs Research, USA); C. Couvreur, Y. Bresler (University of Illinois, Urbana-Champaign. USA) Measures and Algorithms for Best Basis Selection ............................................................................................................ 111-1881 K. Kreutz-Delgado, B. Rao (University of Califomia, San Diego, USA)
Sparse Inverse Solution Methods for Signal and Image Processing Applications ........................................................... 111-1885 B. Jeffs (Brigham Young University, USA) Image Denoising Using Multiple Compaction Domains .................................................................................................... 111-1889 P. Ishwar, K. Ratakonda, P. Moulin, N. Ahuja (University of Illinois, Urbana-Champaign, USA)
SPEC-EDUCATION: FUTURE DIRECTIONS IN SIGNAL PROCESSING EDUCATION A Different First Course in Electrical Engineering ........................................................................................................... 111-1893 D. Johnson, J. Wise, Jr. (Rice University, USA) Integrating Engineering Design, Signal Processing, and Community Service in the EPICS Program ......................... 111-1897 L. Jamieson, E. Coyle, M. Haver, E. Delp (Purdue University, USA) An Interactive Learning Environment for Adaptive Systems .......................................................................................... III-1901 J. Principe, N. Euliano (University of Florida, USA); C. Lefebvre (Neurodimension, USA) Interactive DSP Education Using Java ............................................................................................................................... 111-1905 Y. Cheneval, L Balmelli, P. Prandoni (Swiss Federal Institute of Technology, Switzerland); J. Kovacevic (Bell Labs, Lucent Technologies, USA); M. Vetterli (Swiss Federal Institute of Technology, Switzerland)
E. Patterson (Clemson University, USA); D. Wu (Sony Corporation, USA); J. Gowdy (Clemson University, USA) Multi-Platform CBI Tools Using Linux and Java-Based Solutions for Distance Learning ............................................ 111-1909
xxxv
Authorized licensed use limited to: Univ of Calif San Diego. Downloaded on June 13, 2009 at 11:59 from IEEE Xplore. Restrictions apply.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 4 / 42
Talk Goals
Sparse Signal Recovery is an interesting area with manypotential applications
Tools developed for solving the Sparse Signal Recovery problemare useful for signal processing practitioners to know
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 5 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 6 / 42
Problem Description
t is N × 1 measurement vector
Φ is N ×M Dictionary matrix. M >> N .
w is M×1 desired vector which is sparse with K non-zero entries
ε is the additive noise modeled as additive white Gaussian
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 7 / 42
Problem Statement
Noise Free Case: Given a target signal t and a dictionary Φ, findthe weights w that solve:
minw
M∑i=1
I (wi 6= 0) such that t = Φw
where I (.) is the indicator function
Noisy Case: Given a target signal t and a dictionary Φ, find theweights w that solve:
minw
M∑i=1
I (wi 6= 0) such that ‖t − Φw‖22 ≤ β
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 8 / 42
Complexity
Search over all possible subsets, which would mean a search overa total of (MCK ) subsets. Problem NP hard. WithM = 30,N = 20, and K = 10 there are 3× 107 subsets (VeryComplex)
A branch and bound algorithm can be used to find the optimalsolution. The space of subsets searched is pruned but the searchmay still be very complex.
Indicator function not continuous and so not amenable tostandard optimization tools.
Challenge: Find low complexity methods with acceptableperformance
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 9 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 10 / 42
Applications
Signal Representation (Mallat, Coifman, Wickerhauser, Donoho,...)
EEG/MEG (Leahy, Gorodnitsky, Ioannides, ...)
Functional Approximation and Neural Networks (Chen,Natarajan, Cun, Hassibi, ...)
Bandlimited extrapolations and spectral estimation (Papoulis,Lee, Cabrera, Parks, ...)
Speech Coding (Ozawa, Ono, Kroon, Atal, ...)
Sparse channel equalization (Fevrier, Greenstein, Proakis, )
Compressive Sampling (Donoho, Candes, Tao..)
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 11 / 42
DFT Example
N chosen to be 64 in this example.
Measurement t:
t[n] = cosω0n, n = 0, 1, 2, ..,N − 1
ω0 =2π
64
33
2
Dictionary Elements:
φk = [1, e−jωk , e−j2ωk , .., e−j(N−1)ωk ]T , ωk =2π
M
Consider M = 64, 128, 256 and 512NOTE: The frequency components are included in the dictionary Φfor M = 128, 256, and 512.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 12 / 42
FFT Results with Different MFFT Results
M=64 M=128
M=256 M=512
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 13 / 42
Magnetoencephalography (MEG)
Given measured magnetic fields outside of the head, the goal is tolocate the responsible current sources inside the head.
MEG Example♦ Given measured magnetic fields outside of the head, we would
like to locate the responsible current sources inside the head.
= MEG measurement point= Candidate source location
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 14 / 42
MEG Example
At any given time, typically only a few sources are active (SPARSE).MEG Example♦ At any given time, typically only a few sources are active.
= Candidate source location= MEG measurement point
= Active source location
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 15 / 42
MEG Formulation
Forming the overcomplete dictionary Φ
The number of rows equals the number of sensors.The number of columns equals the number of possible sourcelocations.Φij = the magnetic field measured at sensor i produced by aunit current at location j .We can compute Φ using a boundary element brain model andMaxwells equations.
Many different combinations of current sources can produce thesame observed magnetic field t.
By finding the sparsest signal representation/basis, we find thesmallest number of sources capable of producing the observedfield.
Such a representation is of neurophysiological significance
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 16 / 42
Compressive Sampling
Transform Coding Compressed SensingΨ w b
Ψ w bA A t
Φ
Compressive Sensing
Compressed SensingΨ w b
Ψ w bA A t
Φ
Computation : t → w → b
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 17 / 42
Compressive Sampling
Transform Coding Compressed SensingΨ w b
Ψ w bA A t
Φ
Compressive Sensing
Compressed SensingΨ w b
Ψ w bA A t
Φ
Computation : t → w → b
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 17 / 42
Compressive Sampling
Transform Coding Compressed SensingΨ w b
Ψ w bA A t
Φ
Compressive Sensing
Compressed SensingΨ w b
Ψ w bA A t
Φ
Computation : t → w → b
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 17 / 42
Compressive Sampling
Transform Coding Compressed SensingΨ w b
Ψ w bA A t
Φ
Compressive Sensing
Compressed SensingΨ w b
Ψ w bA A t
Φ
Computation : t → w → bBhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 17 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 18 / 42
Potential Approaches
Problem NP hard and so need alternate strategies
Greedy Search Techniques: Matching Pursuit, OrthogonalMatching Pursuit
Minimizing Diversity Measures: Indicator function notcontinuous. Define Surrogate Cost functions that are moretractable and whose minimization leads to sparse solutions, e.g.`1 minimization
Bayesian Methods: Make appropriate Statistical assumptions onthe solution and apply estimation techniques to identify thedesired sparse solution
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 19 / 42
Greedy Search Methods: Matching Pursuit
Select a column that is most aligned with the current residual
0.7 4 ….‐0.3 ….
t Φ w ε
Remove its contribution
Stop when residual becomes small enough or if we haveexceeded some sparsity threshold.
Some Variations
Matching Pursuit [Mallat & Zhang]Orthogonal Matching Pursuit [Pati et al.]Order Recursive Matching Pursuit (ORMP)
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 20 / 42
Inverse techniques
For the systems of equations Φw = t, the solution set ischaracterized by {ws : ws = Φ+t + v , v ∈ N (Φ)}, where N (Φ)denotes the null space of Φ and Φ+ = ΦT (ΦΦT )−1.
Minimum Norm solution : The minimum `2 norm solutionwmn = Φ+t is a popular solution
Noisy Case: regularized `2 norm solution often employed and is givenby
wreg = ΦT (ΦΦT + λI )−1t
Problem: Solution is not Sparse
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 21 / 42
Diversity Measures
Functionals whose minimization leads to sparse solutions
Many examples are found in the fields of economics, socialscience and information theory
These functionals are concave which leads to difficultoptimization problems
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 22 / 42
Examples of Diversity Measures
`(p≤1) Diversity Measure
E (p)(w) =M∑l=1
|wl |p, p ≤ 1
Gaussian Entropy
HG (w) =M∑l=1
ln |wl |2
Shannon Entropy
HS(w) = −M∑l=1
w̃l ln w̃l . where w̃l =w 2
l
‖w‖2
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 23 / 42
Diversity Minimization
Noiseless Case
minw
E (p)(w) =M∑l=1
|wl |p subject to Φw = t
Noisy Case
minw
(‖t − Φw‖2 + λ
M∑l=1
|wl |p)
p = 1 is a very popular choice because of the convex nature of theoptimization problem (Basis Pursuit and Lasso).
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 24 / 42
Bayesian Methods
Maximum Aposteriori Approach (MAP)
Assume a sparsity inducing prior on the latent variable wDevelop an appropriate MAP estimation algorithm
Empirical Bayes
Assume a parameterized prior on the latent variable w(hyperparameters)Marginalize over the latent variable w and estimate thehyperparametersDetermine the posterior distribution of w and obtain a point asthe mean, mode or median of this density
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 25 / 42
Generalized Gaussian Distribution
Density function: Subgaussian: p > 2 and Supergaussian : p < 2
f (x) =p
2σΓ( 1p
)exp
{−(|x |σ
)p}Generalized Gaussian Distribution
-5 0 50
0.5
1
1.5
2
2.5
3
p=0.5p=1p=2p=10
( )( ) exp
2 1
pxpp x
pμ
σ σ
⎧ ⎫−⎛ ⎞⎪ ⎪= −⎨ ⎬⎜ ⎟Γ ⎝ ⎠⎪ ⎪⎩ ⎭-p < 2: Super Gaussian
-p > 2: Sub Gaussian
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 26 / 42
Student t Distribution
Density function:
f (x) =Γ(ν+1
2)
√πνΓ(ν
2)
(1 +
x2
ν
)( ν+12 )
Student-t Distribution
12 2
12( ) 1
2
xp x
νν
ν ννπ
+⎛ ⎞−⎜ ⎟⎝ ⎠
+⎛ ⎞Γ⎜ ⎟ ⎛ ⎞⎝ ⎠= +⎜ ⎟⎛ ⎞ ⎝ ⎠Γ⎜ ⎟⎝ ⎠
-5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
ν = 1ν = 2ν = 5Gaussian
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 27 / 42
MAP using a Supergaussian prior
Assuming a Gaussian likelihood model for f (t|w), we can find MAPweight estimates
wMAP = arg maxw
log f (w |t)
= arg maxw
(log f (t|w) + log f (w))
= arg minw
(‖Φw − t‖2 + λ
M∑l=1
|wl |p)
This is essentially a regularized LS framework. Interesting range for pis p ≤ 1.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 28 / 42
MAP Estimate: FOCal Underdetermined System
Solver (FOCUSS)
Approach involves solving a sequence of Regularized WeightedMinimum Norm problems
qk+1 = arg minq
(‖Φk+1q − t‖2 + λ‖q‖2
)where Φk+1 = ΦMk+1, and Mk+1 = diag(|wk.l |1−
p2 .
wk+1 = Mk+1qk+1.
p = 0 is the `0 minimization and p = 1 is `1 minimization
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 29 / 42
FOCUSS summary
For p < 1, the solution is initial condition dependent
Prior knowledge can be incorporatedMinimum norm is a suitable choiceCan retry with several initial conditions
Computationally more complex than Matching Pursuitalgorithms
Sparsity versus tolerance tradeoff more involved
Factor p allows a trade-off between the speed of convergenceand the sparsity obtained
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 30 / 42
Convergence Errors vs. Structural ErrorsConvergence Errors vs. Structural Errors
w0w*
-log
p(w
|t)
w0w*-lo
g p(
w|t)
Convergence Error Structural Error
w* = solution we have converged tow0 = maximally sparse solution
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 31 / 42
Shortcomings of these Methods
p = 1
Basis Pursuit/Lasso often suffer from structural errors.
Therefore, regardless of initialization, we may never find the bestsolution.
p < 1
The FOCUSS class of algorithms suffers from numeroussuboptimal local minima and therefore convergence errors.
In the low noise limit, the number of local minima K satisfies
K ∈[(
M − 1N
)+ 1,
(MN
)]At most local minima, the number of nonzero coefficients isequal to N , the number of rows in the dictionary.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 32 / 42
Empirical Bayesian Method
Main Steps
Parameterized prior f (w |γ)
Marginalize
f (t|γ) =
∫f (t,w |γ)dw =
∫f (t|w)f (w |γ)dw
Estimate the hyperparameter γ̂
Determine the posterior density of the latent variable f (w |t, γ̂)
Obtain point estimate of w
Example: Sparse Bayesian Learning (SBL by Tipping)
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 33 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 34 / 42
DFT ExampleDFT Results Using FOCUSS
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 35 / 42
Empirical Tests
Random overcomplete bases Φ and sparse weight vectors w0
were generated and used to create target signals t, i.e.,t = Φw0 + ε
SBL (Empirical Bayes) was compared with Basis Pursuit andFOCUSS (with various p values) in the task of recovering w0.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 36 / 42
Experiment 1: Comparison with Noiseless Data
Randomized Φ (20 rows by 40 columns).
Diversity of the true w0 is 7.
Results are from 1000 independent trials.
NOTE: An error occurs whenever an algorithm converges to asolution w not equal to w0.
Experiment I: Noiseless Comparison
♦ Randomized Φ (20 rows by 40 columns).♦ Diversity of the ‘true’ w0 is 7.♦ Results are from 1000 independent trials.
♦ NOTE: An error occurs whenever an algorithm converges to a solution w not equal to w0.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 37 / 42
Experiment II: Comparison with Noisy Data
Randomized Φ (20 rows by 40 columns).
Diversity of the true w0 is 7.
20 db AWGN
Results are from 1000 independent trials.
NOTE: We no longer distinguish convergence errors from structuralerrors.
Experiment II: Noisy Comparisons
♦ Randomized Φ (20 rows by 40 columns).♦ 20 dB AWGN.♦ Diversity of the ‘true’ w is 7.♦ Results are from 1000 independent trials.
♦ NOTE: We no longer distinguish convergence errors from structural errors.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 38 / 42
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 39 / 42
MEG Example
Data based on CTF MEG system at UCSF with 275 scalpsensors.
Forward model based on 40, 000 points (vertices of a triangulargrid) and 3 different scale factors.
Dimensions of lead field matrix (dictionary): 275 by 120, 000.
Overcompleteness ratio approximately 436.
Up to 40 unknown dipole components were randomly placedthroughout the sample space.
SBL was able to resolve roughly twice as many dipoles as thenext best method (Ramirez 2005).
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 40 / 42
Outline
1 Talk Objective
2 Sparse Signal Recovery Problem
3 Applications
4 Computational Algorithms
5 Performance Evaluation
6 Conclusion
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 41 / 42
Summary
Discussed role of sparseness in linear inverse problems
Discussed Applications of Sparsity
Discussed methods for computing sparse solutions
Matching Pursuit AlgorithmsMAP methods (FOCUSS Algorithm and `1 minimization)Empirical Bayes (Sparse Bayesian Learning (SBL))
Expectation is that there will be continued growth in the applicationdomain as well as in the algorithm development.
Bhaskar RaoDepartment of Electrical and Computer EngineeringUniversity of California, San Diego ()Sparse Signal Recovery: Theory, Applications and Algorithms 42 / 42
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