Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Esa Ollila

S³ Seminar
February 03, 2015

Esa Ollila

(Aalto University, Finland)

https://s3-seminar.github.io/seminars/esa-ollila

Title — Robust approaches to multichannel sparse recovery

Abstract — We consider multichannel sparse recovery problem where the objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vectors (atoms). The model is thus an extension of single measurement vector setting used in compressed sensing (CS). Many popular greedy or convex algorithms proposed for multichannel sparse recovery problem perform poorly under non-Gaussian heavy-tailed noise conditions or in the face of outliers (gross errors), i.e., are not robust. In this talk, we consider different types of mixed robust norms on data fidelity (residual matrix) term and conventional L0-norm constraint on the signal matrix to promote row-sparsity. We devise algorithms based normalized iterative hard thresholding (blumesath and davies, 2010) which is a simple, computationally efficient and scalable approach for solving the simultaneous sparse approximation problem. Performance assessment conducted on simulated data highlights the effectiveness of the proposed approaches to cope with different noise environments (i.i.d., row i.i.d, etc) and outliers. Usefulness of the methods are illustrated in image denoising problem and source localization application with sensor arrays. Finally (if time permits) a (non-robust) Bayesian perspective to multichannel recovery problem is discussed as well.

Biography — Esa Ollila received the M.Sc. degree in mathematics from the University of Oulu, in 1998, Ph.D. degree in statistics with honors from the University of yvaskyla, in 2002, and the D.Sc.(Tech) degree with honors in signal processing from Aalto University, in 2010. From 2004 to 2007 he was a post-doctoral fellow of the Academy of Finland. He has also been a Senior Researcher and a Senior Lecturer at Aalto University and University of Oulu, respectively. Currently, from August 2010, he is appointed as an Academy Research Fellow of the Academy of Finland at the Department of Signal Processing and Acoustics, Aalto University, Finland. He is also adjunct Professor (statistics) of University of Oulu. During the Fall-term 2001 he was a Visiting Researcher with the Department of Statistics, Pennsylvania State University, State College, PA while the academic year 2010-2011 he spent as a Visiting Research Associate with the Department of Electrical Engineering, Princeton University, Princeton, NJ. His research interests focus on theory and methods of statistical signal processing, blind source separation, complex-valued signal processing, array and radar signal processing and robust and non-parametric statistical methods.

S³ Seminar

February 03, 2015
Tweet

More Decks by S³ Seminar

Other Decks in Research

Transcript

  1. Robust multichannel sparse
    recovery
    Esa Ollila
    Department of Signal Processing and Acoustics
    Aalto University, Finland
    SUPELEC, Feb 4th, 2015

    View full-size slide

  2. 1 Introduction
    2 Nonparametric sparse recovery
    3 Simulation studies
    4 Source localization with Sensor Arrays

    View full-size slide

  3. Single Measurement Vector
    Model (SMV)
    y
    M × 1
    = Φ
    M × N
    x
    N × 1
    + e
    N × 1
    Φ = φ1
    · · · φN
    = φ(1)
    · · · φ(M)
    H,
    M × N known measurement (design, system) matrix
    x = (x1, . . . , xN
    )⊤, unobserved signal vector
    e = (e1, . . . , eM
    )⊤, random noise vector.
    Objective
    recover K-sparse or compressible signal from y knowing Φ, K
    NOTE: typically, M < N (ill-posed model) and K ≪ N.
    2/24

    View full-size slide

  4. Single Measurement Vector
    Model (SMV)
    y
    M × 1
    = Φ
    M × N
    x
    N × 1
    + e
    M × 1
    If we can determine Γ = supp(x), then the problem reduces to
    conventional regression problem (more ’responses’ than ’predictors’):
    y Φ x e
    3/24

    View full-size slide

  5. Multiple Measurement Vectors
    Model (MMV)
    yi
    = Φxi
    + ei , i = 1, . . . , Q
    In matrix form
    Y
    M × Q
    = Φ
    M × N
    X
    N × Q
    + E
    M × 1
    Y = y1
    · · · yQ
    , observed measurement matrix
    X = x1
    · · · xQ
    , unobserved signal matrix
    E = e1
    · · · eQ
    , noise matrix
    Objective
    recover K-sparse (or compressible) signal matrix X knowing Y, Φ, K.
    N × Q matrix X is called K-sparse if X 0
    = |supp(X)| ≤ K, where
    Γ = supp(X) =
    Q
    i=1
    supp(xi
    ) = {j ∈ {1, . . . , N} : xjk
    = 0 for some k}
    4/24

    View full-size slide

  6. Multiple Measurement Vectors
    Model (MMV)
    Key idea:
    signals are all predominantly supported on a common support set
    =⇒ Joint estimation can lead both to computational advantages
    and increased reconstruction accuracy
    See [Tropp, 2006, Chen and Huo, 2006, Eldar and Rauhut, 2010,
    Duarte and Eldar, 2011, Blanchard et al., 2014].
    Applications
    EEG/MEG [Gorodnitsky et al., 1995]
    equalization of sparse communications channels [Cotter and Rao, 2002]
    blind source separation [Gribonval and Zibulevsky, 2010]
    source localization using sensor arrays [Malioutov et al., 2005]
    . . .
    5/24

    View full-size slide

  7. Matrix norms
    The mixed ℓp,q
    norm of X for p, q ∈ [1, ∞)
    X p,q
    =
    i j
    |xij
    |p
    q/p 1/q
    =
    i
    x(i)
    q
    p
    1/q
    .
    If p = q =⇒ matrix p-norm X p,p
    = X p
    .
    ℓ2
    (Frobenius) norm · 2
    is denoted shortly as · .
    The mixed ℓ

    -norms of X ∈ CN×Q
    X p,∞
    = maxi∈{1,...,N} x(i) p
    X
    ∞,q
    =
    i
    (maxj∈{1,...,Q}
    |xij
    |)q 1/q.
    The row ℓ0
    quasi-norm of X
    X 0
    = | supp(X)| = nr. of number of nonzero rows of X
    6/24

    View full-size slide

  8. Optimization problem
    min
    X
    Y − ΦX 2 subject to X 0
    ≤ K
    is combinatorial (NP-hard). More feasible approaches:
    Optimization (Convex relaxation)
    Greedy pursuit:
    Orthogonal Matching Pursuit (OMP)
    Normalized Iterative Hard Thresholding (NIHT)
    Normalized Hard Thresholding Pursuit (NHTP)
    Compressive Sampling MP (CoSaMP)
    Guaranteed to perform very well under suitable conditions (restricted
    isometry property on Φ, non impulsive noise e)
    NIHT is simple and scalable, convenient for problems where the
    support is of primary interest
    7/24

    View full-size slide

  9. Optimization problem
    min
    X
    Y − ΦX 2 subject to X 0
    ≤ K
    is combinatorial (NP-hard). More feasible approaches:
    Optimization (Convex relaxation)
    Greedy pursuit:
    Orthogonal Matching Pursuit (OMP)
    Normalized Iterative Hard Thresholding (NIHT)
    Normalized Hard Thresholding Pursuit (NHTP)
    Compressive Sampling MP (CoSaMP)
    Guaranteed to perform very well under suitable conditions (restricted
    isometry property on Φ, non impulsive noise e)
    NIHT is simple and scalable, convenient for problems where the
    support is of primary interest
    7/24

    View full-size slide

  10. Convex relaxation
    IDEA: replace X 0
    by ℓp,1
    norm
    [Tropp, 2006] uses p = ∞:
    min
    X
    Y − ΦX 2 + λ X
    ∞,1
    where X
    ∞,1
    =
    i
    |x(i) ∞
    .
    [Malioutov et al., 2005] uses p = 2:
    min
    X
    Y − ΦX 2 + λ X 2,1
    where X 2,1
    =
    i
    |x(i)
    .
    good recovery depends on the choise of egularization parameter λ > 0
    not robust due to the used data fidelity term (LS-loss)
    not as scalable as greedy methods
    8/24

    View full-size slide

  11. 1 Introduction
    2 Nonparametric sparse recovery
    3 Simulation studies
    4 Source localization with Sensor Arrays

    View full-size slide

  12. Nonparametric approach based
    on mixed norms
    min
    X
    Y − ΦX q
    p,q
    data fidelity
    subject to X 0
    ≤ K
    sparsity constraint
    (Pp,q
    )
    =⇒ mixed ℓ1
    norms provide robustness, i.e., give larger weights on
    small residuals and less weight on large residuals.
    We consider the cases
    (p, q) = (1, 1): Y − ΦX 1
    =
    i j
    |yij
    − φH
    (i)
    xj
    |
    robust norm for errors i.i.d. in space and time
    (p, q) = (2, 1): Y − ΦX 2,1
    =
    i
    y(i)
    − XHφ(i)
    robust norm for errors i.i.d. in ”space” only
    and devise greedy simultaneous NIHT (SNIHT) algorithm
    9/24

    View full-size slide

  13. Normalized iterative hard
    thresholding (NIHT)
    Conventional ℓ2,2
    approach :
    Xn+1 = HK

    hard thresholding
    Xn +
    stepsize
    µn+1 ΦH(Y − ΦXn)

    neg. gradient ∇X∗
    · 2
    ψ2,2
    (X) = X =⇒ Conventional ℓ2,2
    approach
    ψ1,1
    (X) = [sign(xij
    )] (elementwise operation of S(·))
    ψ2,1
    (X) = [sign(x(i)
    )] (rowwise operation of S(·))
    10/24

    View full-size slide

  14. Normalized iterative hard
    thresholding (NIHT)
    In our mixed norm ℓp,q
    approach:
    Xn+1 = HK

    hard thresholding
    Xn +
    stepsize
    µn+1 ΦHψp,q
    (Y − ΦXn)

    neg. gradient ∇X∗
    · q
    p,q
    ψ2,2
    (X) = X =⇒ Conventional ℓ2,2
    approach
    ψ1,1
    (X) = [sign(xij
    )] (elementwise operation of S(·))
    ψ2,1
    (X) = [sign(x(i)
    )] (rowwise operation of S(·))
    10/24

    View full-size slide

  15. Normalized iterative hard
    thresholding (NIHT)
    In our mixed norm ℓp,q
    approach:
    Xn+1 = HK

    hard thresholding
    Xn +
    stepsize
    µn+1 ΦHψp,q
    (Y − ΦXn)

    neg. gradient ∇X∗
    · q
    p,q
    ψ2,2
    (X) = X =⇒ Conventional ℓ2,2
    approach
    ψ1,1
    (X) = [sign(xij
    )] (elementwise operation of S(·))
    ψ2,1
    (X) = [sign(x(i)
    )] (rowwise operation of S(·))
    Spatial sign
    sign(x) =
    x/ x , for x = 0
    0, for x = 0
    10/24

    View full-size slide

  16. Normalized iterative hard
    thresholding (NIHT)
    In our mixed norm ℓp,q
    approach:
    Xn+1 = HK

    hard thresholding
    Xn +
    stepsize
    µn+1 ΦHψp,q
    (Y − ΦXn)

    neg. gradient ∇X∗
    · q
    p,q
    ψ2,2
    (X) = X =⇒ Conventional ℓ2,2
    approach
    ψ1,1
    (X) = [sign(xij
    )] (elementwise operation of S(·))
    ψ2,1
    (X) = [sign(x(i)
    )] (rowwise operation of S(·))
    Spatial sign
    sign(x) =
    x/ x , for x = 0
    0, for x = 0
    10/24

    View full-size slide

  17. SNIHT(p, q) algorithm
    input : Y, Φ, sparsity K, mixed norm indices (p, q)
    output : (Xn+1, Γn+1) as estimates of X and Γ = supp(X)
    initialize: X0 = 0, µ0 = 0, n = 0, Γ0 = ∅
    1 Γ0 = supp HK
    (ΦHψp,q
    (Y))
    while halting criterion false do
    2 Rn
    ψ
    = ψp,q
    (Y − ΦXn)
    3 Gn = ΦHRn
    ψ
    4 µn+1 = CompStepsize(Φ, Gn, Γn, µn, p, q)
    5 Xn+1 = HK
    (Xn + µn+1Gn)
    6 Γn+1 = supp(Xn+1)
    7 n = n + 1
    end
    11/24

    View full-size slide

  18. Computation of the step size µn+1
    For convergence, it is important that the stepsize is adaptively
    controlled at each iteration.
    Given the found support Γn is correct, one can choose µn+1 as the
    min. of the objective fnc for the gradient ascent direction
    Xn + µGn|Γn
    :
    L(µ) = Y − Φ Xn + µGn|Γn
    q
    p,q
    = Rn − µB q
    p,q
    where Rn = Y − ΦXn and B = ΦG(Γn)
    For p = q: simple linear regression problem based on ℓp
    -norm, with
    response r = vec(Rn) and predictor b = vec(B).
    ⇒ µn+1 = Gn
    (Γn)
    2/ ΦGn
    (Γn)
    2 for p = q = 2
    For p = q = 1 and (p, q) = (2, 1), minimizer of L(µ) can not be
    found in closed-form.
    12/24

    View full-size slide

  19. Computation of the step size µn+1
    When p = q = 1, the solution µ verifies a fixed point (FP) equation
    µ = H(µ)
    H(µ) =
    i,j

    rij
    |−1|bij
    |2
    −1
    i,j

    rij
    |−1Re(b∗
    ij
    rij
    ),
    and ˜
    R = Rn − µB (depends on unknown µ) and B = ΦG(Γn)
    .
    We select µn+1 as the 1-step FP iterate
    µn+1 = H(µn),
    where µn is the value of the stepsize at previous nth iteration.
    Often this approximation was within ∼ 1e−3 to minimizer of L(µ).
    Same approach is used in the case (p, q) = (2, 1).
    13/24

    View full-size slide

  20. 1 Introduction
    2 Nonparametric sparse recovery
    3 Simulation studies
    4 Source localization with Sensor Arrays

    View full-size slide

  21. Simulation study
    Y
    M × Q
    = Φ
    M × N
    X
    N × Q
    + E
    M × 1
    Set-up
    Measurement matrix Φ: φij
    ∼ CN(0, 1) with unit-norm columns
    Γ = supp(X) randomly drawn K elements from {1, . . . , N}
    signal matrix X: |xij
    | = 10 and Arg(xij
    ) ∼ Unif (0, 2π) ∀i ∈ Γ.
    For ℓ = 1, . . . , L MC-trials (L = 1000), compute X[ℓ] and Γ[ℓ]
    Performance measures
    mean squared error MSE( ^
    X) = 1
    LQ
    L
    ℓ=1
    ^
    X[ℓ] − X[ℓ] 2.
    probability of exact recovery PER =
    1
    L
    L
    ℓ=1
    I ^
    Γ[ℓ] = Γ[ℓ] .
    14/24

    View full-size slide

  22. IIID Gaussian noise, eij
    ∼ CN (0, σ2)
    MSE vs SNR for Q=16
    0 4 8 12 16 20
    10
    15
    20
    25
    30
    35
    SNR (dB)
    MSE (dB)
    SNIHT(2, 2)
    SNIHT(2, 1)
    SNIHT(1, 1)
    HUB-SNIHT
    PER vs Q at SNR = 5 dB
    2 4 6 8 10 12 14 16 18
    0
    0.2
    0.4
    0.6
    0.8
    1
    Q (number of vectors)
    PER
    SNIHT(2, 2)
    SNIHT(2, 1)
    SNIHT(1, 1)
    HUB-SNIHT
    SNIHT(2, 2)
    but SNIHT(2, 1) and HUB-SNIHT (threshold
    c = 1.517) loose only 0.09 dB in MSE (Q = 16)
    15/24

    View full-size slide

  23. IID t-distributed noise, eij
    ∼ Ctν
    (0, σ2),
    σ2 = MedF
    (|ei
    |2)
    SNR(σ) = 30 dB
    1 1.5 2 3 4 5
    0
    10
    20
    30
    40
    50
    60
    ν (degrees of freedom)
    MSE (dB)
    SNIHT(2, 2)
    SNIHT(2, 1)
    SNIHT(1, 1)
    HUB-SNIHT
    SNR(σ) = 10 dB, ν = 3
    2 4 6 8 10 12 14 16 18
    0
    0.2
    0.4
    0.6
    0.8
    1
    Q (number of vectors)
    PER
    SNIHT(2, 2)
    SNIHT(2, 1)
    SNIHT(1, 1)
    HUB-SNIHT
    SNIHT(1, 1)
    16/24

    View full-size slide

  24. Row IIID compound Gaussian
    inverse Gaussian texture, e(i)
    ∼ CIGλ
    (0, σ2IQ
    )
    0 4 8 12 16 20
    0
    5
    10
    15
    20
    25
    30
    SNR (dB)
    MSE (dB)
    SNIHT(2, 2)
    SNIHT(2, 1)
    SNIHT(1, 1)
    HUB-SNIHT
    SNIHT(2, 1)
    17/24

    View full-size slide

  25. 1 Introduction
    2 Nonparametric sparse recovery
    3 Simulation studies
    4 Source localization with Sensor Arrays

    View full-size slide

  26. Sensor array and narrowband,
    farfield source model
    M sensors, K sources (M > K):
    Model for sensor measurements at time instant t
    y(t) = A(θ)x(t) + e(t),
    - steering matrix A(θ) = a(θ1
    ) . . . a(θK
    )
    - unknown (distinct) Direction of Arrivals (DOA’s) θ1, . . . , θK
    - known array manifold a(θ) (e.g., ULA)
    - unknown signal waveforms x(t) = (x1
    (t), . . . , xK
    (t))⊤
    Objective
    Estimate the DOA’s of the sources given the snapshots y(t1
    ), . . . , y(tQ
    )
    and the number of sources K impinging on the sensor array.
    18/24

    View full-size slide

  27. Multichannel sparse recovery
    approach to source localization
    Construct an M × N overcomplete steering matrix A(˜
    θ) for a grid of
    all source locations ˜
    θ1, . . . , ˜
    θN
    of interest.
    Suppose that ˜
    θ contains the true DOA’s to some accuracy.
    The array model in matrix form then rewrites as
    Y = y(t1
    ) · · · y(tQ
    ) = A(˜
    θ)X + E,
    where X ∈ CN×Q is K-sparse, with K non-zero rows containing the
    source waveforms at time instants t1, . . . , tQ
    .
    =⇒ K-sparse MMV model
    In the multichannel sparce recovery formulation A(˜
    θ) is completely
    known, and we can use SNIHT methods to identify the support.
    19/24

    View full-size slide

  28. Multichannel sparse recovery
    approach to source localization
    Construct an M × N overcomplete steering matrix A(˜
    θ) for a grid of
    all source locations ˜
    θ1, . . . , ˜
    θN
    of interest.
    Suppose that ˜
    θ contains the true DOA’s to some accuracy.
    The array model in matrix form then rewrites as
    Y = y(t1
    ) · · · y(tQ
    ) = A(˜
    θ)X + E,
    where X ∈ CN×Q is K-sparse, with K non-zero rows containing the
    source waveforms at time instants t1, . . . , tQ
    .
    =⇒ K-sparse MMV model
    In the multichannel sparce recovery formulation A(˜
    θ) is completely
    known, and we can use SNIHT methods to identify the support.
    19/24

    View full-size slide

  29. Multichannel sparse recovery
    approach to source localization
    Construct an M × N overcomplete steering matrix A(˜
    θ) for a grid of
    all source locations ˜
    θ1, . . . , ˜
    θN
    of interest.
    Suppose that ˜
    θ contains the true DOA’s to some accuracy.
    The array model in matrix form then rewrites as
    Y = y(t1
    ) · · · y(tQ
    ) = A(˜
    θ)X + E,
    where X ∈ CN×Q is K-sparse, with K non-zero rows containing the
    source waveforms at time instants t1, . . . , tQ
    .
    =⇒ Finding DOA’s is equivalent to identifying Γ = supp(X)
    In the multichannel sparce recovery formulation A(˜
    θ) is completely
    known, and we can use SNIHT methods to identify the support.
    19/24

    View full-size slide

  30. Multichannel sparse recovery
    approach to source localization
    Construct an M × N overcomplete steering matrix A(˜
    θ) for a grid of
    all source locations ˜
    θ1, . . . , ˜
    θN
    of interest.
    Suppose that ˜
    θ contains the true DOA’s to some accuracy.
    The array model in matrix form then rewrites as
    Y = y(t1
    ) · · · y(tQ
    ) = A(˜
    θ)X + E,
    where X ∈ CN×Q is K-sparse, with K non-zero rows containing the
    source waveforms at time instants t1, . . . , tQ
    .
    =⇒ Finding DOA’s is equivalent to identifying Γ = supp(X)
    In the multichannel sparce recovery formulation A(˜
    θ) is completely
    known, and we can use SNIHT methods to identify the support.
    19/24

    View full-size slide

  31. Simul set-up
    ULA of M = 20 sensors with half a wavelength interelement spacing.
    K = 2 independent Gaussian sources from θ1
    = 0o and θ2
    = 8o.
    Temporally/spatially white Gaussian error terms.
    Low SNR = 0 dB.
    Uniform grid ˜
    θ on [−90, 90] with 2o degree spacing
    We compute PER rates and relative frequencies of found DOA
    estimates in the grid for 1000 Monte Carlo runs.
    20/24

    View full-size slide

  32. Number of snapshots Q = 4
    DOA’s at θ1
    = 0o and θ1
    = 8o.
    −4 −2 0 2 4 6 8 10 12 14
    0
    0.2
    0.4
    0.6
    0.8
    1
    DOA θ degrees (2o sampling grid)
    Frequency
    SNIHT(2, 2)
    SNIHT(1, 1)
    SNIHT(2, 1)
    PER rates
    1 SNIHT(1, 1) = 81%
    2 SNIHT(2, 1) = 73.1%
    3 HUB-SNIHT = 42.6%
    4 SNIHT = 37.6%
    SNIHT(1, 1)
    21/24

    View full-size slide

  33. Number of snapshots Q = 40
    DOA’s at θ1
    = 0o and θ1
    = 8o.
    −4 −2 0 2 4 6 8 10 12 14
    0
    0.2
    0.4
    0.6
    0.8
    1
    DOA θ degrees (2o sampling grid)
    Frequency
    SNIHT(2, 2)
    SNIHT(1, 1)
    SNIHT(2, 1)
    PER rates
    1 SNIHT(1, 1) = 100%
    2 SNIHT(2, 1) = 73.1%
    3 HUB-SNIHT = 42.6%
    4 SNIHT = 37.6%
    SNIHT(1, 1)
    22/24

    View full-size slide

  34. Conclusions
    robust and nonparametric appraches for simultaneous sparse recovery
    23/24

    View full-size slide

  35. Conclusions
    robust and nonparametric appraches for simultaneous sparse recovery
    Different method, SNIHT(1, 1), SNIHT(2, 1), HUB-SNIHT, can be
    selected for a problem at hand (iid/correlated noise, high-breakdown, etc)
    23/24

    View full-size slide

  36. Conclusions
    robust and nonparametric appraches for simultaneous sparse recovery
    Different method, SNIHT(1, 1), SNIHT(2, 1), HUB-SNIHT, can be
    selected for a problem at hand (iid/correlated noise, high-breakdown, etc)
    Fast and scalable greedy SNIHT algorithm: increase in robustness does
    not imply increase in computation time!
    23/24

    View full-size slide

  37. Conclusions
    robust and nonparametric appraches for simultaneous sparse recovery
    Different method, SNIHT(1, 1), SNIHT(2, 1), HUB-SNIHT, can be
    selected for a problem at hand (iid/correlated noise, high-breakdown, etc)
    Fast and scalable greedy SNIHT algorithm: increase in robustness does
    not imply increase in computation time!
    Applications in source localization and image denoising, etc.
    23/24

    View full-size slide

  38. This talk based on
    E. Ollila (2015a)
    Nonparametric Simultaneous Sparse Recovery: an Application to
    Source Localization
    PDF (ArXiv): http://arxiv.org/pdf/1502.02441v1
    EUSIPCO’15, submitted.
    E. Ollila (2015b).
    Robust Simultaneous Sparse Approximation
    A Festschrift in Honor of Hannu Oja, Springer, Sept. 2015.
    24/24

    View full-size slide

  39. Blanchard, J. D., Cermak, M., Hanle, D., and Jin, Y. (2014).
    Greedy algorithms for joint sparse recovery.
    IEEE Trans. Signal Processing, 62(7).
    Chen, J. and Huo, X. (2006).
    Theoretical results on sparse representations of multiple-measurement vectors.
    IEEE Trans. Signal Processing, 54(12):4634–4643.
    Cotter, S. and Rao, B. (2002).
    Sparse channel estimation via matching pursuit with application to equalization.
    IEEE Trans. Comm., 50(3):374–377.
    Duarte, M. F. and Eldar, Y. C. (2011).
    Structured compressed sensing: From theory to applications.
    IEEE Trans. Signal Processing, 59(9):4053–4085.
    Eldar, Y. C. and Rauhut, H. (2010).
    Average case analysis of multichannel sparse recovery using convex relaxation.
    IEEE Trans. Inform. Theory, 56(1):505–519.
    Gorodnitsky, I., George, J., and Rao, B. (1995).
    Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm
    algorithm.
    J. Electroencephalogr. Clin. Neurophysiol., 95(4):231–251.

    View full-size slide

  40. Gribonval, R. and Zibulevsky, M. (2010).
    Handbook of Blind Source Separation, chapter Sparse component analysis, pages
    367–420.
    Academic Press, Oxford, UK.
    Malioutov, D., C
    ¸etin, M., and Willsky, A. S. (2005).
    A sparse signal reconstruction perspective for source localization with sensor arrays.
    IEEE Trans. Signal Processing, 53(8):3010–3022.
    Tropp, J. A. (2006).
    Algorithms for simultaneous sparse approximation. part II: Convex relaxation.
    Signal Processing, 86:589–602.
    24/24

    View full-size slide