6,281 research outputs found
Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling
Solving linear regression problems based on the total least-squares (TLS)
criterion has well-documented merits in various applications, where
perturbations appear both in the data vector as well as in the regression
matrix. However, existing TLS approaches do not account for sparsity possibly
present in the unknown vector of regression coefficients. On the other hand,
sparsity is the key attribute exploited by modern compressive sampling and
variable selection approaches to linear regression, which include noise in the
data, but do not account for perturbations in the regression matrix. The
present paper fills this gap by formulating and solving TLS optimization
problems under sparsity constraints. Near-optimum and reduced-complexity
suboptimum sparse (S-) TLS algorithms are developed to address the perturbed
compressive sampling (and the related dictionary learning) challenge, when
there is a mismatch between the true and adopted bases over which the unknown
vector is sparse. The novel S-TLS schemes also allow for perturbations in the
regression matrix of the least-absolute selection and shrinkage selection
operator (Lasso), and endow TLS approaches with ability to cope with sparse,
under-determined "errors-in-variables" models. Interesting generalizations can
further exploit prior knowledge on the perturbations to obtain novel weighted
and structured S-TLS solvers. Analysis and simulations demonstrate the
practical impact of S-TLS in calibrating the mismatch effects of contemporary
grid-based approaches to cognitive radio sensing, and robust
direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal
Processin
Displacement Data Assimilation
We show that modifying a Bayesian data assimilation scheme by incorporating
kinematically-consistent displacement corrections produces a scheme that is
demonstrably better at estimating partially observed state vectors in a setting
where feature information important. While the displacement transformation is
not tied to any particular assimilation scheme, here we implement it within an
ensemble Kalman Filter and demonstrate its effectiveness in tracking
stochastically perturbed vortices.Comment: 26 Pages, 9 figures, 5 table
Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation
In this paper, we aim at recovering an unknown signal x0 from noisy
L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear
operator and w accounts for some noise. To regularize such an ill-posed inverse
problem, we impose an analysis sparsity prior. More precisely, the recovery is
cast as a convex optimization program where the objective is the sum of a
quadratic data fidelity term and a regularization term formed of the L1-norm of
the correlations between the sought after signal and atoms in a given
(generally overcomplete) dictionary. The L1-sparsity analysis prior is weighted
by a regularization parameter lambda>0. In this paper, we prove that any
minimizers of this problem is a piecewise-affine function of the observations y
and the regularization parameter lambda. As a byproduct, we exploit these
properties to get an objectively guided choice of lambda. In particular, we
develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE)
and show that it is an unbiased and reliable estimator of an appropriately
defined risk. The latter encompasses special cases such as the prediction risk,
the projection risk and the estimation risk. We apply these risk estimators to
the special case of L1-sparsity analysis regularization. We also discuss
implementation issues and propose fast algorithms to solve the L1 analysis
minimization problem and to compute the associated GSURE. We finally illustrate
the applicability of our framework to parameter(s) selection on several imaging
problems
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
Imaging spectrometers measure electromagnetic energy scattered in their
instantaneous field view in hundreds or thousands of spectral channels with
higher spectral resolution than multispectral cameras. Imaging spectrometers
are therefore often referred to as hyperspectral cameras (HSCs). Higher
spectral resolution enables material identification via spectroscopic analysis,
which facilitates countless applications that require identifying materials in
scenarios unsuitable for classical spectroscopic analysis. Due to low spatial
resolution of HSCs, microscopic material mixing, and multiple scattering,
spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus,
accurate estimation requires unmixing. Pixels are assumed to be mixtures of a
few materials, called endmembers. Unmixing involves estimating all or some of:
the number of endmembers, their spectral signatures, and their abundances at
each pixel. Unmixing is a challenging, ill-posed inverse problem because of
model inaccuracies, observation noise, environmental conditions, endmember
variability, and data set size. Researchers have devised and investigated many
models searching for robust, stable, tractable, and accurate unmixing
algorithms. This paper presents an overview of unmixing methods from the time
of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models
are first discussed. Signal-subspace, geometrical, statistical, sparsity-based,
and spatial-contextual unmixing algorithms are described. Mathematical problems
and potential solutions are described. Algorithm characteristics are
illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensin
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