89 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
Oracle-order Recovery Performance of Greedy Pursuits with Replacement against General Perturbations
Applying the theory of compressive sensing in practice always takes different
kinds of perturbations into consideration. In this paper, the recovery
performance of greedy pursuits with replacement for sparse recovery is analyzed
when both the measurement vector and the sensing matrix are contaminated with
additive perturbations. Specifically, greedy pursuits with replacement include
three algorithms, compressive sampling matching pursuit (CoSaMP), subspace
pursuit (SP), and iterative hard thresholding (IHT), where the support
estimation is evaluated and updated in each iteration. Based on restricted
isometry property, a unified form of the error bounds of these recovery
algorithms is derived under general perturbations for compressible signals. The
results reveal that the recovery performance is stable against both
perturbations. In addition, these bounds are compared with that of oracle
recovery--- least squares solution with the locations of some largest entries
in magnitude known a priori. The comparison shows that the error bounds of
these algorithms only differ in coefficients from the lower bound of oracle
recovery for some certain signal and perturbations, as reveals that
oracle-order recovery performance of greedy pursuits with replacement is
guaranteed. Numerical simulations are performed to verify the conclusions.Comment: 27 pages, 4 figures, 5 table
Model-Based Calibration of Filter Imperfections in the Random Demodulator for Compressive Sensing
The random demodulator is a recent compressive sensing architecture providing
efficient sub-Nyquist sampling of sparse band-limited signals. The compressive
sensing paradigm requires an accurate model of the analog front-end to enable
correct signal reconstruction in the digital domain. In practice, hardware
devices such as filters deviate from their desired design behavior due to
component variations. Existing reconstruction algorithms are sensitive to such
deviations, which fall into the more general category of measurement matrix
perturbations. This paper proposes a model-based technique that aims to
calibrate filter model mismatches to facilitate improved signal reconstruction
quality. The mismatch is considered to be an additive error in the discretized
impulse response. We identify the error by sampling a known calibrating signal,
enabling least-squares estimation of the impulse response error. The error
estimate and the known system model are used to calibrate the measurement
matrix. Numerical analysis demonstrates the effectiveness of the calibration
method even for highly deviating low-pass filter responses. The proposed method
performance is also compared to a state of the art method based on discrete
Fourier transform trigonometric interpolation.Comment: 10 pages, 8 figures, submitted to IEEE Transactions on Signal
Processin
Signal Recovery in Perturbed Fourier Compressed Sensing
In many applications in compressed sensing, the measurement matrix is a
Fourier matrix, i.e., it measures the Fourier transform of the underlying
signal at some specified `base' frequencies , where is the
number of measurements. However due to system calibration errors, the system
may measure the Fourier transform at frequencies
that are different from the base frequencies and where
are unknown. Ignoring perturbations of this nature can lead to major errors in
signal recovery. In this paper, we present a simple but effective alternating
minimization algorithm to recover the perturbations in the frequencies \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations can be expressed
in terms of a small number of unique parameters . We demonstrate that
in such cases, the method leads to excellent quality results that are several
times better than baseline algorithms (which are based on existing off-grid
methods in the recent literature on direction of arrival (DOA) estimation,
modified to suit the computational problem in this paper). Our results are also
robust to noise in the measurement values. We also provide theoretical results
for (1) the convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included.
More challenging experiments now include
Perturbed Orthogonal Matching Pursuit
Cataloged from PDF version of article.Compressive Sensing theory details how a sparsely
represented signal in a known basis can be reconstructed with
an underdetermined linear measurement model. However, in reality
there is a mismatch between the assumed and the actual
bases due to factors such as discretization of the parameter
space defining basis components, sampling jitter in A/D conversion,
and model errors. Due to this mismatch, a signal may
not be sparse in the assumed basis, which causes significant performance
degradation in sparse reconstruction algorithms. To
eliminate the mismatch problem, this paper presents a novel
perturbed orthogonal matching pursuit (POMP) algorithm that
performs controlled perturbation of selected support vectors to
decrease the orthogonal residual at each iteration. Based on detailed
mathematical analysis, conditions for successful reconstruction
are derived. Simulations show that robust results with much
smaller reconstruction errors in the case of perturbed bases can
be obtained as compared to standard sparse reconstruction techniques
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