7,314 research outputs found
On the Performance Bound of Sparse Estimation with Sensing Matrix Perturbation
This paper focusses on the sparse estimation in the situation where both the
the sensing matrix and the measurement vector are corrupted by additive
Gaussian noises. The performance bound of sparse estimation is analyzed and
discussed in depth. Two types of lower bounds, the constrained Cram\'{e}r-Rao
bound (CCRB) and the Hammersley-Chapman-Robbins bound (HCRB), are discussed. It
is shown that the situation with sensing matrix perturbation is more complex
than the one with only measurement noise. For the CCRB, its closed-form
expression is deduced. It demonstrates a gap between the maximal and nonmaximal
support cases. It is also revealed that a gap lies between the CCRB and the MSE
of the oracle pseudoinverse estimator, but it approaches zero asymptotically
when the problem dimensions tend to infinity. For a tighter bound, the HCRB,
despite of the difficulty in obtaining a simple expression for general sensing
matrix, a closed-form expression in the unit sensing matrix case is derived for
a qualitative study of the performance bound. It is shown that the gap between
the maximal and nonmaximal cases is eliminated for the HCRB. Numerical
simulations are performed to verify the theoretical results in this paper.Comment: 32 pages, 8 Figures, 1 Tabl
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
Sparse Signal Recovery under Poisson Statistics
We are motivated by problems that arise in a number of applications such as
Online Marketing and explosives detection, where the observations are usually
modeled using Poisson statistics. We model each observation as a Poisson random
variable whose mean is a sparse linear superposition of known patterns. Unlike
many conventional problems observations here are not identically distributed
since they are associated with different sensing modalities. We analyze the
performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting
involves a non-linear optimization but yet is computationally tractable. We
derive fundamental sample complexity bounds for sparse recovery when the
measurements are contaminated with Poisson noise. In contrast to the
least-squares linear regression setting with Gaussian noise, we observe that in
addition to sparsity, the scale of the parameters also fundamentally impacts
sample complexity. We introduce a novel notion of Restricted Likelihood
Perturbation (RLP), to jointly account for scale and sparsity. We derive sample
complexity bounds for regularized ML estimators in terms of RLP and
further specialize these results for deterministic and random sensing matrix
designs.Comment: 13 pages, 11 figures, 2 tables, submitted to IEEE Transactions on
Signal Processin
Subspace Methods for Joint Sparse Recovery
We propose robust and efficient algorithms for the joint sparse recovery
problem in compressed sensing, which simultaneously recover the supports of
jointly sparse signals from their multiple measurement vectors obtained through
a common sensing matrix. In a favorable situation, the unknown matrix, which
consists of the jointly sparse signals, has linearly independent nonzero rows.
In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally
proposed by Schmidt for the direction of arrival problem in sensor array
processing and later proposed and analyzed for joint sparse recovery by Feng
and Bresler, provides a guarantee with the minimum number of measurements. We
focus instead on the unfavorable but practically significant case of
rank-defect or ill-conditioning. This situation arises with limited number of
measurement vectors, or with highly correlated signal components. In this case
MUSIC fails, and in practice none of the existing methods can consistently
approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC),
which improves on MUSIC so that the support is reliably recovered under such
unfavorable conditions. Combined with subspace-based greedy algorithms also
proposed and analyzed in this paper, SA-MUSIC provides a computationally
efficient algorithm with a performance guarantee. The performance guarantees
are given in terms of a version of restricted isometry property. In particular,
we also present a non-asymptotic perturbation analysis of the signal subspace
estimation that has been missing in the previous study of MUSIC.Comment: submitted to IEEE transactions on Information Theory, revised versio
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
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
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