234 research outputs found
The Convergence Guarantees of a Non-convex Approach for Sparse Recovery
In the area of sparse recovery, numerous researches hint that non-convex
penalties might induce better sparsity than convex ones, but up until now those
corresponding non-convex algorithms lack convergence guarantees from the
initial solution to the global optimum. This paper aims to provide performance
guarantees of a non-convex approach for sparse recovery. Specifically, the
concept of weak convexity is incorporated into a class of sparsity-inducing
penalties to characterize the non-convexity. Borrowing the idea of the
projected subgradient method, an algorithm is proposed to solve the non-convex
optimization problem. In addition, a uniform approximate projection is adopted
in the projection step to make this algorithm computationally tractable for
large scale problems. The convergence analysis is provided in the noisy
scenario. It is shown that if the non-convexity of the penalty is below a
threshold (which is in inverse proportion to the distance between the initial
solution and the sparse signal), the recovered solution has recovery error
linear in both the step size and the noise term. Numerical simulations are
implemented to test the performance of the proposed approach and verify the
theoretical analysis.Comment: 33 pages, 7 figure
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
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
Local Measurement and Reconstruction for Noisy Graph Signals
The emerging field of signal processing on graph plays a more and more
important role in processing signals and information related to networks.
Existing works have shown that under certain conditions a smooth graph signal
can be uniquely reconstructed from its decimation, i.e., data associated with a
subset of vertices. However, in some potential applications (e.g., sensor
networks with clustering structure), the obtained data may be a combination of
signals associated with several vertices, rather than the decimation. In this
paper, we propose a new concept of local measurement, which is a generalization
of decimation. Using the local measurements, a local-set-based method named
iterative local measurement reconstruction (ILMR) is proposed to reconstruct
bandlimited graph signals. It is proved that ILMR can reconstruct the original
signal perfectly under certain conditions. The performance of ILMR against
noise is theoretically analyzed. The optimal choice of local weights and a
greedy algorithm of local set partition are given in the sense of minimizing
the expected reconstruction error. Compared with decimation, the proposed local
measurement sampling and reconstruction scheme is more robust in noise existing
scenarios.Comment: 24 pages, 6 figures, 2 tables, journal manuscrip
Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-point Attracting Projection
A recursive algorithm named Zero-point Attracting Projection (ZAP) is
proposed recently for sparse signal reconstruction. Compared with the reference
algorithms, ZAP demonstrates rather good performance in recovery precision and
robustness. However, any theoretical analysis about the mentioned algorithm,
even a proof on its convergence, is not available. In this work, a strict proof
on the convergence of ZAP is provided and the condition of convergence is put
forward. Based on the theoretical analysis, it is further proved that ZAP is
non-biased and can approach the sparse solution to any extent, with the proper
choice of step-size. Furthermore, the case of inaccurate measurements in noisy
scenario is also discussed. It is proved that disturbance power linearly
reduces the recovery precision, which is predictable but not preventable. The
reconstruction deviation of -compressible signal is also provided. Finally,
numerical simulations are performed to verify the theoretical analysis.Comment: 29 pages, 6 figure
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