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Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. This line of work shows that $\ell_1$-regularized…
Classical sparsity promoting methods rely on the l0 norm, which treats all nonzero components as equally significant. In practical inverse problems, however, solutions often exhibit many small amplitude components that have little effect on…
Recovering nonlinearly degraded signal in the presence of noise is a challenging problem. In this work, this problem is tackled by minimizing the sum of a non convex least-squares fit criterion and a penalty term. We assume that the…
Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear…
Wideband spectrum sensing detects the unused spectrum holes for dynamic spectrum access (DSA). Too high sampling rate is the main problem. Compressive sensing (CS) can reconstruct sparse signal with much fewer randomized samples than…
This manuscript goes through the fundamental connections between statistical mechanics and estimation theory by focusing on the particular problem of compressive sensing. We first show that the asymptotic analysis of a sparse recovery…
In this paper we study the compressive sensing effects on 2D signals exhibiting sparsity in 2D DFT domain. A simple algorithm for reconstruction of randomly under-sampled data is proposed. It is based on the analytically determined…
Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be…
Spectrum sensing is an important process in cognitive radio. A number of sensing techniques that have been proposed suffer from high processing time, hardware cost and computational complexity. To address these problems, compressive sensing…
From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori…
Sparse phase retrieval aims to recover a $k$-sparse signal from $m$ phaseless measurements. While the theoretically optimal sample complexity for successful recovery is $\Omega(k \log n)$, existing algorithms can only achieve this bound for…
We first propose a novel criterion that guarantees that an $s$-sparse signal is the local minimizer of the $\ell_1/\ell_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition…
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results.…
Detection of a signal under noise is a classical signal processing problem. When monitoring spatial phenomena under a fixed budget, i.e., either physical, economical or computational constraints, the selection of a subset of available…
We propose a new iterative greedy algorithm for reconstructions of sparse signals with or without noisy perturbations in compressed sensing. The proposed algorithm, called \emph{subspace thresholding pursuit} (STP) in this paper, is a…
The generalized approximate message passing (GAMP) algorithm under the Bayesian setting shows advantage in recovering under-sampled sparse signals from corrupted observations. Compared to conventional convex optimization methods, it has a…
Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by…
In this article, we introduce a minimization model via a non-convex transformed $\ell_p$ (TLp) penalty function with two parameters $a\in(0,\infty)$ and $p\in(0,1]$, where the case $p=1$ is known and was established by S. Zhang and J. Xin.…
We demonstrate a simple greedy algorithm that can reliably recover a d-dimensional vector v from incomplete and inaccurate measurements x. Here our measurement matrix is an N by d matrix with N much smaller than d. Our algorithm,…
We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under…