Related papers: Coherence-Based Performance Guarantee of Regulariz…
This paper presents a regularization technique incorporating a non-convex and non-smooth term, $\ell_{1}^{2}-\eta\ell_{2}^{2}$, with parameters $0<\eta\leq 1$ designed to address ill-posed linear problems that yield sparse solutions. We…
This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of…
In this paper, we propose a method for block sparse signal recovery that minimizes the block $q$-ratio sparsity $\left(\lVert z\rVert_{2,1}/\lVert z\rVert_{2,q}\right)^{\frac{q}{q-1}}$ with $q\in[0,\infty]$. For the case of $1<q\leq\infty$,…
Compressed sensing deals with the reconstruction of sparse signals using a small number of linear measurements. One of the main challenges in compressed sensing is to find the support of a sparse signal. In the literature, several bounds on…
Compressed sensing has shown that it is possible to reconstruct sparse high dimensional signals from few linear measurements. In many cases, the solution can be obtained by solving an L1-minimization problem, and this method is accurate…
This short note studies a variation of the Compressed Sensing paradigm introduced recently by Vaswani et al., i.e. the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. The…
In this paper, we show that, under the assumption that $\|\e\|_2\leq \epsilon$, every $k-$sparse signal $\x\in \mathbb{R}^n$ can be stably ($\epsilon\neq0$) or exactly recovered ($\epsilon=0$) from $\y=\A\x+\e$ via $l_p-$mnimization with…
This paper studies the problem of recovering a non-negative sparse signal $\x \in \Re^n$ from highly corrupted linear measurements $\y = A\x + \e \in \Re^m$, where $\e$ is an unknown error vector whose nonzero entries may be unbounded.…
Given an unknown signal $\mathbf{x}_0\in\mathbb{R}^n$ and linear noisy measurements $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\sigma\mathbf{v}\in\mathbb{R}^m$, the generalized $\ell_2^2$-LASSO solves…
Noiseless compressive sensing is a two-steps setting that allows for undersampling a sparse signal and then reconstructing it without loss of information. The LASSO algorithm, based on $\lone$ regularization, provides an efficient and…
We demonstrate that the primal-dual witness proof method may be used to establish variable selection consistency and $\ell_\infty$-bounds for sparse regression problems, even when the loss function and/or regularizer are nonconvex. Using…
We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an $s$-sparse signal from linear measurements…
In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted $\ell_1$ minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into…
It is now well understood that $\ell_1$ minimization algorithm is able to recover sparse signals from incomplete measurements [2], [1], [3] and sharp recoverable sparsity thresholds have also been obtained for the $\ell_1$ minimization…
We consider the following signal recovery problem: given a measurement matrix $\Phi\in \mathbb{R}^{n\times p}$ and a noisy observation vector $c\in \mathbb{R}^{n}$ constructed from $c = \Phi\theta^* + \epsilon$ where $\epsilon\in…
The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurement. We first investigate the minimal number of measurements for the success of the recovery of sparse…
$\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity (the size of the support set), under which with high probability a sparse signal…
The use of generalized LASSO is a common technique for recovery of structured high-dimensional signals. Each generalized LASSO program has a governing parameter whose optimal value depends on properties of the data. At this optimal value,…
In this paper, we consider the problem of signal recovery from 1-bit noisy measurements. We present an efficient method to obtain an estimation of the signal of interest when the measurements are corrupted by white or colored noise. To the…
In this paper we show that if the restricted isometry constant $\delta_k$ of the compressed sensing matrix satisfies \[ \delta_k < 0.307, \] then $k$-sparse signals are guaranteed to be recovered exactly via $\ell_1$ minimization when no…