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 Mathematics , 2008, Abstract: In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless $\ell_1$ problem, namely, $\min \|\beta\|_1 ~ s.t. ~ y=G \beta$ and quantizing the resulting solution. We show that the quantized solution perfectly reconstructs the sign pattern of a sufficiently sparse signal. Specifically, we show that the sign pattern of an arbitrary k-sparse, n-dimensional signal $x$ can be recovered with $SNR=\Omega(\log n)$ and measurements scaling as $m= \Omega(k \log{n/k})$ for all sparsity levels $k$ satisfying $0< k \leq \alpha n$, where $\alpha$ is a sufficiently small positive constant. Surprisingly, this bound matches the optimal \emph{Max-Likelihood} performance bounds in terms of $SNR$, required number of measurements, and admissible sparsity level in an order-wise sense. In contrast to our results, previous results based on LASSO and Max-Correlation techniques either assume significantly larger $SNR$, sublinear sparsity levels or restrictive assumptions on signal sets. Our proof technique is based on noisy perturbation of the noiseless $\ell_1$ problem, in that, we estimate the maximum admissible noise level before sign pattern recovery fails.
 Qiyu Sun Mathematics , 2011, DOI: 10.1109/TSP.2011.2161470 Abstract: In this paper, we introduce a sparse approximation property of order $s$ for a measurement matrix ${\bf A}$: $$\|{\bf x}_s\|_2\le D \|{\bf A}{\bf x}\|_2+ \beta \frac{\sigma_s({\bf x})}{\sqrt{s}} \quad {\rm for\ all} \ {\bf x},$$ where ${\bf x}_s$ is the best $s$-sparse approximation of the vector ${\bf x}$ in $\ell^2$, $\sigma_s({\bf x})$ is the $s$-sparse approximation error of the vector ${\bf x}$ in $\ell^1$, and $D$ and $\beta$ are positive constants. The sparse approximation property for a measurement matrix can be thought of as a weaker version of its restricted isometry property and a stronger version of its null space property. In this paper, we show that the sparse approximation property is an appropriate condition on a measurement matrix to consider stable recovery of any compressible signal from its noisy measurements. In particular, we show that any compressible signalcan be stably recovered from its noisy measurements via solving an $\ell^1$-minimization problem if the measurement matrix has the sparse approximation property with $\beta\in (0,1)$, and conversely the measurement matrix has the sparse approximation property with $\beta\in (0,\infty)$ if any compressible signal can be stably recovered from its noisy measurements via solving an $\ell^1$-minimization problem.
 Mathematics , 2010, Abstract: We study the problem of sparse reconstruction from noisy undersampled measurements when the following two things are available. (1) We are given partial, and partly erroneous, knowledge of the signal's support, denoted by $T$. (2) We are also given an erroneous estimate of the signal values on $T$, denoted by $(\hat{\mu})_T$. In practice, both these may be available from available prior knowledge. Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as $T$ and $(\hat{\mu})_T$. In this work, we introduce regularized modified-BPDN (reg-mod-BPDN) and obtain computable bounds on its reconstruction error. Reg-mod-BPDN tries to find the signal that is sparsest outside the set $T$, while being "close enough" to $(\hat{\mu})_T$ on $T$ and while satisfying the data constraint. Corresponding results for modified-BPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided.
 Mathematics , 2011, Abstract: In this paper, we investigate the theoretical guarantees of penalized $\lun$ minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with non-necessarily random noise, when the sensing operator belongs to the Gaussian ensemble (i.e. random design matrix with i.i.d. Gaussian entries). More precisely, we derive sharp non-asymptotic bounds on the sparsity level and (minimal) signal-to-noise ratio that ensure support identification for most signals and most Gaussian sensing matrices by solving the Lasso problem with an appropriately chosen regularization parameter. Our first purpose is to establish conditions allowing exact sparsity pattern recovery when the signal is strictly sparse. Then, these conditions are extended to cover the compressible or nearly sparse case. In these two results, the role of the minimal signal-to-noise ratio is crucial. Our third main result gets rid of this assumption in the strictly sparse case, but this time, the Lasso allows only partial recovery of the support. We also provide in this case a sharp $\ell_2$-consistency result on the coefficient vector. The results of the present work have several distinctive features compared to previous ones. One of them is that the leading constants involved in all the bounds are sharp and explicit. This is illustrated by some numerical experiments where it is indeed shown that the sharp sparsity level threshold identified by our theoretical results below which sparsistency of the Lasso is guaranteed meets that empirically observed.