Abstract-Finding optimal sparse solutions to estimation problems, particularly in underdetermined regimes has recently gained much attention. Most existing literature study linear models in which the squared error is used as the measure of discrepancy to be minimized. However, in many applications discrepancy is measured in more general forms such as loglikelihood. Regularization by 1-norm has been shown to induce sparse solutions, but their sparsity level can be merely suboptimal. In this paper we present a greedy algorithm, dubbed Gradient Support Pursuit (GraSP), for sparsity-constrained optimization. Quantifiable guarantees are provided for GraSP when cost functions have the "Stable Hessian Property".

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In this paper we analyze the blind deconvolution of an image and an unknown blur in a coded imaging system. The measurements consist of subsampled convolution of an unknown blurring kernel with multiple random binary modulations (coded masks) of the image. To perform the deconvolution, we consider a standard lifting of the image and the blurring kernel that transforms the measurements into a set of linear equations of the matrix formed by their outer product. Any rank-one solution to this system of equations provides a valid pair of an image and a blur. We first express the necessary and sufficient conditions for the uniqueness of a rank-one solution under some additional assumptions (uniform subsampling and no limit on the number of coded masks). These conditions are a special case of a previously established result regarding identifiability in the matrix completion problem. We also characterize a low-dimensional subspace model for the blur kernel that is sufficient to guarantee identifiability, including the interesting instance of "bandpass" blur kernels. Next, assuming the bandpass model for the blur kernel, we show that the image and the blur kernel can be found using nuclear norm minimization. Our main results show that recovery is achieved (with high probability) when the number of masks is on the order of µ log 2 L log Le μ log log(N + 1), where µ is the coherence of the blur, L is the dimension of the image, and N is the number of measured samples per mask. Introduction.The blind deconvolution problem has been encountered in many fields including astronomical, microscopic, and medical imaging, computational photography, and wireless communications. Many blind deconvolution techniques, mostly tailored for particular applications, have been proposed in these communities. These techniques can be divided into two categories based on their general formulation of the problem. The methods of the first category typically reduce the blind deconvolution problem to a regularized least squares problem without imposing stochastic models on either of the convolved signals. High computational cost and sensitivity to noise are the main challenges for these methods. The second category of blind deconvolution methods follow a Bayesian approach and consider prior distributions for either or both of the signals, albeit implicitly (see, e.g., [16,15,28]). Therefore, estimation of the convolved signals is formulated as a maximum a posteriori (MAP) estimation in these approaches. An extensive review of the classic blind deconvolution methods in

In this paper we consider the problem of estimating simultaneously low-rank and row-wise sparse matrices from nested linear measurements where the linear operator consists of the product of a linear operator W and a matrix Ψ . Leveraging the nested structure of the measurement operator, we propose a computationally efficient two-stage algorithm for estimating the simultaneously structured target matrix. Assuming that W is a restricted isometry for low-rank matrices and Ψ is a restricted isometry for row-wise sparse matrices, we establish an accuracy guarantee that holds uniformly for all sufficiently low-rank and row-wise sparse matrices with high probability. Furthermore, using standard tools from information theory, we establish a minimax lower bound for estimation of simultaneously low-rank and row-wise sparse matrices from linear measurements that need not be nested. The accuracy bounds established for the algorithm, that also serve as a minimax upper bound, differ from the derived minimax lower bound merely by a polylogarithmic factor of the dimensions. Therefore, the proposed algorithm is nearly minimax optimal. We also discuss some applications of the proposed observation model and evaluate our algorithm through numerical simulation.

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