sparse recovery

Sharpness and well-conditioning of nonsmooth convex formulations in statistical signal recovery

  • Lijun Ding
  • Alex L. Wang
    • Categories Nonsmooth Optimization, Optimization in Data Science, Statistics Tags , , , , ,

    \(\) We study a sample complexity vs. conditioning tradeoff in modern signal recovery problems where convex optimization problems are built from sampled observations. We begin by introducing a set of condition numbers related to sharpness in \(\ell_p\) or Schatten-p norms (\(p\in[1,2]\)) based on nonsmooth reformulations of a class of convex optimization problems, including sparse recovery, … Read more

    Stable Recovery of Sparse Signals With Non-convex Weighted $r$-Norm Minus $1$-Norm

  • Jianwen Huang
  • Feng Zhang
  • Xinling Liu
  • Jianjun Wang
    • Categories Constrained Nonlinear Optimization, Data-Mining, Nonsmooth Optimization Tags , , ,

    Given the measurement matrix $A$ and the observation signal $y$, the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system $y=Ax+z$, where $x$ is the $s$-sparse signal to be recovered and $z$ is the noise vector. Zhou and Yu \cite{Zhou and Yu 2019} recently proposed a novel … Read more

    Analysis non-sparse recovery for non-convex relaxed $\ell_q$ minimization

  • Jianwen Huang
  • Feng Zhang
  • Xinling Liu
  • Jianjun Wang
    • Categories Data-Mining, Nonsmooth Optimization, Unconstrained Optimization Tags , , , ,

    This paper studies construction of signals, which are sparse or nearly sparse with respect to a tight frame $D$ from underdetermined linear systems. In the paper, we propose a non-convex relaxed $\ell_q(0 Article Download View Analysis non-sparse recovery for non-convex relaxed $ell_q$ minimization

    Sparse Recovery via Partial Regularization: Models, Theory and Algorithms

  • Xiaorui Li
  • Zhaosong Lu
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Statistics Tags , , , , , , ,

    In the context of sparse recovery, it is known that most of existing regularizers such as $\ell_1$ suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We … Read more

    A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem

  • Lin Xiao
  • Tong Zhang
    • Categories Convex and Nonsmooth Optimization Tags , , ,

    We consider solving the $\ell_1$-regularized least-squares ($\ell_1$-LS) problem in the context of sparse recovery, for applications such as compressed sensing. The standard proximal gradient method, also known as iterative soft-thresholding when applied to this problem, has low computational cost per iteration but a rather slow convergence rate. Nevertheless, when the solution is sparse, it often … Read more

    On partially sparse recovery

  • A. S. Bandeira
  • Katya Scheinberg
  • Luis Nunes Vicente
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this paper we consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the l1-norm of the part of the solution vector which is known to be sparse. Such a problem is closely related to the classical problem in Compressed Sensing where the l1-norm of the … Read more

    Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

  • A. S. Bandeira
  • Katya Scheinberg
  • Luis Nunes Vicente
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations … Read more

    On partially sparse recovery

  • A. S. Bandeira
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this paper we consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the l1-norm of the part of the solution vector which is known to be sparse. Such a problem is closely related to the classical problem in Compressed Sensing where the l1-norm of the … Read more

    Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

  • A. S. Bandeira
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations … Read more

    Accuracy guarantees for ℓ1-recovery

  • Anatoli Juditsky
  • Fatma Kılınç-Karzan
  • Arkadi Nemirovski
    • Categories Statistics Tags , , ,

    We discuss two new methods of recovery of sparse signals from noisy observation based on ℓ1- minimization. They are closely related to the well-known techniques such as Lasso and Dantzig Selector. However, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to … Read more