# Sparse Approximate Solutions to Linear Systems

@article{Natarajan1995SparseAS,
title={Sparse Approximate Solutions to Linear Systems},
author={Balas K. Natarajan},
journal={SIAM J. Comput.},
year={1995},
volume={24},
pages={227-234}
}
• B. Natarajan
• Published 1 April 1995
• Mathematics, Computer Science
• SIAM J. Comput.
The following problem is considered: given a matrix $A$ in ${\bf R}^{m \times n}$, ($m$ rows and $n$ columns), a vector $b$ in ${\bf R}^m$, and ${\bf \epsilon} > 0$, compute a vector $x$ satisfying $\| Ax - b \|_2 \leq {\bf \epsilon}$ if such exists, such that $x$ has the fewest number of non-zero entries over all such vectors. It is shown that the problem is NP-hard, but that the well-known greedy heuristic is good in that it computes a solution with at most $\left\lceil 18 \mbox{ Opt} ({\bf… Expand 2,454 Citations #### Topics from this paper A Perturbation Inequality for the Schatten-$p$Quasi-Norm and Its Applications to Low-Rank Matrix Recovery • Mathematics, Computer Science • ArXiv • 2012 A perturbation inequality for the so--called Schatten$p$--quasi--norm is obtained, which allows the validity of a number of previously conjectured conditions for the recovery of low--rank matrices via the popular Schatten p-norm heuristic to be confirmed. Expand Sparse representation of vectors in lattices and semigroups • Mathematics • 2021 We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries,Expand Recovery of sparsest signals via$\ell^q$-minimization In this paper, it is proved that every$s$-sparse vector${\bf x}\in {\mathbb R}^n$can be exactly recovered from the measurement vector${\bf z}={\bf A} {\bf x}\in {\mathbb R}^m$via someExpand Sparse Convex Optimization via Adaptively Regularized Hard Thresholding • Computer Science, Mathematics • ICML • 2020 A new Adaptively Regularized Hard Thresholding (ARHT) algorithm that makes significant progress on this problem by bringing the bound down to$\gamma=O(\kappa)$, which has been shown to be tight for a general class of algorithms including LASSO, OMP, and IHT. Expand Recovery of Sparse Representations by Polytope Faces Pursuit The proposed algorithm, which is based on the geometry of the polar polytope, is called Polytope Faces Pursuit and produces good results on examples that are known to be hard for MP, and it is faster than the interior point method for BP on the experiments presented. Expand Recovery of sparsest signals via ℓq-minimization • Qiyu Sun • Mathematics, Computer Science • ArXiv • 2010 It is proved that every s-sparse vector in R can be exactly recovered from the measurement vector z via some$\ell^q$-minimization with 0< q\le 1$. Expand
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• SWAT
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