# Expander ℓ0-Decoding

@article{MendozaSmith2018Expander,
title={Expander ℓ0-Decoding},
author={Rodrigo Mendoza-Smith and Jared Tanner},
journal={ArXiv},
year={2018},
volume={abs/1508.01256}
}
• Published 6 August 2015
• Mathematics, Computer Science
• ArXiv
Abstract We introduce two new algorithms, Serial- l 0 and Parallel- l 0 for solving a large underdetermined linear system of equations y = A x ∈ R m when it is known that x ∈ R n has at most k m nonzero entries and that A is the adjacency matrix of an unbalanced left d-regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial…
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• Computer Science, Mathematics
IEEE Transactions on Signal and Information Processing over Networks
• 2019
This paper studies the expander recovery performance of the bipartite graph with girth greater than 4, which can be associated with a binary matrix with column correlations equal to either 0 or 1.
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• Computer Science, Mathematics
Front. Appl. Math. Stat.
• 2018
A new reduced sample complexity for the number of nonzeros per column of these matrices is derived, precisely d = \mathcal{O}\left(\log_s(N/s) \right)$; this gives insights into why using small d performed well in numerical experiments involving such matrices. The Permuted Striped Block Model and its Factorization - Algorithms with Recovery Guarantees • Computer Science ArXiv • 2020 The PSB data model is defined as a particular distribution over this class of matrices, motivated by its implications for community detection, provable binary dictionary learning with real valued sparse coding, and blind combinatorial compressed sensing. Weighted sparse recovery with expanders We derived the first sparse recovery guarantees for weighted l1 minimization with sparse random matrices and the class of weighted sparse signals, using a weighted versions of the null space property A Robust Parallel Algorithm for Combinatorial Compressed Sensing • Mathematics, Computer Science IEEE Transactions on Signal Processing • 2018 The robust-<inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-Math></inline- formula> decoding algorithm is presented, which robustifies parallel-< inline-formulas><tex -math notation=LaTeX>$ Â£ell_0$> when the sketch is corrupted by additive noise. Sparse matrices for weighted sparse recovery We derived the first sparse recovery guarantees for weighted$\ell_1$minimization with sparse random matrices and the class of weighted sparse signals, using a weighted versions of the null space #### References SHOWING 1-10 OF 60 REFERENCES Sudocodes ߝ Fast Measurement and Reconstruction of Sparse Signals • Mathematics, Computer Science 2006 IEEE International Symposium on Information Theory • 2006 This work proposes a non-adaptive construction of a sparse Phi comprising only the values 0 and 1; hence the computation of y involves only sums of subsets of the elements of x. Efficient erasure correcting codes • Computer Science, Mathematics IEEE Trans. Inf. Theory • 2001 A simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs is introduced and a simple criterion involving the fractions of nodes of different degrees on both sides of the graph is obtained which is necessary and sufficient for the decoding process to finish successfully with high probability. Vanishingly Sparse Matrices and Expander Graphs, With Application to Compressed Sensing • Mathematics, Computer Science IEEE Transactions on Information Theory • 2013 This work revisits the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random with replacement and presents formulas for the expected cardinality of the set of neighbors for these graphs. Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? • Mathematics, Computer Science IEEE Transactions on Information Theory • 2006 If the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. Neighborly Polytopes And Sparse Solution Of Underdetermined Linear Equations For large d, the overwhelming majority of systems of linear equations with d equations and 4d/3 unknowns have the following property: if there is a solution with fewer than .49d nonzeros, it is the unique minimum ` solution. Compressed sensing • D. Donoho • Computer Science IEEE Transactions on Information Theory • 2006 It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. Verification Decoding of High-Rate LDPC Codes With Applications in Compressed Sensing • Mathematics, Computer Science IEEE Transactions on Information Theory • 2012 The high-rate scaling law for MP decoding of LDPC codes on the binary erasure channel and the q-ary symmetric channel is derived and leads to the result that strictly sparse signals can be reconstructed efficiently with high probability using a constant oversampling ratio. Efficient and Robust Compressed Sensing using High-Quality Expander Graphs • Computer Science, Mathematics ArXiv • 2008 This paper improves upon the result shown earlier by considering expander graphs with expansion coefficient beyond 3/4 and shows that, with the same number of measurements, only$O(k)$recovery iterations are required, which is a significant improvement when$n\$ is large.
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