On sparse representations in arbitrary redundant bases

By Jean-Jacques Fuchs. IEEE Transactions on Information Theory, 2004

616 citations recorded by the source corpus.

Subfields: Computational Mechanics, Control and Systems Engineering, Signal Processing

View Paper: https://www.semanticscholar.org/paper/6cf9b027aa09d042dd13fd9ae848d902240b3d34

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Control and Systems Engineering

In Signal Processing

Citations in the corpus

Approximation accuracy, gradient methods, and error bound for structured convex optimization (2010, background): “where N 0 denotes the optimal value of (1) [42, Theorem 7], [65, Theorem 1], [58], [135, Theorems A and B].”
On the Local Correctness of L^1 Minimization for Dictionary Learning (2011, methodology): “This is due both to the ubiquity of sparsity (or near-sparsity) in practical problems, as well as a line of fundamental theoretical results [DE03, Fuc04, CT05, ZY06, MY09] that assert that if y is known to be sparse in a known basis A satisfying certain technical conditions, the sparse coeﬃcients x…”
For most large underdetermined systems of equations, the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution (2006, background): “In recent years, there has been rapid development in the theory of sparse overcomplete signal representations [5, 13, 14, 29, 30, 16].”
For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution (2006, background): “The fraction O(n−1/2) could be accommodated in results of [7, 12, 8, 14, 26], and O(1/ log(n)) in [2].”
Morphological Component Analysis: An Adaptive Thresholding Strategy (2007, background): “Note that the above inequality has a flavor of the sparsity criterion ensuring unique maximally sparse solut i ns of`1-norm under-determined linear systems in [12], [13], [14], [15].”
An efficient algorithm for compressed MR imaging using total variation and wavelets (2008, background): “…is NP-hard [20], impractical for nearly all real applications, so it is more realistic to solve a computationally tractable alternative such as the 1 -problem min x {{x 1 : Ax = b}, (1) which has also been known to yield sparse solutions under some conditions (see [10] [11] [12] for explains).”
Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) (2005, background): “Representative work along these lines can be found in [5,10–12,15,17,27], where both the exact and the noisy cases are considered.”
Underdetermined blind source separation based on sparse representation (2006, neutral): “From Theorem 1, we can obtain the following corollary, which has appeared in [13] and [14].”
Sparse nonnegative solution of underdetermined linear equations by linear programming. (2005, background): “There is very active research on such problems; for a sampling of different perspectives, see [3, 9, 12, 13, 17, 26, 23] and related work.”
ParNes: a rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals (2009, neutral): “First note the following lasso optimality conditions (see e.g. [20] and [21]).”
Compressed sensing (2006, methodology): “We now describe an alternate abstract approach, allowing us to prove Theorem 1.”
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images (2009, background): “However, if the equation actually has a “suﬃciently sparse” solution, the success of these algorithms in addressing the original objective ( P 0 ) can be guaranteed [46, 86, 79, 156, 133, 134, 135].”
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information (2004, methodology): “In Section II-A, we use duality theory to show that is the unique solution to if and only if a trigonometric polynomial with certain properties exists (a similar duality approach was independently developed in [24] for finding sparse approximations from general dictionaries).”
Inverse problems in spaces of measures (2013, background): “This is most commonly formulated in the discrete setting imposing an 0 constraint [10,15,16,26].”
Iterative Reweighted 1 and 2 Methods for Finding Sparse Solutions (2010, background): “With the advent of compressive sensing and other related applications, there has been growing interest in finding sparse signal representations from redundant dictionaries [7], [8], [12], [13], [24].”
Extensions of compressed sensing (2006, neutral): “Finally we note that at some level of generality, we are discussing here the idea of getting sparse solutions to underdetermined systems of equations using 1 methods, which forms part of a now-extensive body of work: [6, 9, 11, 12, 13, 16, 17, 21, 22, 25, 28, 29].”