A generalized uncertainty principle and sparse representation in pairs of bases

By Michael Elad, Alfred Marcel Bruckstein. IEEE Transactions on Information Theory, 2002

668 citations recorded by the source corpus.

Subfields: Computational Mechanics, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/115221749c9505a8a305ab6df676d9b71ed1438c

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Citations in the corpus

Sparsity and incoherence in compressive sampling (2006, background): “The relationship of the mutual incoherence parameter � to the performance of l1 minimization programs with equality constraints first appeared in the context of Basis Pursuit for sparse approximation, see [12] and also [11, 14 ,16].”
$rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation (2006, background): “Extensive study of these algorithms in recent years has established that if the sought solution, x, is sparse enough, these techniques recover it well in the exact case [16], [26], [27], [28], [29], [30].”
Greed is good: algorithmic results for sparse approximation (2004, methodology): “Note that Theorems A and B unify all of the recent results for BP [7]–[9] and extend them to OMP as well.”
Statistical Learning with Sparsity: The Lasso and Generalizations (2015, background): “Early work by Fazel (2002) studied the use of the nuclear norm as a surrogate for a rank constraint. Srebro, Alon and Jaakkola (2005) studied the nuclear norm as well as related relaxations of rank constraints in the context of matrix completion and collaborative filtering.”
Just relax: convex programming methods for identifying sparse signals in noise (2006, methodology): “See for example [14, 19–23].”
Restricted Eigenvalue Properties for Correlated Gaussian Designs (2010, background): “See the papers (Cohen et al., 2009; Donoho and Huo, 2001; Elad and Bruckstein, 2002; Feuer and Nemirovski, 2003 ) for more discussion of restricted nullspaces and equivalence to exact recovery of basis pursuit.”
Global and Local Uncertainty Principles for Signals on Graphs (2016, background): “As one example, the Elad–Bruckstein uncertainty principle of [28] states that if α and β are the coefficients of a vector f ∈ RN in two different orthonormal bases, then ||α||0 + ||β||0 2 ≥ √ ||α||0 · ||β||0 ≥ 1 μ , (1)”
Stable recovery of sparse overcomplete representations in the presence of noise (2006, background): “Previous work [13, 14, 12, 17] established that both the OGA and the BP approaches can be successful for signals having sparse representations; under appropriate conditions on Φ and y, these algorithms produce the globally optimal solution of (P0).”
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009, background): “We consider the problem of representing a vector y ∈ C L in a given dictionary D of size L × N with L N , so that for a coefﬁcient vector x ∈ C N .”
Recovery of exact sparse representations in the presence of bounded noise (2005, result): “[ 3 ] T. Kailath, L. Ljung, and M. Morf, “Recursive input-output and state-”
Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) (2005, methodology): “Thus, a study of the decay of the mutual incoherence as a function of the rows removed could be of help here (see [ 5 ,10,12,17]).”