Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization

By David L. Donoho, Michael Elad. 2003

3,041 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Radiology, Nuclear Medicine and Imaging

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Citations in the corpus

Best Subset Selection via a Modern Optimization Lens (2015, methodology): “…(2.7) can all be obtained in terms of γk and μ[k − 1], quantities of fundamental interest appearing in the analysis of 1 regularization methods and understanding how close they are to 0 solutions [Candès (2008), Candes and Tao (2006), Donoho and Elad (2003), Donoho and Huo (2001), Tropp (2006)].”
Stability selection (2008, background): “E-mail: meinshausen@stats.ox.ac.uk some discrete optimization problems, e.g. by l1-penalty approaches (Donoho and Elad, 2003; Meinshausen and Bühlmann, 2006; Zhao and Yu, 2006; Wainwright, 2009; Yuan and Lin, 2007) or greedy algorithms (Freund and Schapire, 1996; Tropp, 2004; Zhang, 2009).”
Wavelet Scattering Regression of Quantum Chemical Energies (2016, background): “However, these hypotheses are violated when Φ(x) = (φk(x))k includes highly correlated functions [19, 18, 13].”
Compressing Neural Networks using the Variational Information Bottleneck (2018, background): “It is typically viewed as the canonical or ideal measure of sparse solutions (Donoho & Elad, 2003).”
Inpainting and Zooming Using Sparse Representations (2009, neutral): “A function f is coercive, if limkxk2!þ1 f(x) ¼ þ1.”
L1-L2 Optimization in Signal and Image Processing (2010, background): “Various encouraging theoretical results on the uniqueness of a sufficiently sparse solution to this problem and equivalence between the sparsest solution and the minimal , 1 -norm solution all suggest that finding the signal's representation z is a computationally feasible task [1], [2], [5]–[8].”
Recovering Sparse Signals With a Certain Family of Nonconvex Penalties and DC Programming (2009, positive): “Other works such as those proposed by Donoho and Elad [14], [15] and Tropp [42] have proved that in certain situations the ℓ 1 penalization is able to recover the sparsity profile of the true coefficient β ⋆ .”
Hidden Talents of the Variational Autoencoder. (2017, methodology): “AE -`1: To explicitly encourage sparse latent representations, which could potentially be helpful in learning the correct manifold dimension, we begin with the AE-`2 model from above and replace ‖μz‖(2)2 with the `1 norm ‖μz‖1, a well-known sparsity-promoting penalty function [12].”
A Selective Overview of Variable Selection in High Dimensional Feature Space (2009, methodology): “For properties of various L 1 and related methods see, for example, the Basis Pursuit in Chen, Donoho and Saunders (1999), Donoho and Elad (2003), Donoho (2004), Fuchs (2004), Candes and Tao (2005, 2006), Donoho, Elad and Temlyakov (2006), Tropp (2006), Cand`es, Wakin and Boyd (2008), and Cai, Xu…”
Assumptionless consistency of the Lasso (2013, background): “Much of this work has its origins in the investigations of l1 penalization by David Donoho and coauthors (some of it predating Tibshirani’s original paper) [9, 10, 11, 12, 13].”
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009, background): “It was shown in [5], [12] that appropriate conditions on the coherence guarantee that both BP and OMP recover the sparse vector x.”
On sparse reconstruction from Fourier and Gaussian measurements (2008, background): “1 and in practice, based on the convex relaxation [9, 16, 18, 19, 15, 21, 29, 30, 31, 12, 10, 11, 14, 3, 2, 5, 6, 27, 4, 7, 23].”
Minimization of ℓ1-2 for Compressed Sensing (2015, methodology): “Compressed sensing (CS) has been a rapidly growing field of research in signal processing and mathematics stimulated by the foundational papers [8, 6, 22, 23] and related Bregman iteration methods [57, 33].”