Stable recovery of sparse overcomplete representations in the presence of noise

By David L. Donoho, Michael Elad, Vladimir N. Temlyakov. 2006

2,309 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Signal Processing

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Citations in the corpus

Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “This type of example illustrates the possibility of accommodation of overcomplete (multiple or amalgamated) dictionaries in b, whose use in conjunction with ℓ1− penalization has been advocated by Donoho et al. (2005).”
SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR (2008, background): “If all the diagonal elements of matrix XTX/n are equal to 1 (and thus θ1,1 coincides with the mutual coherence [8]), then a simple sufficient condition for Assumption RE(s, c0) to hold is stated as follows.”
Sparse Overcomplete Word Vector Representations (2015, methodology): “…been motivated in other domains as a way to increase separability and interpretability, with each instance (here, a word) having a small number of active dimensions (Olshausen and Field, 1997; Lewicki and Sejnowski, 2000), and to increase stability in the presence of noise (Donoho et al., 2006).”
Multi-Stage Multi-Task Feature Learning (2012, neutral): “Jalali et al. (2010) also presented a theoretical analysis under the incoherence condition (Donoho et al., 2006; Obozinski et al., 2011) which is more restrictive than RIP (Candes and Tao, 2005; Zhang, 2012).”
Greedy approximation (2008, background): “It is known (see, for instance, Donoho et al. (2006)) that for M -coherent matrices Φ we have u0Φ(Φu) = AΦ(Φu) = u, provided u is k-sparse with k < (1 + 1/M)/2.”
Image Sequence Denoising via Sparse and Redundant Representations (2009, background): “This means that we seek for each patch in the image the sparsest vector to describe it using atoms from .”
Data-driven inversion-based control of nonlinear systems (2015, background): “In step 4, the (cid:96) 1 norm of the coeﬃcient vector β is minimized, leading to a sparse coeﬃcient vector α , i.e. a vector with a “small” number of non-zero elements, Fuchs (2005); Donoho et al. (2006); Candes et al. (2006); Tropp (2006).”
Structured Compressed Sensing: From Theory to Applications (2011, background): “Theorem 5 [59]: Let the signal and write .”
Aggregation for Gaussian regression (2007, background): “, fM can be considered as elements of an (overdetermined) dictionary, see [19], or as “weak learners,” see [37], and our results can be interpreted in such contexts as well.”
Robust Sparse Regression under Adversarial Corruption (2013, background): “Many estimators have been proposed, most notably Basis Pursuit (a.k.a. Lasso) (Tibshirani, 1996; Donoho et al., 2006), which solves an `1-regularized least squares problem min β ‖y −Xβ‖22 + λ ‖β‖1 , as well as Orthogonal Matching Pursuit (OMP) e.g., (Tropp, 2004), which is a greedy algorithm that…”
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009, background): “Interesting avenues of further research on block-sparsity include investigating the stability of BOMP in the presence of noise (as done in [47] for the case) and finding recovery guarantees for mixed -norm algorithms (see [48] and references therein).”