Optimally sparse representation in general (nonorthogonal) dictionaries via l minimization.

By David L. Donoho, Michael Elad. 2003

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Best Subset Selection via a Modern Optimization Lens (2015, methodology): “For k = 1, we obtain the notion of coherence introduced in Donoho and Elad (2003), Donoho and Huo (2001) as a measure of the maximal pairwise correlation in absolute value of the columns of X: μ := μ[1] = maxi =j |〈Xi ,Xj 〉|.”
A unified approach to model selection and sparse recovery using regularized least squares (2009, background): “As mentioned in the Introduction, the minimum L 0 solution to (2) is Donoho and Elad (2003) introduce the concept of spark and show that the uniqueness of β 0 can be characterized by the spark( X ) of X , where τ = spark( X ) is deﬁned as the smallest possible number such that there exists a…”
Stable signal recovery from incomplete and inaccurate measurements (2005, background): “The cone constraint (10) may not hold but a variation does.”
Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal (2008, background): “Clearly, we seek the sparsest solution by minimizing the l1-objective [41][42] within the feasible set C := S1 ∩ S2.”
Stability selection (2008, background): “Keywords: High dimensional data; Resampling; Stability selection; Structure estimation”
A unified Bayesian framework for MEG/EEG source imaging (2009, unknown): “Specifically, conditions are stipulated whereby minimum ℓ 1 -norm solutions are guaranteed to locate some number of nonzero coefficients (or sources) in an underdetermined linear model (Donoho and Elad, 2003; Fuchs, 2004; Gribonval and Nielsen, 2003; Tropp, 2004).”
Challenges of Big Data Analysis (2013, unknown): “For example, the authors of [18] showed that the non-deterministic polynomial-time hard (NP-hard) best subset regression can be recast as an L1-norm penalized least-squares problem which can be solved by the interior point method.”
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions (2004, background): “4 Significance of uncertainty principles In the last three years or so, there has been a series of papers starting with [10] establishing a link between discrete uncertainty principles and sparse approximation [10, 11, 15, 26].”
Nonlinear Extraction of Independent Components of Natural Images Using Radial Gaussianization (2009, background): “…(e.g., Coifman & Wickerhauser, 1992; Mallat & Zhang, 1993; Olshausen & Field, 1997; Chen, Donoho, & Saunders, 1998; Lewicki & Sejnowski, 2000; Donoho & Elad, 2003). the expected value of a higher-order contrast function to avoid the difﬁ-culties associated with entropy estimation (Comon,…”
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches (2012, unknown): “If the system of linear equations has a solution satisfying , where is the smallest number of linearly dependent columns of , it is necessarily the unique solution of (19) [182].”
Necessary and sufficient conditions for linear convergence of ℓ1‐regularization (2011, background): “6) is the sparsest solution of Ax D y , that is, the solution with the smallest number of nonzero coefficients (see [10, 15]).”
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information (2004, background): “For example, a series of beautiful papers [7, 8, 9, 12, 14] is concerned with a special setup where one is given a dictionary D of vectors (waveforms) of C , D = (dk)1≤k≤M and one seeks sparse representations of a signal f ∈ C as a superposition of elements of D f = Dα.”
Iterative Thresholding for Sparse Approximations (2008, background): “For example, a wide range of signal processing applications such as source coding [1], [2], denoising [3], source separation [4] and pattern analysis [5] have beneﬁted from progress made in this area.”
Sparse Coding via Thresholding and Local Competition in Neural Circuits (2008, background): “…which replaces the ℓ 0 norm with the ℓ 1 norm (2) Despite this substitution, BP has the same solution as the optimal sparse approximation problem (Donoho and Elad, 2003) if the signal is sparse compared to the most similar pair of dictionary elements (e.g., || a || 0 min m = n [1 + 1 / h φ…”