Highly sparse representations from dictionaries are unique and independent of the sparseness measure

By Rémi Gribonval, Morten Nielsen. 2007

212 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/6741a376ed3e4b3e7d5da6b3ae12c2c976081f36

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In Applied Mathematics

In Computational Mechanics

In Computer Vision and Pattern Recognition

Citations in the corpus

Sparse solutions to linear inverse problems with multiple measurement vectors (2005, unknown): “S p = I (cid:0) P S p , P S0 = 0 , P ?”
Intrinsic Localization of Frames (2005, methodology): “Other applications deal with the characterization of Banach spaces by means of frames, non-linear approximation properties of such frames [1] [24], the convergence of the frame algorithm in Banach space norms, INTRINSIC LOCALIZATION OF FRAMES and unstructured wavelet frames [1], frames for…”
The Cosparse Analysis Model and Algorithms (2011, unknown): “When measuring sparsity with an l norm, 0 < p ≤ 1, rather than with p = 0, it has been shown [26] that for so-called localized frames the analysis coefficients Ωx obtained with Ω = D† the canonical dual frame of D are near optimally sparse: ‖Ωx‖p ≤ Cp minz|Dz=x ‖z‖p, where the constant Cp does not d”
The null space property for sparse recovery from multiple measurement vectors (2011, unknown): “The proof is a straightforward generalization of the arguments in [21].”
On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements (2008, methodology): “Then we have Given the symmetry of the problem (i.e. the rotaional invariance of the Z i ) it should be noted that, without loss of generality, we can assume z t = [ z t || 2 , 0, 0, . . . of || Further, using the results from [23] we have that η d ( n − m ) − 1 points can be located on the sphere…”
Fast Solution of $\ell _{1}$ -Norm Minimization Problems When the Solution May Be Sparse (2008, methodology): “R ECENTLY, -norm minimization has attracted attention in the signal processing community [5]–[9], [20], [28], [29], [33], [34], [45], as an effective technique for solving underdetermined systems of linear equations, of the following form.”
Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 (2009, background): “Gribonval and Nielsen, see e.g. [10], studied the q -minimization in terms of Coherence.”
A sparse signal reconstruction perspective for source localization with sensor arrays (2005, background): “…code with odd length N, the automorphism group contains the following two subgroups: 0S : The set of permutations 0 1 1, ,..., N , where :k j j k mod N. 1S : The set of permutations 0 1 1, ,..., N , where : 2k kj j mod N and 1m is the smallest positive integer such that 12 1m mod N. 3”
Handbook of Blind Source Separation: Independent Component Analysis and Applications (2010, background): “A body of theoretical work initiated by Donoho and Huo [33] suggests that its solution is often equal or similar to that of the corresponding problems with 0 < τ ≤ 1 [55].”
Iteratively reweighted least squares minimization for sparse recovery (2008, background): “Then the map z 7→ z ℓ Nτ deﬁnes a quasi-norm for R N , in particular the triangle inequality holds up to a constant, i.e.,”
Linear Convergence of Iterative Soft-Thresholding (2007, unknown): “Indeed, solutions of this type of problem admit only ﬁnitely many non-zero coeﬃcients and often coincide with the sparsest solution possible [10,18,20].”