Stable sparse approximations via nonconvex optimization

Saab, Rayan; Yılmaz, Özgür

doi:10.1109/icassp.2008.4518502

Cited by 173 publications

(150 citation statements)

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“…described above gaining in popularity in the literature (see, e.g., [7], [9], [14], [21], [23], [25], [27], [28]). However, there remains a key issue with respect to these choices: to what extent the minimizations (P p ) can achieve the same result as the initial minimization (P 0 ).…”

Section: Introductionmentioning

confidence: 99%

$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

Peng

Yue

2015

IEEE Trans. Inform. Theory

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Section: Introductionmentioning

confidence: 99%

$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

Peng

Yue

2015

IEEE Trans. Inform. Theory

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“…The MOE regularizer has also been recently proposed in the context of compressive sensing [7], where it is conjectured to be better than the 1 -term at recovering sparse signals in compressive sensing applications. 4 This conjecture was partially confirmed recently for non-convex regularizers of the form ψ(a) = a r with 0 < r < 1 in [35], [18], and for a more general family of non-convex regularizers including the one in (17) in [42]. In all cases, it was shown that the conditions on the sensing matrix (here D) can be significantly relaxed to guarantee exact recovery if non-convex regularizers are used instead of the 1 norm, provided that the exact solution to the non-convex optimization problem can be computed.…”

Section: A the Conjugate Priormentioning

confidence: 64%

Universal Sparse Modeling

Ramírez¹,

Sapiro²

2010

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Sparse data models, where data is assumed to be well represented as a linear combination of a few elements from a dictionary, have gained considerable attention in recent years, and their use has led to state-of-the-art results in many signal and image processing tasks. It is now well understood that the choice of the sparsity regularization term is critical in the success of such models. In this work, we use tools from information theory, and in particular universal coding theory, to propose a framework for designing sparsity regularization terms which have several theoretical and practical advantages when compared to the more standard 0 or 1 ones, and which lead to improved coding performance and accuracy in reconstruction and classification tasks. We also report on further improvements obtained by imposing low mutual coherence and Gram matrix norm on the corresponding learned dictionaries.The presentation of the framework and theoretical foundations is complemented with examples in image denoising and classification.

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“…For any S ⊂ T with #S = k, θ(0, z , S ) = k/(L + 1 − k), and therefore (5) holds. If p ∈ (0, 1], recalling the equivalent definition (8), there exists S with #S ≤ k such that γ( p , A, k) = sup…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…The global optimality of (3) has been studied and various conditions have been derived, for example, those based on restricted isometry property [7][8][9]12] and null space property [10,13]. Among them, a necessary and sufficient condition is based on the null space property and its constant [10,13,14].…”

Section: Introductionmentioning

confidence: 99%

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On the Null Space Constant for <formula formulatype="inline"><tex Notation="TeX">${\ell _p}$</tex></formula> Minimization

Chen¹,

Gu²

2015

IEEE Signal Process. Lett.

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The literature on sparse recovery often adopts the p "norm" (p ∈ [0, 1]) as the penalty to induce sparsity of the signal satisfying an underdetermined linear system. The performance of the corresponding p minimization problem can be characterized by its null space constant. In spite of the NP-hardness of computing the constant, its properties can still help in illustrating the performance of p minimization. In this letter, we show the strict increase of the null space constant in the sparsity level k and its continuity in the exponent p. We also indicate that the constant is strictly increasing in p with probability 1 when the sensing matrix A is randomly generated. Finally, we show how these properties can help in demonstrating the performance of p minimization, mainly in the relationship between the the exponent p and the sparsity level k.

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Stable sparse approximations via nonconvex optimization

Cited by 173 publications

References 10 publications

$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

Universal Sparse Modeling

On the Null Space Constant for <formula formulatype="inline"><tex Notation="TeX">${\ell _p}$</tex></formula> Minimization

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