The main triangle projection in matrix spaces and its applications

Kwapień, Stanisław; Pełczyński, A.

doi:10.4064/sm-34-1-43-67

Cited by 131 publications

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“…In this we follow [11]. The heart of the proof of Proposition 3.2 is the following lemma: Lemma 3.3 Let A be an n × n matrix as in Proposition 3.2.…”

Section: Resultsmentioning

confidence: 99%

Error Estimates for Orthogonal Matching Pursuit and Random Dictionaries

Bechler

Wojtaszczyk

2010

Constr Approx

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In this paper we investigate the efficiency of the Orthogonal Matching Pursuit algorithm (OMP) for random dictionaries. We concentrate on dictionaries satisfying the Restricted Isometry Property. We also introduce a stronger Homogenous Restricted Isometry Property which we show is satisfied with overwhelming probability for random dictionaries used in compressed sensing. For these dictionaries we obtain upper estimates for the error of approximation by OMP in terms of the error of the best n-term approximation (Lebesgue-type inequalities). We also present and discuss some open problems about OMP. This is a development of recent results obtained by D.L. Donoho, M. Elad and V.N. Temlyakov.

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“…In this we follow [11]. The heart of the proof of Proposition 3.2 is the following lemma: Lemma 3.3 Let A be an n × n matrix as in Proposition 3.2.…”

Section: Resultsmentioning

confidence: 99%

Error Estimates for Orthogonal Matching Pursuit and Random Dictionaries

Bechler

Wojtaszczyk

2010

Constr Approx

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“…The existence of a basis for the full tensor product was proved by Gelbaum and Gil de Lamadrid [7] who also showed that the unconditionality of the basis for E does not necessarily imply the same property for the tensor product basis. This was taken further by Kwapień and Pe lczyński [8] who treated this issue in the context of spaces of matrices and by Pisier [9] and Schütt [12]. The dual problem, whether the monomials are a basis in the space of homogeneous polynomials, was dealt with by Dimant in her thesis [1], as well as in two other articles, together with Dineen [2] and Zalduendo [3].…”

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confidence: 99%

Schauder Bases for Symmetric Tensor Products

Grecu¹,

Ryan²

2005

Publ. Res. Inst. Math. Sci.

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For a Banach space E with Schauder basis, we prove that the n fold symmetric tensor product⊗ n µ, s E has a Schauder basis for all symmetric uniform crossnorms µ. This is done by modifying the square ordering on N n and showing that the new ordering gives tensor product bases in both⊗The main purpose of this article is to prove that the n-fold symmetric tensor product of a (real or complex) Banach space E has a Schauder basis whenever E does. The result was stated without proof in Ryan's thesis [11] and has been constantly referred to in the literature. In the particular case of a shrinking Schauder basis for a complex Banach space E, an implicit proof was given by Dimant and Dineen [2]. The existence of a basis for the full tensor product was proved by Gelbaum and Gil de Lamadrid [7] who also showed that the unconditionality of the basis for E does not necessarily imply the same property for the tensor product basis. This was taken further by Kwapień and Pe lczyński [8] who treated this issue in the context of spaces of matrices and by Pisier [9] and Schütt [12]. The dual problem, whether the monomials are a basis in the space of homogeneous polynomials, was dealt with by Dimant in her thesis [1], as well as in two other articles, together with Dineen [2] and Zalduendo [3]. The unconditionality (or lack thereof) of the monomial basis was extensively analysed by Defant, Díaz, Garcia and Maestre [4].

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“…For example, we can show that the Lipschitz-continuity of the map A -»\A\ Implies the boundedness (relative to the relevant norm) of the triangle projection ( [17]). Therefore, we can use Arazy's theorem, [4], stating that a symmetrically normed ideal possesses the above-mentioned Interpolation property if and only if the triangle projection Is bounded.…”

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confidence: 99%

Unitarily Invariant Norms under Which the Map $A → |A|$ Is Lipschitz Continuous

Kosaki¹

1992

Publ. Res. Inst. Math. Sci.

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We will characterize the unitarily invariant norms (for compact operators) under which the map A -»\A\ = (A*A) 112 is Lipschitz-continuous. Although the map is not Lipschitz-continuous for the trace class norm, we will obtain a certain Lipschitz-type estimate by making use of the Macaev ideal. § 0. Introduction In [9] E. B. Davies showed the following Lipschitz-type estimates in the Schatten p-norm (1 < p < +00): An obvious next problem is to characterize unitarily invariant norms (of compact operators) under which the map A->|4| = (A*A) 1/2 is Lipschitzcontinuous. In the present article we will obtain quite a complete solution to this problem based on very powerful analysis in [9] and Arazy's result, [4].One of the difficulties of dealing with the map A -> \A\ is its non-linearity. A very clever trick in [9] is to reduce the desired Lipschitz-continuity to the boundedness of a certain linear operator (Schur-Hadamard multiplier, etc.).

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The main triangle projection in matrix spaces and its applications

Cited by 131 publications

References 0 publications

Error Estimates for Orthogonal Matching Pursuit and Random Dictionaries

Error Estimates for Orthogonal Matching Pursuit and Random Dictionaries

Schauder Bases for Symmetric Tensor Products

Unitarily Invariant Norms under Which the Map $A → |A|$ Is Lipschitz Continuous

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