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Optimal reconstruction systems for erasures and for the qpotential
, 2008
"... We introduce the qpotential as an extension of the BenedettoFickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols ..."
Abstract

Cited by 9 (4 self)
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We introduce the qpotential as an extension of the BenedettoFickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call qfundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko’s and Fulton’s theory on sums of hermitian operators.
Author manuscript, published in "SAMPTA'09, Marseille: France (2009)" Gradient descent of the frame potential
, 2010
"... Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential ..."
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Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential. We consider this minimization problem from a numerical perspective. In particular, we discuss how by descending the gradient of the frame potential, one, under certain conditions, is guaranteed to produce a sequence of unit norm frames which converge to a unit norm tight frame at a geometric rate. This makes the gradient descent of the frame potential a viable method for numerically constructing unit norm tight frames. 1.
Acta Applicandae Mathematicae manuscript No. (will be inserted by the editor) Minimizing Fusion Frame Potential
"... Abstract Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as t ..."
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Abstract Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the fusion frame’s subspaces are large in number but small in dimension compared to the dimension of the underlying space, then fusion frames will always exists, with each being a minimizer of the fusion frame potential. Key words frames, fusion, potential, tight
Minimizers of the fusion frame potential
, 2008
"... In this paper we study the fusion frame potential, that is a generalization of the BenedettoFickus (vectorial) frame potential to the finitedimensional fusion frame setting. Local and global minimizers of this potential are studied, when we restrict it to a suitable set of fusion frames. These min ..."
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In this paper we study the fusion frame potential, that is a generalization of the BenedettoFickus (vectorial) frame potential to the finitedimensional fusion frame setting. Local and global minimizers of this potential are studied, when we restrict it to a suitable set of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. Thus, we exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the socalled HornKlyachko’s compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, varying the sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different
Frame Theory: A Complete Introduction to Overcompleteness. Contents
"... Abstract. In this chapter we survey two topics that have recently been investigated in frame theory. First, we give an overview of the class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in such a way that the resulting frames are tight. This ..."
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Abstract. In this chapter we survey two topics that have recently been investigated in frame theory. First, we give an overview of the class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in such a way that the resulting frames are tight. This process can be thought of as a preconditioning method for finite frames. In particular, we: (1) describe the class of scalable frames; (2) formulate various equivalent characterizations of scalable frames, and relate the scalability problem to the Fritz John ellipsoid theorem. Next, we discuss some results on a probabilistic interpretation of frames. In this setting, we: (4) define probabilistic frames as a generalization of frames and as a subclass of continuous frames; (5) review the properties of certain potential functions whose minimizers are frames with certain optimality properties. The chapter