Minimization of convex functionals over frame operators

Massey, Pedro; Ruiz, Mariano Andrés

doi:10.1007/s10444-008-9092-5

Cited by 30 publications

(111 citation statements)

References 17 publications

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“…Below, we show that if two sequences of vectors are close with respect to (14), then the projections they generate are necessarily close with respect to (13). However, the appropriate converse statement is more complicated.…”

Section: Proposition 1 For Any Sequence Of Positive Integersmentioning

confidence: 91%

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Minimizing Fusion Frame Potential

Casazza

Fickus

2008

Acta Appl Math

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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 desired number of fusion frame subspaces is large, and if the desired dimension of these subspaces is small compared to the dimension of the underlying space, then a tight fusion frame of those dimensions will necessarily exist, being a minimizer of the fusion frame potential.

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Section: Proposition 1 For Any Sequence Of Positive Integersmentioning

confidence: 91%

“…It has been used to characterize tight filter bank frames [10,11]. Recently, generalized frame potentials have been a subject of interest [2,14].…”

Section: Introductionmentioning

confidence: 99%

Minimizing Fusion Frame Potential

Casazza

Fickus

2008

Acta Appl Math

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“…Hence, the matrix nearness problem in Eq. (20) coincides with the problem of computing global minimizers on Θ (N , S , a) in T d (a). Similarly, the study of local minimizers of the matrix nearness problem corresponds to the study of local minimizers of Θ (N , S , a) .…”

Section: Generalized Frame Operator Distancesmentioning

confidence: 97%

Local Lidskii's theorems for unitarily invariant norms

Massey

Rios

Stojanoff

2018

Linear Algebra and its Applications

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Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.

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“…It turns out that both the MSE and the FP lie within the class of convex potentials introduced in [23]. It is shown in [23] that there are solutions F op to the frame design problem which are structural in the sense that they are minimizers of every convex potential (e.g. MSE and FP) among frames with squared norms prescribed by α.…”

Section: Introductionmentioning

confidence: 99%

Optimal frame designs for multitasking devices with weight restrictions

et al. 2020

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Let d = (d j ) j∈Im ∈ N m be a finite sequence (of dimensions) and α = (α i ) i∈In be a sequence of positive numbers (weights), where I k = {1, . . . , k} for k ∈ N. We introduce the (α , d)-designs i.e., families Φ = (F j ) j∈Im such that F j = {f ij } i∈In is a frame for C dj , j ∈ I m , and such that the sequence of non-negative numbers ( f ij 2 ) j∈Im forms a partition of α i , i ∈ I n . We show, by means of a finite-step algorithm, that there exist (α , d)-designs Φ op = (F op j ) that are universally optimal; that is, for every convex function ϕ : [0, ∞) → [0, ∞) then Φ op minimizes the joint convex potential induced by ϕ among (α , d)-designs, namely j∈Im

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Minimization of convex functionals over frame operators

Cited by 30 publications

References 17 publications

Minimizing Fusion Frame Potential

Minimizing Fusion Frame Potential

Local Lidskii's theorems for unitarily invariant norms

Optimal frame designs for multitasking devices with weight restrictions

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