Minimizing Fusion Frame Potential

Casazza, Peter G.; Fickus, Matthew

doi:10.1007/s10440-008-9377-1

Cited by 42 publications

(56 citation statements)

References 15 publications

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“…Analogous as in [, Proposition 1] we have: Proposition If

W \in {scriptS}_{K} false(boldL false)

, then

F F P_{boldw} false(boldW false) \geq \frac{1}{d} {((), \sum_{k = 1}^{K} w_{k}^{2} L_{k})}^{2},

with equality holding in if and only if

(W, w)

is a tight fusion frame for

F^{d}

.…”

Section: Preliminariesmentioning

confidence: 79%

“…Following we define the fusion frame potential

F F P_{boldw} : {scriptS}_{K} false(boldL false) \to R

F F P_{boldw} false(boldW false) = tr (S_{boldW, boldw}^{2}) .

We will consider local minimizers of

F F P_{boldw} false(boldW false)

in the topology of

{scriptS}_{K} false(boldL false)

induced by the distance

d (W, V)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

boldW

be a local minimizer of

F F P_{boldw}

. Theorem 4 in says that there exists a generating family

{true{{false{f_{k, l} false}}_{l = 1}^{L_{k}} true}}_{k = 1}^{K}

{true{π_{W_{k}} true}}_{k = 1}^{K}

(i.e. for each

k = 1, \dots, K

{true{f_{k, l} true}}_{l = 1}^{L_{k}}

is an orthonormal basis for

W_{k}

) that consists of eigenvectors of

S_{boldW, boldw}

, and that the elements of this family which are in

E_{j}

are a

λ_{j}

‐tight frame for

E_{j}

.…”

Section: The Orthogonal Projections As Eigenoperators Of the Fusion Fmentioning

confidence: 99%

See 2 more Smart Citations

On the minimizers of the fusion frame potential

Heineken

Llarena

Morillas

2017

Mathematische Nachrichten

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We study the minimizers of the fusion frame potential in the case that both the weights and the dimensions of the subspaces are fixed and not necessarily equal. Using a concept of irregularity we provide a description of the local (that are also global) minimizers which projections are eigenoperators of the fusion frame operator. This result will be related to the existence of tight fusion frames. In this way we generalize results known for the classical vector frame potential.

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“…Analogous as in [, Proposition 1] we have: Proposition If

W \in {scriptS}_{K} false(boldL false)

, then

F F P_{boldw} false(boldW false) \geq \frac{1}{d} {((), \sum_{k = 1}^{K} w_{k}^{2} L_{k})}^{2},

with equality holding in if and only if

(W, w)

is a tight fusion frame for

F^{d}

.…”

Section: Preliminariesmentioning

confidence: 79%

“…Following we define the fusion frame potential

F F P_{boldw} : {scriptS}_{K} false(boldL false) \to R

F F P_{boldw} false(boldW false) = tr (S_{boldW, boldw}^{2}) .

We will consider local minimizers of

F F P_{boldw} false(boldW false)

in the topology of

{scriptS}_{K} false(boldL false)

induced by the distance

d (W, V)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

boldW

be a local minimizer of

F F P_{boldw}

. Theorem 4 in says that there exists a generating family

{true{{false{f_{k, l} false}}_{l = 1}^{L_{k}} true}}_{k = 1}^{K}

{true{π_{W_{k}} true}}_{k = 1}^{K}

(i.e. for each

k = 1, \dots, K

{true{f_{k, l} true}}_{l = 1}^{L_{k}}

is an orthonormal basis for

W_{k}

) that consists of eigenvectors of

S_{boldW, boldw}

, and that the elements of this family which are in

E_{j}

are a

λ_{j}

‐tight frame for

E_{j}

.…”

Section: The Orthogonal Projections As Eigenoperators Of the Fusion Fmentioning

confidence: 99%

See 1 more Smart Citation

On the minimizers of the fusion frame potential

Heineken

Llarena

Morillas

2017

Mathematische Nachrichten

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show abstract

“…Note that the frame potential of { f i } M i=1 is the trace of the square of the frame operator S. Considering k∈K P k instead of the frame operator S, the fusion frame potential of {W k } k∈K is defined as [17] FFP({W k } k∈K ) := tr k∈K P k 2 = i∈K j∈K tr(P i P j ).…”

Section: Tight Frames For Whichmentioning

confidence: 99%

Welch bounds for cross correlation of subspaces and generalizations

Datta

2015

Linear and Multilinear Algebra

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Lower bounds on the maximal cross correlation between vectors in a set were first given by Welch and then studied by several others. In this work, this is extended to obtaining lower bounds on the maximal cross correlation between subspaces of a given Hilbert space. Two different notions of cross correlation among spaces have been considered. The study of such bounds is done in terms of fusion frames, including generalized fusion frames. In addition, results on the expectation of the cross correlation among random vectors have been obtained.

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“…Two recent papers 7,21 have introduced and studied fusion frame potentials to address the existence of fusion frames, but with limited success. The problem here is that minimizers of the fusion frame potential are not necessarily tight fusion frames.…”

Section: Introductionmentioning

confidence: 99%

Constructing fusion frames with desired parameters

et al. 2009

Self Cite

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A fusion frame is a frame-like collection of subspaces in a Hilbert space. It generalizes the concept of a frame system for signal representation. In this paper, we study the existence and construction of fusion frames. We first introduce two general methods, namely the spatial complement and the Naimark complement, for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given M, N, m ∈ N and {λ j } M j=1 , does there exist a fusion frame in R M with N subspaces of dimension m for which {λ j } M j=1 are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m ∈ N and {λ j } M j=1 . Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become tight.

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Minimizing Fusion Frame Potential

Cited by 42 publications

References 15 publications

On the minimizers of the fusion frame potential

On the minimizers of the fusion frame potential

Welch bounds for cross correlation of subspaces and generalizations

Constructing fusion frames with desired parameters

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