Robustness and surgery of frames

Narayan, Seema; Radzwion, Eileen; Rimer, Sara P.; Tomasino, Rachael; Wolfe, Jennifer; Zimmer, Andrew

doi:10.1016/j.laa.2010.11.049

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…The next theorem describes when (p, q)-length surgery resulting in a tight frame set is possible. This result is a restatement of Theorem 4.9 in [14] which gives a condition for (0, q)-length surgery resulting in a tight frame set; the proof given here is different from that of [14]. In the Appendix we provide an algorithm for computing the support of the null space of a matrix A.…”

Section: Length Surgerymentioning

confidence: 86%

“…(1) There exists a unit-norm tight subframe {f j } j∈I ⊆ F where I = {i 1 , · · · , i m }. Next, we provide a generalization of the necessary condition for (p, q)surgery presented in [14].…”

Section: Definition 32 ([7])mentioning

confidence: 99%

“…In [14], the authors characterize frames in R n that are robust to k erasures and a necessary and sufficient condition is given for performing (r, k)-surgery on unit-norm tight frames in R 2 . In this paper, we present a method to determine the maximum robustness of any given frame, and generalize results on surgery from [14]. We also answer the question of when length surgery resulting in a tight frame set for H n is possible.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Maximum robustness and surgery of frames in finite dimensions

Copenhaver

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et al. 2013

Linear Algebra and its Applications

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Section: Length Surgerymentioning

confidence: 86%

Section: Definition 32 ([7])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximum robustness and surgery of frames in finite dimensions

Copenhaver

Kim

Logan

et al. 2013

Linear Algebra and its Applications

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Uniform Excess Frames in Hilbert Spaces

2018

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On structural decompositions of finite frames

Chan

Copenhaver²,

Narayan

et al. 2015

Adv Comput Math

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A frame in an n-dimensional Hilbert space Hn is a possibly redundant collection of vectors {fi}i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {fi}i∈I is said to be scalable if there exist nonnegative scalars {ci}i∈I such that {cifi}i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {fi}i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J ⊆ I is in the factor poset if and only if {fj }j∈J is a tight frame for Hn. A similar definition is given for the scalability poset of a frame. We prove conditions which factor posets satisfy and use these to study the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P . We determine a necessary condition for solving the inverse factor poset problem in Hn which is also sufficient for H2. We describe how factor poset structure of frames is preserved under orthogonal projections. We also consider the enumeration of the number of possible factor posets and bounds on the size of factors posets. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset.

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Robustness and surgery of frames

Cited by 4 publications

References 7 publications

Maximum robustness and surgery of frames in finite dimensions

Maximum robustness and surgery of frames in finite dimensions

Uniform Excess Frames in Hilbert Spaces

On structural decompositions of finite frames

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