The Paulsen Problem in operator theory

Casazza, Peter G.; Cahill, Jameson

doi:10.7153/oam-07-06

Cited by 29 publications

(44 citation statements)

References 20 publications

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“…We have proved that the bound in the Paulsen problem is independent of the number of vectors, and through the reduction in [10] we have also proved the projection conjecture with the same bound. We hope that our results and techniques will find applications in other problems, in particular the dynamical system and the new method in proving capacity lower bound.…”

Section: Conclusion and Discussionmentioning

confidence: 55%

The Paulsen problem, continuous operator scaling, and smoothed analysis

Kwok

Lau

Lee

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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The Paulsen problem is a basic open problem in operator theory: Given vectors u 1 , . . . , u n ∈ R d that are ǫ-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors v 1 , . . . , v n ∈ R d that exactly satisfy the Parseval's condition and the equal norm condition? Given u 1 , . . . , u n , the squared distance (to the set of exact solutions) is defined as inf v * University of Waterloo.

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Section: Conclusion and Discussionmentioning

confidence: 55%

The Paulsen problem, continuous operator scaling, and smoothed analysis

Kwok

Lau

Lee

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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“…One could answer this question by seeking a metric that best describes the distance of a frame to the set of tight frames. This is similar to the Paulsen problem [3], in that, after we have solved one of the formulations above, we produce a scaling and subsequent new frame and wish to determine the distance of this new frame to the canonical Parseval frame associated to our original frame. In [8], the question of distance to Parseval frames was generalized to include frames that could be made tight with a diagonal scaling, resulting in the distance between a frame and the set of scalable frames:…”

Section: But the Feasibility Conditionmentioning

confidence: 99%

“…Example 3. Consider the graph G complete graph on four nodes with the edge (3,4) removed. Then G was rescaled and conditioned via GraphCondition; both graphs are shown in Figure 2.…”

Section: Minimizing Condition Number Of Graphsmentioning

confidence: 99%

Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling

Bălan

Mathew

Clark

et al. 2017

Frames and Other Bases in Abstract and Function Spaces

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A frame is scalable if each of its vectors can be rescaled in such a way that the resulting set becomes a Parseval frame. In this paper, we consider four different optimization problems for determining if a frame is scalable. We offer some algorithms to solve these problems. We then apply and extend our methods to the problem of reweighing (finite) graph so as to minimize the condition number of the resulting Laplacian.

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“…In Section 2, and for the sake of completeness, we give a new and simple proof of this result and we refer to [2,3,4] for related results.…”

Section: Introductionmentioning

confidence: 99%

Optimal Properties of the Canonical Tight Probabilistic Frame

Cheng

Okoudjou

2019

Numerical Functional Analysis and Optimization

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A probabilistic frame is a Borel probability measure with finite second moment whose support spans R d . A Parseval probabilistic frame is one for which the associated matrix of second moment is the identity matrix in R d . Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the 2−Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabilistic frame can be approximated in the 2−Wasserstein metric with (compactly supported) finite frames whose bounds can be controlled. On the other hand we establish some fine continuity properties of the function that maps a probabilistic frame to its canonical Parseval probabilistic frame. Our results generalize similar ones for finite frames and their associated Parseval frames.

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The Paulsen Problem in operator theory

Cited by 29 publications

References 20 publications

The Paulsen problem, continuous operator scaling, and smoothed analysis

The Paulsen problem, continuous operator scaling, and smoothed analysis

Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling

Optimal Properties of the Canonical Tight Probabilistic Frame

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