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Optimal frame completions

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Abstract

Given a finite sequence of vectors \(\mathcal F_{0}\) in ℂd we describe the spectral and geometrical structure of optimal frame completions of \(\mathcal F_{0}\) obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to a general convex potential. In particular, our analysis includes the so-called Mean Square Error (MSE) and the Benedetto-Fickus’ frame potential. On a first step, we reduce the problem of finding the optimal completions to the computation of the minimum of a convex function in a convex compact polytope in ℝd. As a second step, we show that there exists a finite set (that can be explicitly computed in terms of a finite step algorithm that depends on \(\mathcal F_{0}\) and the sequence of prescribed norms) such that the optimal frame completions with respect to a given convex potential can be described in terms of a distinguished element of this set. As a byproduct we characterize the cases of equality in Lidskii’s inequality from matrix theory.

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Authors and Affiliations

  1. Departamento de Matemática, FCE-UNLP, La Plata, Bs. As., Argentina and IAM-CONICET, Buenos Aires, Argentina

    Pedro G. Massey, Mariano A. Ruiz & Demetrio Stojanoff

Authors
  1. Pedro G. Massey
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  2. Mariano A. Ruiz
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  3. Demetrio Stojanoff
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Correspondence to Mariano A. Ruiz.

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Communicated by: Yang Wang

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Massey, P.G., Ruiz, M.A. & Stojanoff, D. Optimal frame completions. Adv Comput Math 40, 1011–1042 (2014). https://doi.org/10.1007/s10444-013-9339-7

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  • DOI: https://doi.org/10.1007/s10444-013-9339-7

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