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On structural decompositions of finite frames

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Abstract

A frame in an n-dimensional Hilbert space H n is a possibly redundant collection of vectors {f i } i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f i } i∈I is said to be scalable if there exist nonnegative scalars {c i } i∈I such that {c i f i } 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 {f i } i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J⊆I is in the factor poset iff {f j } j∈J is a tight frame for H n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. 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; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.

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Authors and Affiliations

  1. Department of Mathematics, UC, San Diego, CA, USA

    Alice Z.-Y. Chan

  2. Operations Research Center, MIT, E40-149, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Martin S. Copenhaver

  3. Department of Mathematics, Central Michigan University, Mt Pleasant, MI, USA

    Sivaram K. Narayan

  4. Department of Mathematics, UT, Austin, TX, USA

    Logan Stokols

  5. Department of Statistics, Montana State University, Bozeman, MT, USA

    Allison Theobold

Authors
  1. Alice Z.-Y. Chan
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  2. Martin S. Copenhaver
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  3. Sivaram K. Narayan
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  4. Logan Stokols
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  5. Allison Theobold
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Corresponding author

Correspondence to Martin S. Copenhaver.

Additional information

Communicated by: Yang Wang

Research supported by NSF-REU Grant DMS 11-56890.

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Chan, A.ZY., Copenhaver, M.S., Narayan, S.K. et al. On structural decompositions of finite frames. Adv Comput Math 42, 721–756 (2016). https://doi.org/10.1007/s10444-015-9440-1

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  • DOI: https://doi.org/10.1007/s10444-015-9440-1

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