On structural decompositions of finite frames
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Author(s)
Stokols, Logan
Theobold, Allison
Chan, Alice Z.-Y.
Narayan, Sivaram K.
Copenhaver, Martin Steven
Date Issued
October 2015
Journal
Advances in Computational Mathematics
Publisher
Springer US
Citation
Chan, Alice Z.-Y., Martin S. Copenhaver, Sivaram K. Narayan, Logan Stokols, and Allison Theobold. “On Structural Decompositions of Finite Frames.” Adv Comput Math 42, no. 3 (October 30, 2015): 721–756.
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Author's final manuscript
Abstract
A frame in an n-dimensional Hilbert space Hn is a possibly redundant collection of vectors {f[subscript i]}[subscript i∈I] that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f[subscript i]}[subscript i∈I] is said to be scalable if there exist nonnegative scalars {c[subscript i]}[subscript i∈I] such that {c[subscript i]f[subscript i]}[subscript 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[subscript i]}[subscript 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[subscript j]}[subscript j∈J] is a tight frame for Hn. 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.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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DOI of Published Version
https://doi.org/10.1007/s10444-015-9440-1
