Polytopes of Eigensteps of Finite Equal Norm Tight Frames

Abstract

Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand–Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of facets of the polytope. While studying the polytope, we find two affine isomorphisms and show how they relate to operations on the underlying frames.

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Correspondence to Tim Haga.

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Editor in Charge: Kenneth Clarkson

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Haga, T., Pegel, C. Polytopes of Eigensteps of Finite Equal Norm Tight Frames. Discrete Comput Geom 56, 727–742 (2016). https://doi.org/10.1007/s00454-016-9799-x

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Keywords

  • Hilbert space frames
  • Polytopes
  • Convex geometry
  • Combinatorics

Mathematics Subject Classification

  • 52B05
  • 42C15