## 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.

Original language | English |
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Pages (from-to) | 721-756 |

Number of pages | 36 |

Journal | Advances in Computational Mathematics |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2016 |

## Keywords

- Erasure resilience
- Erasures
- Frame decompositions
- Frames
- Prime frames
- Tight frames