On structural decompositions of finite frames

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_if_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.