Diagram vectors and tight frame scaling in finite dimensions

Abstract. For a unit-norm frame F = { f i } k i=1 in R n , a scaling is a vector c = (c(1),... ,c(kis a Parseval frame in R n . If such a scaling exists, F is said to be scalable. A scaling c is a minimal scaling if { f i : c(i) > 0} has no proper scalable subframe. In this paper, we provide an algorithm to find all possible contact points for the John's decomposition of the identity by applying the b-rule algorithm to a linear system which is associated with a scalable frame. We also give an estimate of the number of minimal scalings of a scalable frame. We provide a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings c = (c(1),... ,c(k)) ∈ R k >0 of F have the same structural property. That is, the collections of all tight subframes of strictly scaled frames are the same up to a permutation of the frame elements. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame. Mathematics subject classification (2010): Primary 42C15, 05B20, 15A03.

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“…We present a few results that will be used later in the paper. For basic facts about scalable frames we refer to [15,4,14,9,8,7].…”

Section: Preliminariesmentioning

confidence: 99%

“…In the last couple of years the theme of scalable frames have been developed as a method of constructing tight frames from general frames by manipulating the length of frame vectors. Scalable frames maintain erasure resilience and sparse expansion properties of frames [15,4,14,9,8]. In this paper, we further explore scalable frames.…”

Section: Introductionmentioning

confidence: 99%

Minimal scalings and structural properties of scalable frames

Chan

Domagalski

Kim³

et al. 2017

Operators and Matrices

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“…The diagram vectors of a tight frame in R 2 can be placed tip-to-tail to demonstrate that they sum to zero. In [13] the notion of diagram vectors was extended to R n and C n . (1) .…”

Section: Preliminariesmentioning

confidence: 99%

“…Let P be the polytope consisting of all scalings of a given frame in R n . Using the interpretation of scalings in [13], this polytope is the intersection of the vector space ker( G), the null space of the Gramian of the diagram vectors for the frame, with the convex region R…”

Section: Scalable Framesmentioning

confidence: 99%

On structural decompositions of finite frames

Chan

Copenhaver²,

Narayan

et al. 2015

Adv Comput Math

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A frame in an n-dimensional Hilbert space Hn is a possibly redundant collection of vectors {fi}i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {fi}i∈I is said to be scalable if there exist nonnegative scalars {ci}i∈I such that {cifi}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 {fi}i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J ⊆ I is in the factor poset if and only if {fj }j∈J is a tight frame for Hn. A similar definition is given for the scalability poset of a frame. We prove conditions which factor posets satisfy and use these to study the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P . We determine a necessary condition for solving the inverse factor poset problem in Hn which is also sufficient for H2. We describe how factor poset structure of frames is preserved under orthogonal projections. We also consider the enumeration of the number of possible factor posets and bounds on the size of factors posets. 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.

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“…The notion of scalable frame has been investigated in recent years [10,16,4,15], where the focus was more on characterizing frames whose vectors can be rescaled resulting in a tight frame. For completeness, we recall that a set of vectors…”

Section: Introductionmentioning

confidence: 99%

Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling

Bălan

Mathew

Clark

et al. 2017

Frames and Other Bases in Abstract and Function Spaces

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A frame is scalable if each of its vectors can be rescaled in such a way that the resulting set becomes a Parseval frame. In this paper, we consider four different optimization problems for determining if a frame is scalable. We offer some algorithms to solve these problems. We then apply and extend our methods to the problem of reweighing (finite) graph so as to minimize the condition number of the resulting Laplacian.

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Diagram vectors and tight frame scaling in finite dimensions

Cited by 26 publications

References 9 publications

Minimal scalings and structural properties of scalable frames

Minimal scalings and structural properties of scalable frames

On structural decompositions of finite frames

Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling

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