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Non-Orthogonal Fusion Frames

Jameson Cahill a Peter G. Casazzab and Shidong Li c

a,bDepartment of Mathematics, University of Missouri, Columbia Missouri 65211;cDepartment of Mathematics, San Francisco State University, San Framcisco, DA 94132

ABSTRACT

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusionframe applications in distributed systems is reflected in the efficiency of the end fusion process. This requiresthe inversion of the fusion frame operator which is difficult or impossible in practice. What we want is for thefusion frame operator to be the identity. But in most applications, especially to sensor networks, this almostnever occurs. We will solve this problem by introducing the notion of non-orthogonal fusion frames which havethe property that in most cases we can turn a family of subspaces of a Hilbert space into a non-orthogonal fusionframe which has a fusion frame operator which is the identity.

Keywords: Sensor Networks, Distributed Processing, Fusion Frames and Non-orthogonal Fusion Frames

1. INTRODUCTION

Fusion frames were introduced in11 (under the name frames of subspaces), were further developed in,12 and havealready turned into an industry (see www.fusionframes.org; www.framerc.org). Recent developments include ap-plications to sensor networks,13 filter bank fusion frames,14 applications to coding theory,1 compressed sensing,2

construction methods,5–9 sparsity for fusion frames,10 and frame potentials and fusion frames.17 Until now, mostof the work on fusion frames has centered on developing their basic properties and on constructing fusion frameswith specific properties. We now know that there are very few tight fusion frames without weights.

A major stumbling block for the application of fusion frame theory is that in practice, we generally do not getto construct the fusion frame, but instead it is thrust upon us by the application. In a majority of fusion frameapplications, such as in sensor network data processing, each sensor spans a fixed subspace Wi of H generatedby the spatial reversal and the translates of the sensor’s impulse response function.15, 16 There is no opportunitythen for subspace transformation, manipulation and/or selection. As a result, the fusion frame operator SW isalways non-sparse with an extremely high probability. The lack of sparsity of SW is a significant hinderancein computing SW and its inverse, which is necessary to apply the theory. So the central issue in the effectiveapplication of fusion frames is to have sparsity for the fusion frame operator - preferably for it to be a diagonaloperator.

We will address these problems by introducing non-orthogonal fusion frames. The main idea is to replace theorthogonal projections onto the fusion frame subspaces by non-orthogonal projections. We will see that this willallow us to turn most families of subspaces of a Hilbert space into a non-rothogonal fusion frame which has theidentity as its fusion frame operator.

2. FUSION FRAMES

In contrast to frame theory where a signal is represented by a collection of scalars which measure the amplitudesof the projections of the signal onto the frame vectors, in fusion frame theory the signal is represented by acollection of vectors which are the projections of the signal onto the fusion frame subspaces. This allows, forexample, a two-stage data processing procedure where the projections are locally processed data which are latercombined to reconstruct the signal.

Send correspondence to P.G. Casazza: E-mail: [email protected],JC: E-mail: [email protected]: [email protected]

Wavelets and Sparsity XIV, edited by Manos Papadakis, Dimitri Van De Ville, Vivek K. Goyal, Proc. of SPIE Vol. 8138, 81380C · © 2011 SPIE · CCC code: 0277-786X/11/$18

doi: 10.1117/12.892015

Proc. of SPIE Vol. 8138 81380C-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/07/2013 Terms of Use: http://spiedl.org/terms

Definition 2.1. Given a Hilbert space H and a family of closed subspaces {Wi}i∈I with associated positiveweights {vi}i∈I , the family W = {Wi, vi}i∈I is a fusion frame with fusion frame bounds 0 < A ≤ B < ∞ if

A‖f‖2 ≤∑i∈I

v2i ‖πif‖2 ≤ B‖f‖2, for all f ∈ H,

where πi is the orthogonal projection onto Wi, for all i ∈ I.

The fusion frame is tight if A = B and it is Parseval if A = B = 1. In particular, the fusion frame is tight ifand only if

A · Id =∑i∈I

v2i πi.

If vi = 1 for all i ∈ I, we will write the fusion frame as W = {Wi}i∈I . The analysis operator of the fusion frameis

T : H →(∑

i∈I

⊕Wi

)�2

,

whereT (f) = {viπi(f)}i∈I .

The synthesis operator is T ∗ and satisfies

T ∗(x) =∑i∈I

xi, where x =∑i∈I

⊕xi ∈(∑

i∈I

⊕Wi

)�2

.

The fusion frame operator is the positive, self-adjoint, invertible operator SW = T ∗T : H → H given by

SW (f) =∑i∈I

v2i πi(f).

The fusion frame operator satisfies A · Id ≤ SW ≤ B · Id. The decomposition of a signal f ∈ H with respect to afusion frame W = {Wi, vi}i∈I is given by the fusion frame measurements {viπi(f)}i∈I . This uniquely identifiesthe signal which can be reconstructed via

f =∑i∈I

viS−1W (viπi(f)).

A fundamental result in fusion frame theory11 is

Theorem 2.2. Let {Wi}i∈I be subspaces of H and let {vi}i∈I be positive constants. For each i ∈ I let {eij}Ji

j=1

be an orthonormal basis for Wi. The following are equivalent:

(1) {vieij}i∈I,j∈Ji is a frame for H with frame bounds A, B.

(2) {Wi, vi}i∈I is a fusion frame for H with fusion frame bounds A, B.

So {Wivi}i∈I is a Parseval fusion frame if and only if {vieij}i∈I,j∈Ji is a Parseval frame for H. Moreover,in this case, SW = Id.

A complete characterization of the weights and dimensions of the subspaces for which fusion frames existcan be found in.17 This is a deep piece of mathematics. It has the disadvantage that it is an existence proofwithout any algorithms for its implementation. In the next section we will look at algorithms for constructingfusion frames.



3. FUSION FRAME CONSTRUCTIONS

Much effort has been put into constructing fusion frames. A major breakthrough occurred recently with theintroduction of spectral tetris which is an algorithm for constructing tight frames and tight fusion frames.? Herewe will be working with weights vi = 1 and equi-dimensional subspaces. We say that a sequence of projections{πk}K

k=1 of N × N orthogonal projection matrices of rank L is a (K, L, N)-tight fusion frame if there exists anA > 0 so that

A · Id =K∑

k=1

πk.

Now we can state the main results from.?

Theorem 3.1. Let K, L, N ∈ N satisfy L ≤ N . If L divides N then (K, L, N)-tight fusion frames exist if andonly if K ≥ N

L . Otherwise, if 2L < N and L does not divide N then

(1) If (K, L, N)-tight fusion frames exist then K ≥ NL + 1.

(2) If K ≥ NL + 2 then (K, L, N)-tight fusion frames exist.

Using this theorem, in? the existence of (K, L, N)-tight fusion frames is completely resolved.

These results were generalized in?, 5 to construct fusion frames with prescribed eigenvalues for the fusionframe operator. First we need a definition.

Definition 3.2. A fusion frame W = {Wi}i∈I is k-sparse with respect to an orthonormal basis {ej}Mj=1 for HM

if each subspace Wi is spanned by an orthonormal basis {eij}mi

j=1 so that for each j = 1, 2, . . . , mi, eij ∈ span{e�}�∈J with |J | ≤ k.

Theorem 3.3. Suppose real numbers λ1 ≥ λ2 ≥ λM > 0 are given and satisfy:

(1)∑M

j=1 λj = N .

(2) λM ≥ 2.

(3) If j0 is the first integer in {1, 2, . . . , M} for which λj0 is not an integer, then �λj0� ≤ N − 3.

Then there is a fusion frame W = {Wi}Ni=1 whose fusion frame operator has {λj}M

j=1. Moreover, this fusionframe is 2-sparse.

A further generalization of spectral tetris which gives the first algorithm for constructing tight fusion frameswith 1 < A = B < 2 was given in.9 This paper also gives the first constructions of tight fusion frames withprescribed fusion frame operator and at the same time prescribing the dimensions of the subspaces. To statethis result we need another definition.

Definition 3.4. Given a = (an)Nn=1 ∈ RN , denote by a↓ ∈ RN the vector obtained by rearranging the coordinates

of a in decreasing order. If (an)Nn=1, (bn)N

n=1 ∈ RN , we say (an)Nn=1 majorizes (bn)N

n=1, denoted by (an) (bn),if∑m

n=1 a↓n ≥ ∑m

n=1 b↓n for all m = 1, . . . , N − 1 and∑N

n=1 an =∑N

n=1 bn. We will also use the notion ofmajorization between tuples of different length, by agreeing to add zero entries to the shorter tuple, in order tohave tuples of the same length.

One main result from? is

Theorem 3.5. Let M ≥ 2N be natural numbers and (dn)Dn=1 ∈ ND such that

∑Dn=1 dn = M . Let (Vn)t

n=1

be the reference fusion frame for (λn)Nn=1 = (M

N , . . . , MN ). Then there exists a tight spectral tetris fusion frame

(Wn)Dn=1 for RN with dim Wn = dn for n = 1, . . . , D if and only if (dimVn) (dn).

The other main result from? is too technical to describe here. The idea is to construct fusion frames withprescribed dimensions of the subspaces and prescribed eigenvalues for the fusion frame operator. The theoremdoes not classify all cases when this can be done. Instead, it classifies all cases where this can be done usingspectral tetris.



4. NON-ORTHOGONAL FUSION FRAMESWe will now replace the orthogonal projections in the definition of a fusion frame by non-orthogonal projectionsto get non-orthogonal fusion frames. A non-orthogonal projection P onto a subspace W of H is a linear operatorP : H → W which is onto and satisfies P 2 = P . In this case, P ∗ is also a non-orthogonal projection onto N(P )⊥,W⊥ = N(P ∗) and N(P ) = {f ∈ H : Pf = 0}. It follows that N(P ) ∩ W = {0} and so N(P ) ⊕ W = H. Nowwe are ready for the definition of a non-orthogonal fusion frame.

Definition 4.1. Let {Wi}Mi=1 be a family of closed subspaces of HN which span HN and let {vi}M

i=1 be a familyof positive weights. For each i = 1, 2, . . . , M , let Pi be a (possibly) non-orthogonal projection onto Wi. Then{Pi, vi}M

i=1 is a non-orthogonal fusion frame for HN if there are constants 0 < A ≤ B < ∞ satisfying

A‖f‖2 ≤M∑i=1

v2i ‖Pi(f)‖2 ≤ B‖f‖2, for all f ∈ HN .

The analysis operator for the fusion frame is the operator

T : HN →(

M∑i=1

⊕Wi

)�M2

,

given byT (f) = {viPi(f)}M

i=1,

and the synthesis operator is

T ∗({fi}Mi=1) =

M∑i=1

vifi, for all {fi}Mi=1 ∈

(M∑i=1

⊕Wi

)M

�2

.

The non-orthogonal fusion frame operator is

SW = T ∗T =M∑i=1

v2i P ∗

i Pi,

and satisfies A · Id ≤ SW ≤ B · Id.

The basic properties of non-orthogonal projections are summarized in the following theorem.Theorem 4.2. Let P be a projection on HN with W = Im P , W ∗ = (ker P )⊥ = [(I − P )(HN )]⊥. Then:

(1) W ∗ = Im P ∗P .(2) The restriction of P to W ∗ is an invertible operator mapping W ∗ onto W .

(3) If λ is a non-zero eigenvalue of P ∗P then λ ≥ 1. Moreover, λ = 1 if and only if the correspondingeigenvector is in W ∩ W ∗.

(4) Let {ei}ki=1 be an orthonormal basis of W ∗ consisting of eigenvectors of P ∗P with corresponding nonzero

eigenvalues {λi}ki=1. Then {Pei}k

i=1 is an orthogonal basis for W .Proof. (1) and (2) are clear.

(3) Let W = im P , W ∗ = [kerP ]⊥. Then W ∗ = im P ∗P , so all eigenvectors of P ∗P corresponding tononzero eigenvalues are in W ∗. Let f ∈ W ∗ and write Pf = f + (P − I)f . Since f ⊥ (I − P )f ,

‖Pf‖2 = ‖f‖2 + ‖(P − I)f‖2 ≥ ‖f‖2. (1)

By the same argument we get ‖P ∗Pf‖ ≥ ‖Pf‖ ≥ ‖f‖. Therefore, if P ∗Pf = λf we have that λ ≥ 1.Finally, by Equation 1, λ = 1 if and only if (I − P )f = 0. i.e. Pf = f ∈ W ∩ W ∗.

(4) We compute for any i �= j,

〈Pei, P ej〉 = 〈P ∗Pei, ej〉 = λi〈ei, ej〉 = 0.



5. CLASSIFYING POSITIVE SELF-ADJOINT OPERATORS BY PROJECTIONS

An important result from3 involves classifying positive self-adjoint operators in terms of non-orthogonal projec-tions.

Let T : RN → RN be a positive, self adjoint, linear operator. We would like to classify the set

Ω(T ) = {P : P 2 = P, P ∗P = T }.The spectral theorem tells us that T =

∑Mi=1 λiπi where the λi are the eigenvalues of T and πi is the orthogonal

projection onto the one dimensional span of the ith eigenvector of T . Therefore P ∈ Ω(T ) if and only if P ∗Phas the same eigenvalues and eigenvectors as T . Also, note that if P ∈ Ω(T ) then ker(P ) = im(T )⊥. Since aprojection is uniquely determined by its kernel and its image we can make the association

Ω(T ) = {W ⊆ RN : there exists P, P 2 = P, im(P ) = W, P ∗P = T }.

We observe that the only projection of rank N is Id, so we can assume rank(T ) < N , otherwise Ω(T ) = ∅.Theorem 5.1. Let T : RN → RN be a positive, self-adjoint operator of rank k ≤ N

2 . Let {λi}ki=1 be the

nonzero eigenvalues of T and suppose λi ≥ 1 for i = 1, ..., k and suppose {ei}ki=1 is an orthonormal basis of

im(T ) consisting of eigenvectors of T . Then

Ω(T ) = W = span { 1√λi

ei +√

λi − 1λi

ei+k} : where {ei}2ki=1 is orthonormal}.

Moreover, the projection P with T = P ∗P is the projection onto W along (Im T )⊥. We refer the reader to3 forthe proof. As a consequence of this result we have the classification of positive self-adjoint operators in terms ofnon-orthogonal projections.

Theorem 5.2. If T is a positive self adjoint operator of rank ≤ N2 on RN , then there is a projection P and a

weight v so that T = v2P ∗P .

Proof. Let λk be the smallest non-zero eigenvalue of T . So all nonzero eigenvalues of 1λk

T are ≥ 1 and byTheorem 5.1 there is a projection P so that P ∗P = 1

λkT . Let v =

√λk to finish the proof.

6. CONSTRUCTING TIGHT NON-ORTHOGONAL FUSION FRAMES

In this section we will see that non-orthogonal projections give us much greater flexibility for constructing tightfusion frames.

Theorem 6.1. Assume n1 + · · · + nM ≥ N , ni ≤ N2 . Then there exists a tight nonorthogonal fusion frame

{Pi, vi}Mi=1 (vi = 1) for RN such that rank(Pi) = ni, for i = 1, ..., M .

Proof. Choose an orthonormal basis {ej}Nj=1 for RN and choose a collection of subspaces {Wi}M

i=1 such that:1) Wi = span {ej}j∈Ji with |Ji| = ni for each i = 1, ..., M , and2) W1 + · · · + WM = RN .Let πi be the orthogonal projection onto Wi and let S =

∑Mi=1 πi. Observe that Id = S−1S =

∑Mi=1 S−1πi.

Since each πi is diagonal with respect to {ej}Nj=1 it follows that S−1 commutes with πi, so S−1πi is positive and

self adjoint for every i = 1, ..., M . If γ is the smallest nonzero eigenvalue of any S−1πi, then 1γ S−1πi satisfies the

hypotheses of Theorem 5.1 so there is a projection Pi so that P ∗i Pi = 1

γ S−1πi, and we have

M∑i=1

P ∗i Pi =

1γ

Id.

Theorem 6.1 already shows that we have many more tight non-orthogonal fusion frames than standard fusionframes. Now we will see that if we are willing to use two projections on each subspace, then we can constructParseval non-orthogonal fusion frames for all cases. The proof can be found in.4



Theorem 6.2. Let T : RN → RN be a positive, self adjoint operator of rank k > N2 whose nonzero eigenvalues

are all greater than or equal to 1. The following are equivalent:

(1) There are two projections P1 and P2 whose eigenvectors correspond with the eigenvectors of T and suchthat T = P ∗

1 P1 + P ∗2 P2.

(2) Either N is even,

or

N is odd and T has at least one eigenvalue in the set {0, 1, 2}.As a consequence of the above theorem, we will see that we can construct very large classes of non-orthogonal

Parseval fusion frames.

We have immediately,

Theorem 6.3. Let T : RN → RN be a positive, self adjoint operator of rank k > N2 . There is a weight v and

projections {Pi}2i=1 so that

T = v2(P ∗1 P1 + P ∗

2 P2).

Proof. Let T have eigenvectors {ei}Ni=1 with respective eigenvalues {λ1 ≥ λ2 ≥ λk > 0 = λk+1 = . . . = λn}.

If N is even, we are done by Proposition 6.2. If N is odd, let T1 = 1λk

T . Then the smallest eigenvalue of T1

equals 1. By Proposition 6.2, we can find projections {Pi}2i=1 so that

1λk

T = T1 = P ∗1 P1 + P ∗

2 P2.

Letting v =√

λk finishes the proof.

Now we can construct Parseval fusion frames for all choices of dimensions for the subspaces if we are allowedto use two projections for each subspace.

Theorem 6.4. Suppose n1 + · · ·+ nM ≥ N . Then there exists subspaces {Wi}Mi=1 of RN with dim Wi = ni and

projections {Pi, Qi}Mi=1 onto Wi and weights {vi}M

i=1 so that

M∑i=1

v2i (P ∗

i Pi + Q∗i Qi) = I.

Proof. Choose an orthonormal basis {ej}Nj=1 for RN and choose a collection of subspaces {Wi}M

i=1, such that:(1) Wi = span {ej}j∈Ji with |Ji| = ni, for each i = 1, ..., M ,and(2) W1 + · · · + WM = RN .Now, we can proceed as in the proof of Theorem 5.1. Let πi be the orthogonal projection onto Wi and letS =

∑Mi=1 πi. Observe that Id = S−1S =

∑ki=1 S−1πi. Since each πi is diagonal with respect to {ej}n

j=1 itfollows that S−1 commutes with πi. Hence, S−1πi is positive and self adjoint for every i = 1, ..., M . Let γ bethe smallest nonzero eigenvalue of any S−1πi. Now, 1

γ S−1πi satisfies the hypotheses of either Corollary 5.2 orCorollary 6.3 so there is are projections Pi, Qi and weights ui so that u2

i (P∗i Pi + Q∗

i Qi) = 1γ S−1πi, and we have

N∑i=1

u2i γ(P ∗

i Pi + Q∗i Qi) = I.

Letting vi = ui√

γ completes the proof.

ACKNOWLEDGMENTS

J. Cahill was supported by NSF DMS 1008183 ; P.G. Casazza was supported by NSF DMS 1008183, DTRA/NSF1042701 and AFOSR FA9550-11-1-0245; S. Li was supported by NSF DMS 0709384, NSF DMS 1010058 andAFOSR FA9550-11-1-0245.



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