grassmannian fusion frames and its use in block sparse recovery

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Signal Processing

Signal Processing 94 (2014) 498–502

0165-16http://d

n CorrE-m

bs.adiga

journal homepage: www.elsevier.com/locate/sigpro

Grassmannian fusion frames and its use in blocksparse recovery

N. Mukund Sriram a,n, B.S. Adiga b, K.V.S. Hari a

a Statistical Signal Processing Lab, Department of Electrical Communication Engineering, Indian Institute of Science,Bangalore 560012, Indiab Tata Consultancy Services: Innovation Labs, Bangalore 560066, India

a r t i c l e i n f o

Article history:Received 4 April 2013Received in revised form8 July 2013Accepted 17 July 2013Available online 26 July 2013

Keywords:Fusion frameOptimal Grassmannian packingGrassmannian fusion frameBlock sparsityBlock coherenceSimplex bound

84/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.sigpro.2013.07.016

esponding author. Tel.: +91 9731093578, +9ail addresses: mukundns@ece.iisc.ernet.in (N@tcs.com (B.S. Adiga), hari@ece.iisc.ernet.in

a b s t r a c t

Tight fusion frames which form optimal packings in Grassmannian manifolds are of interestin signal processing and communication applications. In this paper, we study optimalpackings and fusion frames having a specific structure for use in block sparse recoveryproblems. The paper starts with a sufficient condition for a set of subspaces to be an optimalpacking. Further, a method of using optimal Grassmannian frames to construct tight fusionframes which form optimal packings is given. Then, we derive a lower bound on the blockcoherence of dictionaries used in block sparse recovery. From this result, we conclude thatthe Grassmannian fusion frames considered in this paper are optimal from the blockcoherence point of view.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Frames provide redundant representation of signals.The concept of frames occurs when we are dealing withovercomplete spanning systems [1,2]. In recent years, theconcept of fusion frames which are frames of subspaceshas been introduced [3]. Frames represent a signal by acollection of scalars, while fusion frames represent a signalby a collection of vectors which are projections of thesignal onto the fusion frame subspaces.

Fusion frames occur in many applications like packetencoding, distributed sensing, error correction, sparse signalprocessing and neurology [3–5]. In these applications, itis required that the frames have some desirable properties.One such property is that the fusion frame is an equidi-mensional, equidistant tight fusion frame. These frames also

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1 23489915.. Mukund Sriram),(K.V.S. Hari).

form optimal packings and have been called as Grassman-nian fusion frames [4,6].

The Grassmannian packing problem is concerned witharrangement of n subspaces of dimension d in K dimen-sional Euclidean space so that the subspaces are asfar apart as possible [7]. It is important to note thatGrassmannian fusion frames exist only for certain valuesof n, K and d.

The construction of Grassmannian fusion frames is acomplex problem and only a few methods are available inthe literature. Grassmannian fusion frames can be con-structed using either algebraic methods or numericalmethods. Some examples of algebraic methods are usingHadamard matrix [6], partial Fourier matrix [5], specialgroups like Clifford groups and lattices [4]. Though thesemethods are accurate, they are applicable only for certainsizes and dimensions. The alternating projection algorithmis a numerical method to construct Grassmannian fusionframes of arbitrary size and dimension [8]. However, thisalgorithm may give suboptimal results which are not veryaccurate and have convergence issues.

N. Mukund Sriram et al. / Signal Processing 94 (2014) 498–502 499

The block sparse recovery problem is concerned withthe solution to an underdetermined system of equationswhich is the result of observation of an unknown blocksparse vector through a measurement matrix (dictionary)[9]. A block sparse vector has a few nonzero entries whichoccur in clusters. Such a formulation has applications insampling from union of subspaces, multiple measurementvector (MMV) problem and burst error correction [5,10,11].The recovery property of numerical algorithms used toobtain the solution depends on the properties of thedictionary. Hence it is of interest to find dictionaries withgood recovery properties. One metric which quantifies thesuitability of dictionary is block coherence [9].

Grassmannian fusion frames are natural dictionaries forblock sparse recovery. In this paper, we give a procedurefor constructing optimal block sparse dictionaries. A con-ceptually simple algebraic method of constructing Grass-mannian fusion frames which is more suitable than theones mentioned earlier [4–6] is given. The method usesGrassmannian frames to construct Grassmannian fusionframes. So, we can exploit the variety of methods whichexist for designing Grassmannian frames [12,13] to achieveour goal.

We will derive a lower bound for the block coherenceof dictionaries (measurement matrices) used in blocksparse signal processing and show that the Grassmannianfusion frames constructed here meet this bound. Hence,these Grassmannian fusion frames minimize the blockcoherence and are optimal dictionaries. The bound forblock coherence derived here is a counterpart of the Welchbound on the coherence of sparse dictionaries [12].

2. Preliminaries

2.1. Fusion frames and optimal Grassmannian packings

In this section, we briefly review the existing conceptsand results which will be used in the paper. Throughoutthe paper, the norm under consideration is the l2 norm.

Definition 1 (Kutyniok et al. [4]). A collection of subspacesfW igMi ¼ 1⊂C

K is a fusion frame if there exist constants0o A≤Bo1 such that

A‖x‖2≤ ∑M

i ¼ 1‖Pix‖2≤B‖x‖2; ∀x∈CK ; ð1Þ

where Pi is the orthogonal projection onto W i.

If an orthonormal basis is chosen for each subspace W i

and represented as columns of a matrix W½i�, then thefusion frame can be written as W¼ ½W½1�W½2�……W½M��.

Frames can be viewed as a special case of the abovedefinition where each subspace is spanned by a singlevector. A frame/fusion frame is said to be tight if A¼B.

A Grassmannian manifold GðK; dÞ is the set of alld-dimensional subspaces of CK . Among the various meth-ods of measuring the distance between subspaces, we willbe concerned with the chordal distance [7]. The chordaldistance between two subspacesW i andW j is calculated as

dcði; jÞ2 ¼ 12 ‖Pi�Pj‖2F ¼ tr Pið Þ�tr PiPj

� �; ð2Þ

(tr denotes trace of a square matrix). An optimal packing inGðK ; dÞ involves finding a set fW igni ¼ 1 of d-dimensionalsubspaces such that mini≠jdcði; jÞ is as large as possible. Thefollowing result about chordal distance is very useful inpacking problems.

Theorem 1 (Simplex bound, Conway et al. [7]). For any setof n, d-dimensional subspaces of CK the chordal distance dcsatisfies

dcði; jÞ2≤dðK�dÞ

Kn

n�1; ð3Þ

if equality holds, the subspaces form an optimal packing forthe manifold GðK; dÞ.

It was shown in [4] that equidimensional equidistanttight fusion frames are optimal packings.

A Grassmannian frame is a solution to the packingproblem for GðK;1Þ [6]. Let V¼ ½v1v2…vn�∈Ck�n denote thesynthesis matrix [1] of the frame V. In the work thatfollows, the frame and its synthesis matrix will be repre-sented by the same symbol V. Then, a Grassmannian frameis a solution to

minF

maxi≠j

j⟨vi;vj⟩j� �

; ð4Þ

where F denotes the set of unit norm frames fvigni ¼ 1∈Ckðn4kÞ. We will be concerned with optimal Grassman-nian frames [12,13] which meet the Welch bound withequality.

Theorem 2 (Welch bound, Strohmer and Heath [12]). Let Vbe an unit norm frame. Define coherence of V, μðVÞ ¼maxi≠jj⟨vi; vj⟩j. Then

μ Vð Þ≥ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�k

kðn�1Þ

sð5Þ

where equality holds iff fvigni ¼ 1 is an equiangular tight frame,i.e., a tight frame where j⟨vi; vj⟩j ¼ c; ∀i≠j.

Frames which meet the Welch bound with equality areoptimal Grassmannian frames and the correlation betweenany two frame vectors is given by the value on the right-hand side of (5). Simple non-iterative methods for theirconstruction are available in [12,13].

2.2. Relation to block sparse recovery

Fusion frames and subspace packings are useful inblock sparse signal processing. To see this, let us considerthe problem [9] of representing a vector x∈CN in adictionary W∈CK�N with KoN so that

y¼Wx: ð6ÞIt is desired to recover block sparse vector x from measure-ment vector y. In the block sparse signal processing context,x is represented as a concatenation of blocks x½i�∈Cd whereN¼nd. A vector is said to be block p sparse if

‖x‖2;0 ¼ ∑n

i ¼ 1Ið‖x½i�‖240Þ≤p;

(IðÞ is the indicator function). The dictionary W (assumed tobe full rank), also referred to as the measurement matrix,

N. Mukund Sriram et al. / Signal Processing 94 (2014) 498–502500

can be similarly partitioned into blocks of size d:

W¼ ½w1……wd|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}W½1�

wdþ1…w2d|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}W½2�

……wN�dþ1…wN|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}W½n�

�;

where W½i�∈CK�d.Since W is a full rank matrix and due to the fact that (6)

is underdetermined, W forms an overcomplete spanningsystem for CK . This is where frames enter the picture as Wcan be interpreted as a fusion frame where the columns ofW½i� span the subspaces of the fusion frame [9]. Forrecovery of x from measurement, certain conditions whichdepend on the block coherence structure of the matrix Wmust be satisfied. The lower the block coherence, thehigher will be the sparsity level that can be recovered.

3. Construction of Grassmannian fusion frames

Designing Grassmannian fusion frames is a challengingproblem. Algebraic methods of construction are restrictedto specific values of n, K and d. The difficulties are bestillustrated by the following design procedure which usesWalsh–Hadamard matrices (a Hadamard matrix is anorthogonal matrix with 71 entries).

Theorem 3 (King [6]). Let Hn be a 2n � 2n Walsh–Hada-mard matrix indexed by 0;1;……;2n�1. Then

Wp ¼ span0≤r≤2m�1

fðHnðp; qþ r2n�mÞÞ2m≤q≤2n�1g( )

0≤p≤2n�m�1

;

is a tight Grassmannian fusion frame for R2n�2m consisting of2n�m;2m dimensional subspaces.

It can be observed that the above technique cannot beapplied to obtain Grassmannian fusion frames for n, K andd which are not powers of two. We attempt to overcomethis difficulty by giving a less restrictive method.

Consider an equidimensional fusion frame fW igni ¼ 1with dimðW iÞ ¼ d; ∀i. Let the local frame elements (vectorsspanning the subspaces) [3] form an orthonormal basis forthe respective subspaces. Then we can write the fusionframe as a K�N matrix W¼ ½W½1�W½2�……W½n�� ðKoNÞ,where W½i�∈CK�d and W i ¼ rangeðW½i�Þ. The bases of thesubspaces of an equidimensional fusion frame would bepoints in the complex Stiefel manifold VdðCK Þ [8].

In the work that follows, we assume N¼nd and K¼kd.Our results cover a large class of fusion frames that haveapplications in block sparse recovery [9]. Based on theconcepts in [5,14], we give a sufficient condition forfW igni ¼ 1 to form an optimal Grassmannian packing whichwill be used for proving theorems stated in the paper.

Lemma 1. If the fusion frame W satisfies

W½i�HW j½ � ¼Id if i¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n�kkðn�1Þ

sexp jθij

� �Id if i≠j

8>><>>: ð7Þ

0≤θi;j≤2π, then the subspaces frangeðW½i�Þgni ¼ 1 satisfy thesimplex bound and form an optimal Grassmannian packing.

Proof. The orthogonal projection for W i is Pi ¼W½i�W½i�H .Consider expression (2) for some i≠j:

dcði; jÞ2 ¼ tr Pið Þ�tr PiPj� �

¼ d�tr W i½ �W½i�HW j½ �W½j�H� �¼ d�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�k

kðn�1Þ

sexp jθij

� �tr W i½ �W½j�H� �

¼ d� n�kkðn�1Þ d:

Simplifying the previous equation we get

dcði; jÞ2 ¼dk�dk

� n

n�1¼ dðK�dÞ

Kn

n�1

�;

which is exactly the value of the upper bound in (3). HenceW is an optimal packing. □

A simple method for constructing Grassmannian fusionframes from optimal Grassmannian frames is now given.The example given in [9] for obtaining an optimal dic-tionary with n¼2 is the motivation for this method.

Theorem 4. Consider a k�n optimal Grassmannian frame Vand an arbitrary d�d unitary matrix Ud. Then W¼V⊗Ud isa Grassmannian fusion frame.

Proof. Consider the Gramian of W:

WHW¼ ðV⊗UdÞHðV⊗UdÞ¼ ðVH⊗UH

d ÞðV⊗UdÞ¼ VHV⊗UH

dUd

¼ R⊗Id; ð8Þwhere R¼ VHV is the Gramian of V. Since V is an optimalGrassmannian frame [12], from Theorem 2 we have

Ri;j ¼1 if i¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n�kkðn�1Þ

sexp jθij

� �if i≠j;

8>><>>: ð9Þ

where θij is arbitrary. We can partition W into blocks ofsize K�d and rewrite (8) as

W½i�HW j½ � ¼ Ri;jId ¼Id if i¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n�kkðn�1Þ

sexp jθij

� �Id if i≠j:

8>><>>: ð10Þ

Now using Lemma 1, we can conclude thatW is an optimalGrassmannian packing. To prove that W is a tight fusionframe it is sufficient to show that ∑n

i ¼ 1Pi ¼ AIK [4].The summation can be rewritten as

∑n

i ¼ 1Pi ¼ ∑

n

i ¼ 1W½i�W½i�H ¼WWH

¼ ðV⊗UdÞðV⊗UdÞH

¼ VVH⊗UdUHd : ð11Þ

Since Grassmannian frames are equiangular tight frames[12], VVH ¼ ðn=kÞIk. Substituting in (11) we get

∑n

i ¼ 1Pi ¼WWH

¼ nkIk⊗Id ¼

nkIK :

N. Mukund Sriram et al. / Signal Processing 94 (2014) 498–502 501

Hence W is a tight fusion frame with frame constant A¼n/k. Since W is an optimal Grassmannian packing and atight fusion frame, it is a Grassmannian fusion frame. □

It is worth noting that the above procedure is depen-dent on preexisting constructions of optimal Grassman-nian frames.

As an illustrative example, we consider the case whereV is a partial Fourier matrix constructed using a differenceset (DS) and Ud is an identity matrix. The details aboutdifference sets along with some examples can be found in[13]. It was shown in [13] that by choosing k rowscorresponding to the elements of DS U from a n�ndiscrete Fourier transform (DFT) matrix F, we obtain ak�n partial Fourier matrix FU which forms an optimalGrassmannian frame.

Examples: Suppose we want to design a Grassmannianfusion frame with N¼55 for Gð25;5Þ. Since N and d are notpowers of two, the Hadamard construction of Theorem 3cannot be used. However, we can apply the Kroneckermethod as follows. We select 5 rows corresponding toelements of the quadratic DS U ¼ f1;3;4;5;9g from a11�11 DFT matrix to get FU . The Grassmannian fusionframe is given by WU ¼ FU⊗I5.

Another example where the Hadamard technique failsis when we want a Grassmannian fusion frame with N¼45for Gð21;3Þ. In this case the present method can be appliedusing a twin prime DS U ¼ f0;1;2;4;5;8;10g and a 15�15DFT matrix along the same lines as the previous example.

While the method given here is suitable for construct-ing optimal dictionaries, it can also be used to obtainoptimal packings of interest in applications besides sparserecovery. In such applications, however, the method maysometimes not be useful. For example, when K is prime,the method cannot be used. However, this particularcondition does not occur in the block sparse recoverysetting.

4. Optimality in block sparse recovery

Grassmannian frames are useful in sparse signal pro-cessing applications as they have minimum coherence.Grassmannian fusion frames play a similar role in theblock sparse signal processing [15]. The packings con-structed in Section 3 can be used as optimal dictionaries.In order to substantiate this, we first define the concept ofblock coherence.

Definition 2 (Eldar et al. [9]). Block coherence of a dic-tionary W is defined as

μB Wð Þ ¼maxi;j≠i

ρðW½i; j�Þd

: ð12Þ

W½i; j� ¼W½i�HW½j� is the ði; jÞth d� d block of WHW∈CN�N

and ρðAÞ is the spectral radius of A, ρðAÞ ¼ λð1=2Þmax ðAHAÞ.Usually in the block sparse recovery setting (6), algo-

rithms like block orthogonal matching pursuit (BOMP) andblock matching pursuit (BMP) are used to recover signal xfrom measurement y [9]. Recovery guarantees for thesealgorithms are given in terms of block coherence. Anexample from [9] is given below.

Theorem 5 (Eldar et al. [9]). If W½i�HW½i� ¼ Id, ∀i¼ 1;2;…;nand if the block coherence is such that

pdo μ�1B ðWÞ þ d

2; ð13Þ

then guaranteed recovery of p block sparse vector in (6) ispossible using BOMP.

From this theorem, it is evident that a dictionary with alow value of block coherence is desirable as signals withhigher sparsity level can be recovered. Grassmannianfusion frames are optimal dictionaries in the above con-text. The following result gives a lower bound on the blockcoherence of the matrices used in block sparse recovery.

Theorem 6. If W∈CK�NðN¼ nd;K ¼ kdÞ is such thatW½i�HW½i� ¼ Id, then

μB Wð Þ≥1d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�k

kðn�1Þ

s: ð14Þ

Equality holds iff (7) is satisfied.

Proof. According to (12), μB is given by

μB Wð Þ ¼maxi;j≠i

ρðW½i; j�Þd

¼maxi;j≠i

λ1=2maxðW½j�HW½i�W½i�HW½j�Þd

¼maxi;j≠i

λ1=2maxðΦi;jÞd

; ð15Þ

where Φi;j ¼ ðW½j�HW½i�W½i�HW½j�Þ. Now, consider the chor-dal distance between the subspaces spanned by any twoblocks

d2c ði; jÞ ¼ trðW½i�W½i�HÞ�trðW½i�W½i�HW½j�W½j�HÞ¼ d�trðW½j�HW½i�W½i�HW½j�Þ ¼ d�trðΦi;jÞ¼ d�∑

lλlðΦi;jÞ≥dð1�λmaxðΦi;jÞÞ: ð16Þ

Using simplex bound (3), we have

dðK�dÞK

nn�1

≥d 1�λmax Φi;j� �� �

λmax Φi;j� �

≥1� ðK�dÞnKðn�1Þ

λmax Φi;j� �

≥n�k

kðn�1Þ : ð17Þ

Using the above inequality in (15), we get

μB Wð Þ≥1d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin�k

kðn�1Þ

s:

Lemma 1 can be shown to be a sufficient condition tomeet the bound by substituting (7) for (15). To show that itis also a necessary condition, observe that if μB takes thevalue of the lower bound, then λmaxðΦi;jÞ ¼ ðn�kÞ=kðn�1Þfor some i; j. By the definition of block coherence, thisimplies that λmaxðΦi;jÞ is the same for all (i,j). Now for λmax

to take the desired value, (16) and (17) must be met withequality. This implies that every Φi;j is a matrix with equaleigenvalues for all values of i and j. Since Φi;j is Hermitian,from the above arguments we can conclude that (7) issatisfied. □

N. Mukund Sriram et al. / Signal Processing 94 (2014) 498–502502

The above result clearly implies that Grassmannianfusion frames constructed according to Lemma 1 andTheorem 4 achieve minimum block coherence and arehence optimal.

5. Conclusion

The main focus of this paper is the design of Grass-mannian fusion frames and their implication in blocksparse recovery. A method is given for constructing Grass-mannian fusion frames from Grassmannian frames whichmeet the Welch bound. The method, while fairly general,requires that the dimension of the vector space (K) bedivisible by the dimension of the subspaces (d). Thestructure of the fusion frames constructed conforms tothose of dictionaries used in block sparse approximation.A lower bound which is an analog of the Welch bound isderived for the block coherence of dictionaries, which weuse to conclude that the matrices constructed by us areoptimal for block sparse recovery since they minimize theblock coherence.

References

[1] J. Kovacevic, A. Chebira, Life beyond bases: the advent of frames(part I), IEEE Signal Processing Magazine 24 (4) (2007) 86–104.

[2] J. Kovacevic, A. Chebira, Life beyond bases: the advent of frames(part II), IEEE Signal Processing Magazine 24 (5) (2007) 115–125.

[3] P.G. Casazza, G. Kutyniok, S. Li, Fusion frames and distributedprocessing, Applied and Computational Harmonic Analysis 25 (1)(2008) 114–132.

[4] G. Kutyniok, A. Pezeshki, R. Calderbank, T. Liu, Robust dimensionreduction, fusion frames and Grassmannian packings, Applied andComputational Harmonic Analysis 26 (1) (2009) 64–76.

[5] N. Mukund Sriram, B.S. Adiga, K.V.S. Hari, Burst error correctionusing partial Fourier matrices and block sparse representation, in:National Conference on Communications (NCC), 2012, pp. 1–5.

[6] E.J. King, Grassmannian Fusion Frames, arXiv:1004.1086, 5 April2010.

[7] J.H. Conway, R.H. Hardin, N.J.A. Sloane, Packing lines, planes, etc.:packings in Grassmannian spaces, Experimental Mathematics 5 (2)(1996) 139–159.

[8] I.S. Dhillon, R.W. Heath, T. Strohmer, J.A. Tropp, Constructing pack-ings in Grassmannian manifolds via alternating projection, Experi-mental Mathematics 17 (1) (2008) 9–35.

[9] Y.C. Eldar, P. Kuppinger, H. Bolcskei, Block-sparse signals: uncer-tainty relations and efficient recovery, IEEE Transactions on SignalProcessing 58 (6) (2010) 3042–3054.

[10] Y.C. Eldar, M. Mishali, Robust recovery of signals from a structuredunion of subspaces, IEEE Transactions on Information Theory 55 (11)(2009) 5302–5316.

[11] L. Lampe, Bursty impulse noise detection by compressed sensing, in:IEEE International Symposium on Power Line Communications andits Applications (ISPLC), 2011, pp. 29–34.

[12] T. Strohmer, R.W. Heath, Grassmannian frames with applications tocoding and communication, Applied and Computational HarmonicAnalysis 14 (3) (2003) 257–275.

[13] P. Xia, S. Zhou, G.B. Giannakis, Achieving the Welch bound withdifference sets, IEEE Transactions on Information Theory 51 (5)(2005) 1900–1907.

[14] B.G. Bodmann, Optimal linear transmission by loss-insensitivepacket encoding, Applied and Computational Harmonic Analysis22 (3) (2007) 274–285.

[15] P. Boufounos, G. Kutyniok, H. Rauhut, Sparse recovery from com-bined fusion frame measurements, IEEE Transactions on InformationTheory 57 (6) (2011) 3864–3876.