# Support Recovery of Greedy Block Coordinate Descent Using the Near Orthogonality Property

Authors: Haifeng Li
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this paper, using the near orthogonal property, we analyze the performance of greedy block coordinate descent (GBCD) algorithm when both the measurements and the measurement matrix are perturbed by some errors. An improved sufficient condition is presented to guarantee that the support of the sparse matrix is recovered exactly. A counterexample is provided to show that GBCD fails. It improves the existing result. By experiments, we also point out that GBCD is robust under these perturbations.


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## Body

## 1. Introduction

Greedy block coordinate descent (GBCD) algorithm was presented by [1] for direction of arrival (DOA) estimation. In the work of [1], the DOA estimation is treated as the multiple measurement vectors (MMV) model that recovers a common support shared by multiple unknown vectors from multiple measurements. The authors provided a sufficient condition, based on mutual coherence, to guarantee that GBCD exactly recover the nonzero supports with noiseless measurements.Recently, the work of [2] discussed the following method: (1)minXX2,1s.t.Y^=AX+N,A^=A+E,with inputs Y^∈Rm×L and A^∈Rm×N. N denotes the measurement noise and E denotes the system perturbation. The perturbations E and N are quantified with the following relative bounds:(2)NFYF≤ε,E2KA2K≤εA,where A2(K) and YF are nonzero. Here, A2(K) denotes the largest spectral norm taken over all K-column submatrices of A. Throughout the paper, we are only interested in the case where ε and εA are far less than 1. In (1), X is a K-group sparse matrix; that is, it has no more than K nonzero rows, and X2,1=∑i=1Nxi2, xi is the ith row of X. It is assumed that all columns of A^ are normalized to be of unit-norm [3]. Both Y=AX and A are totally perturbed in (1). This case can be found in source separation [4], radar [5], remote sensing [6], and countless other problems. In addition, the total perturbations have also been discussed in [7–9].One of the most commonly known conditions is the restricted isometry property (RIP). A matrixA satisfies RIP of the order K if there exists a constant δ∈(0,1) such that (3)1-δh22≤Ah22≤1+δh22for all K-sparse vector h. In particular, the minimum of all constants δ satisfying (3) is called the restricted isometry constant (RIC) δK.There are many papers [8, 10–14] discussing the sufficient condition for orthogonal matching pursuit (OMP) that is one of the widely greedy algorithms for sparse recovery. In [3], using the near orthogonality property, the authors improved the sufficient condition of OMP. As cited in [3], the near orthogonality property can further develop the orthogonality characterization of columns in A; it will play a fundamental role in the study of the signal reconstruction performance in compressed sensing. In the noiseless case, the work of [15] analyzed the performance of GBCD using near orthogonality property and improved the results in [2].In this paper, under the total perturbations, we use near orthogonality property to improve the theoretical guarantee for the GBCD algorithm. In [2], the authors stated that δK+1<1/(K+1) is a sufficient condition for GBCD. We improve this condition to δK+1<(4K+1-1)/2K. We also present a counterexample to show that GBCD fails. The example is superior to that in [2]. Under the total perturbations, the robustness of GBCD is shown by experiments.Now we give some notations that will be used in this paper.ai denotes the ith column of a matrix A. A′ denotes the transpose of A. IM denotes an M×M identity matrix. The symbol vec denotes the vectorization operator by stacking the columns of a matrix one underneath the other. The cardinality of a finite set Γ is denoted by |Γ|. Let Ω≔{1,2,…,N}. Γc=Ω∖Γ={i∣i∈Ω,andi∉Γ}. The support of X is denoted by supp⁡(X) (supp⁡(X)={i∣xi≠0}). A2(K) denotes the largest spectral norm taken over all K-column submatrices of A. Let A2,∞ denote the maximum l2 norm of the rows of A. We write AΓ for the column submatrix of A whose indices are listed in set of Γ and XΓ for the row submatrix of X whose indices are listed in the set Γ. ei∈RN denotes the ith unit standard vector.


## 2. Problem Formulation

Analogous to [1], (1) can be rewritten as(4)minX⁡12Y^-A^XF+λX2,1.Assume thatΓ≔supp⁡(X). Obviously, |Γ|=K. The objective function in (4) can be written as (5)FX=GX+HX,where G(X)=(1/2)Y^-A^XF2=(1/2)vec⁡(Y^)-IL⊗A^vec⁡(X)F2 with ⊗ denoting the Kronecker product and H(X)=λX2,1=λ∑i=1Nxi2. Combining the quadratic approximation for G(X) and standard BCD algorithm, the solution to the ith subproblem can be given by a soft-thresholding operator. The authors in [1] only update the block that yields the greatest descent distance. Now, we list GBCD algorithm (Algorithm 1).Algorithm 1:GBCD: greedy block coordinate descent algorithm [1].
Input:
A
^
,
Y
^
,
X
(
0
)
=
0
,
n
=
1
,
λ
>
0
,
β
>
0.
(1) Repeat until “stopping criterion" is met
(2) fori=1:N
(3) pi(n-1)=xi(n-1)-βa^i′(A^X(n-1)-Y^)
(4) xi(n)=(pin-1/pin-12)max⁡(0,pi(n-1)2-λβ)
(5) comp(i)=xi(n)-xi(n-1)
(6) end for
(7) Choose the index i0 such that comp(i0)=max⁡(comp)
(8) X(n)←[x1(n-1);…;xi0-1(n-1);
(9)      xi0(n);xi0+1(n-1);…;xN(n-1)]
(10) n←n+1
(11) End RepeatSuppose thatA satisfies the Kth order RIC δK∈(0,1). Recall that X has no more than K nonzero rows. According to the fact XF2=∑i=1Lxi22, we can obtain (6)1-δKXF2≤AXF2≤1+δKXF2from (3).Combining Lemma2.4 in [3] and (6), we have (7)AΓ′AXF2=∑i=1LAΓ′Axi22(8)≥∑i=1L1-δKAxi22(9)=1-δK∑i=1LAxi22(10)=1-δKAXF2.Lemma 1 (near orthogonality property, see [3]).
Letu and v be two orthogonal sparse vectors with supports Tu and Tv fulfilling Tu∪Tv≤K. Suppose that A satisfies RIP of order K with RIC δK. Then we have (11)cos⁡∡Au,Av≤K,where ∡(Au,Av) denotes the angle between Au and Av.Lemma 2 (see [3]).
Under the same assumptions as in Lemma1, we have (12)Au,Av≤δAu2Av2.Lemma 3.
For finite setsΓ- and Γ~, let supp⁡(X-)=Γ- and supp⁡(X~)=Γ~. Here, Γ-∩Γ~=∅, and Γ-∪Γ~≤K. If A satisfies the RIP condition (3) with δK∈(0,1), then we have (13)AX-,AX~F≤δKAX-FAX~F.Proof.
Note that the Frobenius norm ofA is derived from the Frobenius inner product. (14)AX-,AX~F=∑i=1LAx-i,Ax~i(15)≤δK∑i=1LAx-i2Ax~i2(16)=δK∑i=1LAx-i2Ax~i22(17)≤δK∑i=1LAx-i22∑i=1LAx~i22(18)=δKAX-FAX~F,where (15) and (17) follow from Lemma 2 and Cauchy-Schwarz inequality, respectively.


## 3. RIP Based Recovery Condition

In this section, we firstly present the upper bound of the noise matrix-EX+N and provide the recovery condition for GBCD.Lemma 4 (see [2]).
Suppose thatA^ satisfies the Kth order RIC δ^K∈(0,0.5). Then we have (19)-EX+NF<εA1/3-1+1/3εA+εY^F1-ε.According to steps (7) and (8) of Algorithm 1, at the nth iteration, GBCD can obtain a correct index if (20)maxi∈Γ⁡Xin-Xin-12>maxj∈ΓcXjn-Xjn-12.Theorem 5.
Consider model (4). Let t0=mini∈Γ⁡xi2. If the matrix A^ satisfies RIP of order K+1 with(21)δ^K+1<4K+1-12K,(22)t0>1+δ^K+1+Kε01-δ^K+1-1-δ^K+1Kδ^K+1,where ε0=εA/1/3-1+1/3εA+ε(Y^F/(1-ε)), then GBCD can exactly recover the support set Γ.Proof.
Considern=1. The initial value is X(0)=0. In order to guarantee that GBCD selects a correct index i0∈Γ, combining step (4) of Algorithm 1 and (20), we should verify the following inequality: (23)maxi∈Γ⁡xi1-xi02(24)=maxi∈Γpi0pi02max⁡0,pi02-λβ2(25)>maxj∈Γc⁡xj1-xj02(26)=maxj∈Γcpj0pj02max0,pj02-λβ2.If pj(0)2-λβ≤0(j∈Γc), the right-hand-side is 0. Then inequality (26) holds. Thus, we only consider pj(0)2-λβ>0. Using Remark 1 in [2], inequality (26) is true when (27)maxi∈Γ⁡pi02>maxj∈Γc⁡pj02.
Now, it is sufficient to verify (27). Let us construct an upper bound for maxj∈Γc⁡pj(0)2. By step (3) of Algorithm 1, we have (28)maxj∈Γc⁡pj02(29)=maxj∈Γc⁡xj0-βa^j′A^X0-Y^2(30)=maxj∈Γc⁡βa^j′Y^2(31)=βA^Γc′A^X-EX+N2,∞(32)≤maxj∈Γc⁡βa^j′A^X2+βA^Γc′2,∞-EX+NF(33)=βmaxj∈Γc⁡A^ej,A^XF+βA^Γc′2,∞-EX+NF(34)≤βδ^K+1A^XF+βε0,where (32) is from the property of norm and (34) follows from each column of A^ which is of unit-norm, Lemmas 3 and 4.
To prove (27), we only need to prove (35)maxi∈Γpi02>βδ^K+1A^XF+βε0.We then go on to show by contradiction that (35) is true. For all i∈Γ, assume that (36)pi02≤βδ^K+1A^XF+βε0.Then we have (37)PΓ0F=∑i∈Tpi022≤βKδ^K+1A^XF+ε0.
Using the triangle inequality, we can get(38)PΓ0F=XΓ0-βA^Γ′A^X0-Y^F=βA^Γ′Y^F(39)≥βA^Γ′A^XF-βA^Γ′-EX+NF(40)≥β1-δ^K+1A^XF-1+δ^K+1ε0,where (40) is from (10) and the property of norm.
After straightforward manipulations, we have(41)PΓ0F≥βKδ^K+1A^XF+ε0+β1-δ^K+1-Kδ^K+1︸≥0A^XF-β1+δ^K+1+Kε0(42)≥βKδ^K+1A^XF+ε0+β1-δ^K+1-Kδ^K+11-δ^K+1XF-β1+δ^K+1+Kε0(43)>βKδ^K+1A^XF+ε0+K-11+δ^K+1ε0+K-Kε0(44)≥βKδ^K+1A^XF+ε0,where (41) follows from (21) and (43) follows from XF≥Kt0 and (22).
Obviously, (44) contradicts (37), so this fact guarantees (27).
Assume that GBCD always picks up indices from the supportΓ for n≤k(k≥1 is an integer). Consider n=k+1. In order to prove that GBCD can choose a correct index i0∈Γ, analogous to [2], inequality (46) should be verified.(45)maxi∈Γ⁡pin-1-xin-12(46)>maxj∈Γc⁡pjn-1-xjn-12.
Combining step (3) of Algorithm 1 with (46) yields (47)maxi∈Γ⁡a^i′A^Xn-1-Y^2(48)>maxj∈Γc⁡a^j′A^Xn-1-Y^2.
It is sufficient to prove that (48) holds. Note that supp⁡(X(n-1))⊆Γ; we have (49)maxj∈Γc⁡a^j′A^Xn-1-Y^2(50)≤δ^K+1A^Xn-1-XF+ε0.Now, we only need to prove (51)maxi∈Γ⁡a^i′A^Xn-1-Y^2(52)>δ^K+1A^Xn-1-XF+ε0.
We then show that (52) is true by contradiction. For all i∈Γ, assume that (53)a^i′A^Xn-1-Y^2≤δ^K+1A^Xn-1-XF+ε0.
Using the definition of Frobenius norm, we have(54)A^Γ′A^Xn-1-Y^F=∑i∈Γa^i′A^Xn-1-Y^22(55)≤Kδ^K+1A^Xn-1-XF+ε0.Combining X(n-1)-XF≥t0, (21), and (22), we have (56)A^Γ′A^Xn-1-Y^F(57)≥1-δ^K+1A^Xn-1-XF-1+δ^K+1ε0(58)=Kδ^K+1A^Xn-1-XF+ε0+1-δ^K+1-Kδ^K+1A^Xn-1-XF-1+δ^K+1+Kε0(59)>Kδ^K+1A^Xn-1-XF+ε0,where (59) follows from (60)1-δ^K+1-Kδ^K+1A^Xn-1-XF(61)>1+δ^K+1+Kε0.
This contradicts (53). Thus, (48) is true.Remark 6.
The weaker the RIC bound is, the less required number of measurements we need, and the improved RIC results can be used in many CS-based applications [16]. In the work of [2], the authors provided that the condition for GBCD is δK+1<1/(K+1). Obviously, it is smaller than the bound (4K+1-1)/2K in (21).


## 4. The Counterexample

Consider the measurements(62)Y^=A^-EX+N=A^X-EX+N.In this section, giving a matrix A^, whose RIC is a slight relaxation of 1/K+1, we will verify that GBCD can fail to recover the support of sparse matrix from (62).Let(63)X=22t022t022t022t0⋮⋮22t022t000K+1×2,E=000⋯00.500⋯0000⋯0⋮⋮⋮⋯⋮000⋯0K+1×K+1,N=0024t024t000⋮⋮0022ϵ22ϵK+1×2,where supp⁡(X)={1,2,…,K}=Γ, Γc={K+1} and t0/ϵ>1+1/K (the value of ϵ is far less than 1; this is reasonable).The matrixA^ is constructed as(64)A^=a00⋯0s0a0⋯0s00a⋯0s⋮⋮⋮⋮⋮⋮000⋯as000⋯01K+1×K+1,where (65)s=δK,a=1-δ2.Set(66)δ=1-ϵ/t0K+1.The eigenvaluesλii=1K+1 of A^′A^ are (67)λi=1-δ2,1≤i≤K-1,λK=1-δ,λK+1=1+δ.Thus, the RIC ofA^ is δ-K+1(A^)=δ.Recall that condition (27) is the criterion of recovery for GBCD. Note that pi(0)2=βa^i′Y^2. One can obtain (68)maxi∈Γ⁡a^i′Y^2=22t0a2,22t0a22=a2t0.On the other hand, we have (69)maxj∈Γc⁡a^j′Y^2=22t0Kas+22ϵ,22t0Kas+22ϵ2=Kast0+ϵ.It can be derived that(70)Kast0+ϵ-t0a2=t0aKs-a2+ϵ(71)=t01-δ2Kδ-1+δ2+ϵ(72)>0,where (71) and (72) follow from (65) and (66).It is obviously in contradiction to (27). Thus, GBCD fails to recover support Γ.Remark 7.
In the work of [2], the authors presented a matrix A- whose RIC is δK+1(A-)=1/K-ϵ2/K(t02K+ϵ2). They showed that the GBCD algorithm fails when using A- as measurement matrix. After a simple calculation, we can get (73)1-ϵ/t0K+1<1K-ϵ2Kt02K+ϵ2.Thus, our result improves this existing result.


## 5. Experimental Results

In this section, under the total perturbations, we test the performance of the GBCD algorithm for solving the DOA estimation problem.ConsiderK narrowband far-field point source signals impinging on an m-element uniform linear array. The steering vector of the matrix A is (74)ai=1e-jπcos⁡θi-1⋯e-jm-1πcos⁡θi-1′,where 1≤i≤N. L is the number of snapshots.Using the sparse optimization approach in [1], the DOA estimation problem can be rewritten as model (1). Then the aim is hence to find out which row of the matrix X is nonzero, that is, the support of the matrix X.Analogous to the simulation of [1], we have the following assumptions:(i)
The number of the array elements ism=11.(ii)
The number of snapshots isL=200.(iii)
The grid spacing is1∘ from 0∘ to 180∘. Then N=181.(iv)
Five (K=5) uncorrelated signals impinge from θl1=30∘, θl2=80∘, θl3=100∘, θl4=120∘, and θl5=145∘.(v)
Both the signals and the noise are white and follow Gaussian distributions. The power of nonzero entries ofX is σ2, and the power of each entry of N is σN2.(vi)
Use the following SNR1 and SNR2 to measure noisesE and N, respectively: (75)SNR1=10log10⁡σ2σN2.(76)SNR2=AFEF.Define the root mean square error (RMSE) of 500 Monte Carlo trials as the performance index:(77)RMSE=∑i=1500∑k=1Kθ^lki-θlk2500K,where θ^lk(i) is the estimate of θlk at the ith trial.Figure1, fixing matrix E, describes the performance of GBCD. The results show that RMSE decreases as SNR1 increases. Figure 2, fixing matrix N, describes the performance of GBCD. The results show that RMSE decreases as SNR2 increases. Thus, the performance of GBCD still is robust under the total perturbations.Figure 1
For SNR2 = 10. The RMSE of GBCD versus input SNR1.Figure 2
For SNR1 = 2. RMSE of GBCD versus input SNR2.


## 6. Conclusion

In this paper, using the near orthogonality property, we provide a recovery condition for GBCD under the total perturbations. A counterexample is presented to show that GBCD fails. By experiments, we point out that GBCD is robust under the total perturbations.



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*Source: 4903791-2017-04-27.xml*