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

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 sparsematrix 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.


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: with inputs Ŷ ∈ R × and Â ∈ R × .N denotes the measurement noise and E denotes the system perturbation.
The perturbations E and N are quantified with the following relative bounds: where ‖A‖ () 2 and ‖Y‖  are nonzero.Here, ‖A‖ () 2 denotes the largest spectral norm taken over all -column submatrices of A. Throughout the paper, we are only interested in the case where  and   are far less than 1.In (1), X is a -group sparse matrix; that is, it has no more than  nonzero rows, and ‖X‖ 2,1 = ∑  =1 ‖x  ‖ 2 , x  is the th row of X.It is assumed that all columns of Â 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][8][9].
One of the most commonly known conditions is the restricted isometry property (RIP).A matrix A satisfies RIP of the order  if there exists a constant  ∈ (0, 1) such that for all -sparse vector h.In particular, the minimum of all constants  satisfying (3) is called the restricted isometry constant (RIC)   .There are many papers [8,[10][11][12][13][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  +1 < 1/( √  + 1) is a sufficient condition for GBCD.We improve this condition to  +1 < ( √ 4 + 1 − 1)/2.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.a  denotes the th column of a matrix A. A  denotes the transpose of A. I  denotes an  ×  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 fl {1, 2, . . ., }.Γ  = Ω \ Γ = { |  ∈ Ω, and  ∉ Γ}.The support of X is denoted by supp(X) (supp(X) = { | x  ̸ = 0}).‖A‖ () 2 denotes the largest spectral norm taken over all column submatrices of A. Let ‖A‖ 2,∞ denote the maximum ℓ 2 norm of the rows of A. We write A Γ for the column submatrix of A whose indices are listed in set of Γ and X Γ for the row submatrix of X whose indices are listed in the set Γ. e  ∈ R  denotes the th unit standard vector.

Problem Formulation
Analogous to [1], (1) can be rewritten as min Assume that Γ fl supp(X).Obviously, |Γ| = .The objective function in (4) can be written as where with ⊗ denoting the Kronecker product and (X) = ‖X‖ 2,1 =  ∑  =1 ‖x  ‖ 2 .Combining the quadratic approximation for (X) and standard BCD algorithm, the solution to the th subproblem can be given by a softthresholding operator.The authors in [1] only update the block that yields the greatest descent distance.Now, we list GBCD algorithm (Algorithm 1).
Suppose that A satisfies the th order RIC   ∈ (0, 1).Recall that X has no more than  nonzero rows.According to the fact ‖X‖ 2  = ∑  =1 ‖x  ‖ 2 2 , we can obtain from (3).Combining Lemma 2.4 in [3] and (6), we have Lemma 1 (near orthogonality property, see [3]).Let u and v be two orthogonal sparse vectors with supports   and  V fulfilling |  ∪  V | ≤ .Suppose that A satisfies RIP of order  with RIC   .Then we have where ∡(Au, Ak) denotes the angle between Au and Ak.

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.
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  +1 < 1/( √  + 1).Obviously, it is smaller than the bound ( √ 4 + 1 − 1)/2 in (21).
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  +1 (A) = 1/ √  −  2 / √ ( 2 0  +  2 ).They showed that the GBCD algorithm fails when using A as measurement matrix.After a simple calculation, we can get Thus, our result improves this existing result.

Experimental Results
In this section, under the total perturbations, we test the performance of the GBCD algorithm for solving the DOA estimation problem.Consider  narrowband far-field point source signals impinging on an -element uniform linear array.The steering vector of the matrix A is where 1 ≤  ≤ . is the number of snapshots.Using the sparse optimization approach in [1], the DOA estimation problem can be 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 is  = 11.
(ii) The number of snapshots is  = 200.
where θℓ  () is the estimate of  ℓ  at the th trial.Figure 1, 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.

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.