Sparse Recovery by Semi-Iterative Hard Thresholding Algorithm

Abstract

We propose a computationally simple and efficient method for sparse recovery termed as the semi-iterative hard thresholding (SIHT). Unlike the existing iterative-shrinkage algorithms, which rely crucially on using negative gradient as the search direction, the proposed algorithm uses the linear combination of the current gradient and directions of few previous steps as the search direction. Compared to other iterative shrinkage algorithms, the performances of the proposed method show a clear improvement in iterations and error in noiseless, whilst the computational complexity does not increase.

1. Introduction

Compressed sensing (CS) [1–3] is a new framework for acquiring sparse signals based on the revelation that a small number of linear measurements of the signal contain enough information for its reconstruction. CS relies on the fact that many natural signals are sparse or compressible when expressed in the proper basis and frame. The model of CS can be written as a linear sampling operator by a matrix Φ yielding a measurement vector 𝐲=Φ𝐱, (1) where Φ is an 𝑀×𝑁 matrix, 𝐱 is 𝑆-sparse vector, and 𝑀≪𝑁. Since the linear sampling operator Φ is not bijection and therefore has infinitely many solutions. Efficient algorithms to find sparse solutions are becoming very important. This leads to solving the 𝑙₀-minimization problem min‖𝐱‖₀s.t.𝐲=Φ𝐱. (2)

Unfortunately, this minimization problem is NP-hard [2]. As alternatives, approximation algorithms are often considered. Approximation algorithms to find sparse solutions may be classified into greedy pursuits algorithms, convex relaxation algorithms, Bayesian framework, and nonconvex optimization. In this paper, we will focus on greedy pursuits algorithms and convex relaxation algorithms; thus more details of Bayesian framework and nonconvex optimization methods can be found in [4, 5]. Greedy pursuits algorithms include orthogonal matching pursuit (OMP) [6], stagewise OMP (StOMP) [7], regularized OMP (ROMP) [8], compressive sampling matching pursuit (CoSaMP) [9], iterative hard thresholding (IHT) [10], and gradient descent with sparsification (GraDeS) [11]. Convex relaxation algorithms include gradient projection for sparse reconstruction (GPSR) [12] and sparse reconstruction by separable approximation (SpaRSA) [13]. For more details about convex relaxation algorithms, see, for example [14]. Convex relaxation algorithms succeed with a very small number of measurements, but they tend to be computationally burdensome [15]. An alternative family of numerical algorithms has gradually built, addressing the optimization problems very effectively [15]. This family is the iterative-shrinkage algorithms. Iterative-shrinkage algorithms include iterative hard thresholding (IHT) [10] and gradient descent with sparsification (GraDeS) [11], parallel coordinate descent (PCD) [16], and fast iterative-shrinkage thresholding algorithm (FISTA) [17]. In these methods, each iteration consists of a multiplication by Φ and its transpose, along with a scalar shrinkage step on the obtained 𝐱. For iterative-shrinkage algorithms, IHT and GraDeS use a negative gradient as the search direction, that is, Landweber iteration [18], but the main drawback of Landweber iteration is its slow performance, that is, a large number of iterations need to obtain the optimal convergence rates [18]. Inspired by the semi-iterative method [18] and hard thresholding, we present an algorithm for solving sparse recovery, which requires less time and fewer iterations.

2. Background on Compressed Sensing

2.1. Sensing Matrix

Without further information, it is impossible to recover 𝐱 from 𝐲, since 𝐲=Φ𝐱 is highly underdetermined. In order to recover a good estimate of 𝐱 from 𝑀 measurements, the measurement matrix Φ must obey the restricted isometry property (RIP) [2], 1−𝛿_𝑠‖𝐱‖²₂≤‖Φ𝐱‖²₂≤1+𝛿_𝑠‖𝐱‖²₂, (3) for all 𝐱∈𝚺_𝑠, 𝚺_𝑠={𝐱∈𝑅^𝑁∶‖𝐱‖₀≤𝑆} denotes the set of 𝑆-sparse vectors, 𝛿_𝑠∈(0,1) is restricted isometry constant, 𝑆∈{1,2,…,𝑁}, provided that 𝑀≥𝐶⋅𝑆⋅log(𝑁/𝑆), where 𝐶 is some constant depending on each instance. It is difficult to verify the RIP conditions for a given matrix. A widely used technique for avoiding checking the RIP directly is to generate the matrix randomly, such as Gaussian matrix, symmetric Bernoulli matrix, and partial Fourier matrix [1–3], and to show that the resulting random matrix satisfies the RIP with high probability. In this paper, we will use Gaussian matrix as the measurement matrix.

2.2. Sparse Recovery

An alternative approach to sparse signal recovery is based on the idea of iterative greedy pursuit and tries to approximate the solution to (2) directly. In this case, the problem (2) is closely related to the following optimization problem: 1min2‖𝐲−Φ𝐱‖²₂s.t.‖𝐱‖₀≤𝑠, (4) where 𝑆 denotes the sparse level of the vector 𝐱.

The second one is convex relaxation. In this case, the problem (2) is closely related to the following optimization problem: min‖𝐱‖₁s.t.𝐲=Φ𝐱. (5) However, these methods are often inefficient, requiring many iterations and excessive central processing unit time to reach their solutions [15].

An alternative family of numerical algorithms has gradually built, addressing the above optimization problems very effectively [15]. This family is the iterative-shrinkage algorithms. We will discuss iterative-shrinkage algorithms in the next section.

3. Semi-Iterative Hard Thresholding

The main drawback of Landweber iteration is its comparatively slow rate of convergence while for Landweber iteration only information about the last iterate 𝐱^[𝑘−1] is used to construct the new approximation 𝐱^[𝑘]. In order to overcome the drawback, more sophisticated iteration methods have been developed on the basis of the so-called semi-iterative methods. A basic step of a semi-iterative method (polynomial acceleration methods) consists of one step of iteration, followed by an averaging process over all or some of the previously obtained approximations. A basic step of a semi-iterative method has the form 𝐱^[𝑘]=𝜇_1,𝑘𝐱^[𝑘−1]+𝜇_2,𝑘𝐱^[𝑘−2]+⋯+𝜇_𝑘,𝑘𝐱^[0]+𝜔_𝑘Φ^T𝐲−Φ𝐱^[𝑘−1], (6) where ∑^𝑘_𝑖=1𝜇_𝑖,𝑘=1,𝜔_𝑘≠0,𝑘≥1. An example for semi-iterative methods with optimal rate of convergence are the 𝛾-methods (two-step methods) by [18], which are defined by 𝐱^[𝑘]=𝐱^[𝑘−1]+𝜇_𝑘𝐱^[𝑘−1]−𝐱^[𝑘−2]+𝜔_𝑘Φ^T𝐲−Φ𝐱^[𝑘−1], (7) where 𝜇₁=0,𝜔₁=4𝛾+2,𝜔4𝛾+1_𝑘=42𝑘+2𝛾−1𝑘+𝛾−1,𝜇𝑘+2𝛾−12𝑘+2𝛾−1_𝑘=(𝑘−1)(2𝑘−3)2𝑘+2𝛾−1,𝐱𝑘+2𝛾−12𝑘+4𝛾−12𝑘+2𝛾−3_[−1]=𝐱^[0]=𝐱₀. (8)

From (4), the gradient of the cost function 𝑓(𝐱)=(1/2)‖𝐲−Φ𝐱‖²₂ is given by ∇𝑓(𝐱)=Φ^T(Φ𝐱−𝐲) and easy to compute the step length 𝛼 that minimizes 𝑓(𝐱^[𝑘]−𝛼∇𝑓_𝑘). By differentiating the function 𝜑(𝛼)=𝑓(𝐱^[𝑘]−𝛼∇𝑓_𝑘) with respect to 𝛼, we obtain 𝜑^′(𝛼)=𝑓𝐱^[𝑘]−𝛼∇𝑓_𝑘^T∇𝑓_𝑘. (9) By setting the derivative to zero, we obtain 𝛼_𝑘=∇𝑓^T_𝑘∇𝑓_𝑘∇𝑓^T_𝑘Φ^TΦ∇𝑓_𝑘. (10) If we choose the step lengths by (10), thus 𝑓(𝐱^[𝑘]−𝛼∇𝑓_𝑘)^T∇𝑓_𝑘=0, that is, the search direction ∇𝑓_𝑘+1=∇𝑓(𝐱^[𝑘]−𝛼∇𝑓_𝑘) is orthogonal to the gradient ∇𝑓_𝑘 (previous search direction). In this case, the sequence of iterations is subject to zigzags. Since IHT and GraDeS use the negative gradient of the cost function 𝑓(𝐱)=(1/2)‖𝐲−Φ𝐱‖²₂ as the search direction, and sampling matrix Φ must obey the RIP, that is, ‖Φ‖²₂≈1, which means 𝛼_𝑘≈1, thus the iteration zigzag toward the solution. As a result, a large number of iterations need to obtain the optimal solution.

In order to avoid zigzagging toward solution and find the sparse solution for (4), inspired by the 𝛾-methods [18] as mentioned above, we present the semi-iterative hard thresholding method, which has the form 𝐱^[𝑘]=𝑃_𝑠𝐱^[𝑘−1]+𝜇_𝑘𝐱^[𝑘−1]−𝐱^[𝑘−2]+𝜔_𝑘Φ^T𝐲−Φ𝐱^[𝑘−1], (11) where 𝑃_𝑠(⋅) is the nonlinear operator that sets all but the largest (in magnitude) 𝑆 elements of a vector to zero. From (11), we use the linear combination of the current negative gradient Φ^T(𝐲−Φ𝐱^[𝑘−1]) and the search direction of the previous step (𝐱^[𝑘−1]−𝐱^[𝑘−2]) as the new search direction. In this case, the search direction 𝜇_𝑘(𝐱^[𝑘−1]−𝐱^[𝑘−2])+𝜔_𝑘Φ^T(𝐲−Φ𝐱^[𝑘−1]) dose not tend to become orthogonal to the gradient ∇𝑓_𝑘; thus SIHT avoids zigzagging toward solution. The algorithm is summarized as in Algorithm 1.

Step 1. Initialize 𝐱[−1]=𝐱[0]=𝐱0 Step 2. Compute for 𝑘≥1, for a given 𝛾.    𝜇1=0,   𝜔1=(4𝛾+2)/(4𝛾+1)    𝜔𝑘=42𝑘+2𝛾−1𝑘+𝛾−1/𝑘+2𝛾−12𝑘+2𝛾−1, 𝑘≥2    𝜇𝑘=(𝑘−1)(2𝑘−3)2𝑘+2𝛾−1/𝑘+2𝛾−12𝑘+4𝛾−12𝑘+2𝛾−3, 𝑘≥2 𝐚[𝑘]=𝐱[𝑘−1]+𝜇𝑘(𝐱[𝑘−1]−𝐱[𝑘−2])+𝜔𝑘ΦT(𝑦−Φ𝐱[𝑘−1]) Step 3. Compute 𝐱[𝑘]=𝑃𝑠(𝐚[𝑘]) Step 4. Repeat step2-step3, until stopping criterion    𝐲−Φ𝐱[𝑘]2≤𝜀‖𝐲‖2 is satisfied.

Algorithm 1 

Semi-iterative hard thresholding algorithm.

As mentioned above, the semi-iterative hard thresholding algorithm is easy to implement. It involves the application of the matrix Φ and Φ^T at each iteration as well as two vector additions. The storage requirements are small. Apart from storage of 𝐲, we only require the storage of the vector 𝐱^[𝑘−1] and 𝐱^[𝑘−2], which require two 𝑆 elements to be stored. The choice of the parameter 𝛾 will be discussed in the next section.

4. Experimental Results

This section describes some experiments testifying to the performances of the proposed algorithm. All the experiments were carried out on HP z600 workstation with eight Intel Xeon 2.13 GHz processors and 16 GB of memory, using a MATLAB implementation under Windows XP.

4.1. Choice of the Parameter 𝛾

In our experiment, we consider a typical CS scenario, where the goal is to reconstruct a length-𝑁 sparse vector from 𝑀 measurements. In this case, first, the 𝑀×𝑁 random matrix Φ is created by filling it with entries generated independently and identically distribution and then orthogonalizing the rows. Second, original vector 𝐱 contains 𝑆 randomly placed ±1 spikes, and the measurement 𝐲 is generated according to (1). Unless otherwise stated, we terminate the iteration after ‖𝐲−Φ𝐱^[𝑘]‖²₂≤𝜀‖𝐲‖²₂, with 𝜀=10⁻¹⁰.

The experiment assesses how the running time of the proposed algorithm grows with the parameter 𝛾. In order to find a better optimization parameter 𝛾 in the experiment, we set the parameter 𝛾, respectively, to 𝛾∈{2,3,…,19,20}, whilst the running time of our method is computed. Figure 1 shows the running time of our algorithm as the parameter 𝛾 is varied. The label 𝑀/𝑁/𝑆 stands for 𝑀 measurements and 𝑆 sparse length-𝑁 vector in our experiments. A careful examination reveals that as parameter 𝛾 is increased, the running time of our method is minimized with respect to 𝛾=7. For 4≤𝛾≤20, the running time increases only marginally as 𝛾 is increased, that is, the choice of parameter 𝛾 appears to give good performance for a wide range of problems.

Figure 1 

Influence of different parameter gamma on running time.

4.2. Comparison in Recovery Rate

In this experiment, we compared the empirical performance of GraDes, IHT, SpaRSA, FISTA, and SIHT solutions to the sparse recovery. We generated a Gaussian 𝐍(0,1) random matrix Φ∈𝑅^256×512 and generated 𝑆 sparse ±1 spikes vector. The reconstruction is considered to be exact when the 2 norm of the difference between the reconstruction and original vector is below 10⁻². We repeated the experiment 100 times for each value of 𝑆 from 2 to 128 (in steps of 2). Figure 2 shows that SIHT algorithm provides higher probability of perfect recovery than GraDes, IHT, SpaRSA, and FISTA, when the sparse vectors are drawn ±1 spikes. Furthermore, in the perfect recovery case, we observe that the GraDes and SpaRSA algorithms perform similarly. While it reveals that measurements of SIHT require less than those of IHT, GraDes, SpaRSA, and FISTA to recover the sparse vector for a given 𝑁 and 𝑆.

Figure 2 

Simulation of the exact recovery rate.

4.3. Comparison in Running Time

In order to evaluate running time of the proposed algorithm, these experiments include comparisons with OMP, StOMP, ROMP, IHT, and GraDeS. Now, the sampling matrix Φ, the measurement vector 𝐲, and the sparsity level 𝑆 are given to each of the algorithms as inputs. For the proposed algorithm, we set 𝛾=7; the performance is insensitive to the choices. Table 1 compares the running time of the MATLAB implementation of SIHT and the five existing methods. The symbol “∞” indicates the algorithm fails.

Table 1 

The running time of different algorithm.

Table 1 shows that the iterative-shrinkage algorithms are significantly faster than match pursuit algorithms. For simplicity, we will compare the performance of SIHT with IHT and GraDes in the next experiments.

4.4. Comparison in Sparsity

In this experiment, we show the dependence of the 2-norm errors of different algorithms in different sparsity level 𝑆. In Figure 3, we show the 2-norm errors of SIHT comparison with IHT, GraDes, SpaRSA, and FISTA in different sparsity levels. We generated a Gaussian 𝐍(0,1) random matrix Φ∈𝑅^512×1024 and generated 𝑆 sparse ±1 spikes vector or Gaussian vector. We repeated the experiment 100 times for each value of 𝑆 from 2 to 120 (in steps of 10). Both GraDes and SpaRSA begin to fail when sparsity level is above 120; thus the failed results are omitted from the figure.

(a)

(b)

Figure 3 

Recover error versus sparsity with fixed 𝑀=512and𝑁=1024: (a) Gaussian sparse vectors and (b) sparse ±1 spikes vectors.

Figure 3 shows that GraDes, IHT, and SIHT algorithms perform similarly for Gaussian sparse vectors, and GraDes algorithm is to fail in recovery for sparse ±1 spikes vectors when sparsity level 𝑆 is above 70, that is, GraDes algorithm requires more measurements to recover the sparse vectors. It reveals that FISTA, IHT, and SIHT algorithms are insensitive to the sparsity level 𝑆, whilst GraDes and SpaRSA algorithms are sensitive to the sparsity level 𝑆. In addition, SIHT algorithm outperforms other algorithms in 2-norm errors for sparse ±1 spikes or Gaussian vector.

4.5. Comparison in Number of Iterations

In the experiment, we show the number of iterations required by SIHT algorithm in comparison with four algorithms, namely, IHT, GraDes, SpaRSA, and FISTA for sparse ±1 spikes vector or Gaussian vector. We generated a Gaussian 𝐍(0,1) random matrix Φ∈𝑅^512×1024 and generated 𝑆 sparse ±1 spikes or Gaussian vector. Figures 4 and 5 show the number of iterations needed by the algorithms as mentioned above for 𝑀=512, 𝑁=1024, and 𝑆=20.

(a)

(b)

Figure 4 

Recover error versus number of iteration with fixed 𝑀=512, 𝑁=1024, and 𝑆=20 for Gaussian sparse vectors: (a) iteration between 2 and 120 and (b) iteration between 2 and 20.

(a)

(b)

Figure 5 

Recover error versus number of iteration with fixed 𝑀=512, 𝑁=1024, and 𝑆=20 for sparse ±1 spikes vectors: (a) iteration between 1 and 120 and (b) iteration between 1 and 20.

Figures 4 and 5 depict that IHT and GraDes algorithms show a faster rate of convergence, when the number of iteration is less than 4. However, when the number of iteration is above 6, owing to polynomial acceleration, FISTA and SIHT algorithms show a faster rate of convergence than the others. In addition, from Figures 4 and 5, for each 𝑆≥7, FISTA and IHT algorithms are roughly similar in terms of number of iterations. In that SIHT algorithm uses the linear combination of the current gradient and directions of a few previous steps as the new search direction, SIHT algorithm shows a faster rate of convergence than the others. While GraDes algorithm exhibits poorer performance than the others in rate of convergence.

From Figures 4 and 5, the 2-norm errors of those algorithms except SpaRSA algorithm are insensitive to the number of iterations, that is, 2-norm errors are strictly monotone reduced as iteration is increased.

As expected, these results suggest that SIHT outperforms other iterative-shrinkage algorithms in iterations and 2-norm errors.

5. Conclusions

In this paper, semi-iterative hard thresholding recovery algorithm for sparse recovery was proposed in this work. The proposed algorithm uses the linear combination of the current gradient and directions of a few previous steps as the new search direction and avoids zigzagging toward solution. Owing to using the new search direction, the performance of SIHT is improved compared with iterative-shrinkage algorithms.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant no. 61271294). The authors would like to thank Arvind Ganesh, Allen Y. Yang, and Zihan Zhou for sharing their software packages (L1benchmark) with us.