A Note on Approximating Curve with 1-Norm Regularization Method for the Split Feasibility Problem

Abstract

Inspired by the very recent results of Wang and Xu (2010), we study properties of the approximating curve with 1-norm regularization method for the split feasibility problem (SFP). The concept of the minimum-norm solution set of SFP in the sense of 1-norm is proposed, and the relationship between the approximating curve and the minimum-norm solution set is obtained.

1. Introduction

Let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively. The problem under consideration in this paper is formulated as finding a point 𝑥 satisfying the property: 𝑥∈𝐶,𝐴𝑥∈𝑄,(1.1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. Problem (1.1), referred to by Censor and Elfving [1] as the split feasibility problem (SFP), attracts many authors’ attention due to its application in signal processing [1]. Various algorithms have been invented to solve it (see [2–13] and references therein).

Using the idea of Tikhonov's regularization, Wang and Xu [14] studied the properties of the approximating curve for the SFP. They gave the concept of the minimum-norm solution of the SFP (1.1) and proved that the approximating curve converges strongly to the minimum-norm solution of the SFP (1.1). Together with some properties of this approximating curve, they introduced a modification of Byrne’s CQ algorithm [2] so that strong convergence is guaranteed and its limit is the minimum-norm solution of SFP (1.1).

In the practical application, 𝐻₁ and 𝐻₂ are often ℝ^𝑁 and ℝ^𝑀, respectively. Moreover, scientists and engineers are more willing to use 1-norm regularization method in the calculation process (see, e.g., [15–18]). Inspired by the above results of Wang and Xu [14], we study properties of the approximating curve with 1-norm regularization method. We also define the concept of the minimum-norm solution set of SFP (1.1) in the sense of 1-norm. The relationship between the approximating curve and the minimum-norm solution set is obtained.

2. Preliminaries

Let 𝑋 be a normed linear space with norm ‖⋅‖, and let 𝑋^∗ be the dual space of 𝑋. We use the notation ⟨𝑥,𝑓⟩ to denote the value of 𝑓∈𝑋^∗ at 𝑥∈𝑋. In particular, if 𝑋 is a Hilbert space, we will denote it by 𝐻, and ⟨⋅,⋅⟩ and ‖⋅‖ are the inner product and its induced norm, respectively.

We recall some definitions and facts that are needed in our study.

Let 𝑃_𝐶 denote the projection from 𝐻 onto a nonempty closed convex subset 𝐶 of 𝐻; that is, 𝑃_𝐶𝑥=argmin_𝑦∈𝐶‖𝑥−𝑦‖,𝑥∈𝐻.(2.1) It is well known that 𝑃_𝐶𝑥 is characterized by the inequality ⟨𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0,∀𝑦∈𝐶.(2.2)

Definition 2.1. Let 𝜑∶𝑋→ℝ∪{+∞} be a convex functional, 𝑥₀∈dom(𝜑)={𝑥∈𝑋∶𝜑(𝑥)<+∞}. Set 𝜕𝜑𝑥₀=𝜉∈𝑋^∗∶𝜑(𝑥)≥𝜑𝑥₀+⟨𝑥−𝑥₀,𝜉⟩,∀𝑥∈𝑋.(2.3) If 𝜕𝜑(𝑥₀)≠∅, 𝜑 is said to be subdifferentiable at 𝑥₀ and 𝜕𝜑(𝑥₀) is called the subdifferential of 𝜑 at 𝑥₀. For any 𝜉∈𝜕𝜑(𝑥₀), we say 𝜉 is a subgradient of 𝜑 at 𝑥₀.

Lemma 2.2. There holds the following property: 𝜕(‖𝑥‖)=𝑥^∗𝑥∈𝑋^∗∶‖𝑥^∗‖=1,⟨𝑥,𝑥^∗⟩=‖𝑥‖,𝑥≠0,_∗∈𝑋^∗∶‖𝑥^∗‖≤1,𝑥=0,(2.4) where 𝜕(‖𝑥‖) denotes the subdifferential of the functional ‖𝑥‖ at 𝑥∈𝑋.

Proof. The process of the proof will be divided into two parts.
Case  1. In the case of 𝑥=0, for any 𝑥^∗∈𝑋^∗ such that ‖𝑥^∗‖≤1 and any 𝑦∈𝑋, there holds the inequality ‖𝑦‖≥⟨𝑦,𝑥^∗⟩=‖𝑥‖+⟨𝑦−𝑥,𝑥^∗⟩,(2.5) so we have 𝑥^∗∈𝜕(‖𝑥‖), and thus, 𝑥^∗∈𝑋^∗∶‖𝑥^∗‖≤1⊂𝜕(‖𝑥‖).(2.6) Conversely, for any 𝑥^∗∈𝜕(‖𝑥‖), we have from the definition of subdifferential that ‖𝑦‖‖≥𝑦‖‖𝑥=‖‖+−⟨𝑦𝑦‖−≥𝑥,⟨𝑥−^∗⟩𝑦=,𝑥⟨_∗𝑦,𝑥^∗⟩,∀𝑦∈𝑋,⟩=−⟨𝑦,𝑥^∗⟩.(2.7) Consequently, ||⟨𝑦,𝑥^∗⟩||≤‖𝑦‖,∀𝑦∈𝑋,(2.8) and this implies that ‖𝑥^∗‖≤1. Thus, we have verified that 𝜕(‖𝑥‖)⊂𝑥^∗∈𝑋^∗∶‖𝑥^∗‖≤1.(2.9) Combining (2.6) and (2.9), we immediately obtain 𝜕(‖𝑥‖)=𝑥^∗∈𝑋^∗∶‖𝑥^∗‖≤1.(2.10)
Case  2. If 𝑥≠0, for any 𝑥^∗∈{𝑥^∗∈𝑋^∗∶‖𝑥^∗‖=1,⟨𝑥,𝑥^∗⟩=‖𝑥‖}, we obviously have ⟨𝑦−𝑥,𝑥^∗⟩=⟨𝑦,𝑥^∗⟩−‖𝑥‖≤‖𝑦‖−‖𝑥‖,∀𝑦∈𝑋,(2.11) which means that 𝑥^∗∈𝜕(‖𝑥‖), and thus, 𝑥^∗∈𝑋^∗∶‖𝑥^∗‖=1,⟨𝑥,𝑥^∗⟩=‖𝑥‖⊂𝜕(‖𝑥‖).(2.12) Conversely, if 𝑥^∗∈𝜕(‖𝑥‖), we have ⟨−𝑥,𝑥^∗⟩≤0−‖𝑥‖=−‖𝑥‖,⟨𝑥,𝑥^∗⟩≤2‖𝑥‖−‖𝑥‖=‖𝑥‖;(2.13) hence, ⟨𝑥,𝑥^∗⟩=‖𝑥‖.(2.14) On the other hand, using (2.14), we get ‖𝑦‖≥‖𝑥‖+⟨𝑦−𝑥,𝑥^∗⟩=‖𝑥‖+⟨𝑦,𝑥^∗⟩−⟨𝑥,𝑥^∗⟩=⟨𝑦,𝑥^∗⟩,∀𝑦∈𝑋,(2.15) and consequently, ‖𝑦‖==‖‖𝑥−‖𝑦−‖⟨≥𝑦,𝑥‖_∗𝑥‖+⟨−𝑦−𝑥,𝑥^∗⟩=−⟨𝑦,𝑥_∗⟩−⟨𝑥,𝑥^∗⟩⟩;(2.16) that is, −‖𝑦‖≤⟨𝑦,𝑥^∗⟩.(2.17) Equation (2.17) together with (2.15) implies that ||⟨𝑦,𝑥^∗⟩||≤‖𝑦‖,∀𝑦∈𝑋;(2.18) hence, ‖𝑥^∗‖≤1. Note that (2.14) implies that ‖𝑥^∗‖≥⟨𝑥,𝑥^∗⟩/‖𝑥‖=1; we assert that ‖𝑥^∗‖=1.(2.19) Thus we have from (2.14) and (2.19) that 𝑥^∗∈𝑋^∗∶‖𝑥^∗‖=1,⟨𝑥,𝑥^∗⟩=‖𝑥‖⊃𝜕(‖𝑥‖).(2.20) The proof is finished by combining (2.12) and (2.20).

‖⋅‖_∞ and ‖⋅‖₁ will stand for ∞-norm and 1-norm of any Euclidean space; respectively, that is, for any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑙)∈ℝ^𝑙, we have ‖𝑥‖_∞=max_{1≤𝑗≤𝑙}||𝑥_𝑗||,‖𝑥‖₁=^𝑙_𝑗=1||𝑥_𝑗||.(2.21)

Corollary 2.3. In l-dimensional Euclidean space ℝ^𝑙, there holds the following result: 𝜕‖𝑥‖₁=𝜉∈ℝ^𝑙𝜉∈ℝ_𝑙∶‖𝜉‖_∞=1,⟨𝑥,𝜉⟩=‖𝑥‖₁,𝑥≠0,=⎧⎪⎨⎪⎩𝜉∈ℝ_𝑙∶‖𝜉‖_∞≤1,𝑥=0,∶𝜉_𝑖=𝑥^𝑖||𝑥_𝑖||,if𝑥^𝑖𝜉∈ℝ_𝑙≠0;𝜉_𝑖∈[−1,1],if𝑥_𝑖=0,𝑥≠0,∶‖𝜉‖_∞≤1,𝑥=0.(2.22) Let 𝐻 be a Hilbert space and 𝑓∶𝐻→ℝ a functional. Recall that (i)𝑓is convex if 𝑓(𝜆𝑥+(1−𝜆)𝑦)≤𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦∈𝐻;(ii)𝑓 is strictly convex if 𝑓(𝜆𝑥+(1−𝜆)𝑦)<𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦∈𝐻 with 𝑥≠𝑦;(iii)𝑓 is coercive if 𝑓(𝑥)→∞ whenever ‖𝑥‖→∞. See [19] for more details about convex functions.

The following lemma gives the optimality condition for the minimizer of a convex functional over a closed convex subset.

Lemma 2.4 (see [20]). Let 𝐻 be a Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑓∶𝐻→ℝ be a convex and subdifferentiable functional. Then 𝑥∈𝐶 is a solution of the problem min_𝑥∈𝐶𝑓(𝑥)(2.23) if and only if there exists some 𝜉∈𝜕𝑓(𝑥) satisfying the following optimality condition: ⟨𝜉,𝑣−𝑥⟩≥0,∀𝑣∈𝐶.(2.24)

3. Main Results

It is well known that SFP (1.1) is equivalent to the minimization problem min_𝑥∈𝐶‖‖𝐼−𝑃_𝑄𝐴𝑥‖‖².(3.1) Using the idea of Tikhonov's regularization method, Wang and Xu [14] studied the minimization problem in Hilbert spaces: min_𝑥∈𝐶‖‖𝐼−𝑃_𝑄𝐴𝑥‖‖²+𝛼‖𝑥‖²,(3.2) where 𝛼>0 is the regularization parameter.

In what follows, 𝐻₁ and 𝐻₂ in SFP (1.1) are restricted to ℝ^𝑁 and ℝ^𝑀, respectively, and ‖⋅‖ will stand for the usual 2-norm of any Euclidean space ℝ^𝑙; that is, for any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑙)∈ℝ^𝑙, ‖𝑥‖=𝑥²¹+⋯+𝑥^2𝑙.(3.3) Inspired by the above work of Wang and Xu, we study properties of the approximating curve with 1-norm regularization scheme for the SFP, that is, the following minimization problem: min_𝑥∈𝐶12‖‖𝐼−𝑃^𝑄𝐴𝑥‖‖²+𝛼‖𝑥‖₁,(3.4) where 𝛼>0 is the regularization parameter. Let 𝑓_𝛼(𝑥)=12‖‖𝐼−𝑃^𝑄𝐴𝑥‖‖²+𝛼‖𝑥‖₁.(3.5) It is easy to see that 𝑓_𝛼 is convex and coercive, so problem (3.4) has at least one solution. However, the solution of problem (3.4) may not be unique since 𝑓_𝛼 is not necessarily strictly convex. Denote by 𝑆_𝛼 the solution set of problem (3.4); thus we can assert that 𝑆_𝛼 is a nonempty closed convex set but may contain more than one element. The following simple example illustrates this fact.

Example 3.1. Let 𝐶={(𝑥,𝑦)∶𝑥+𝑦=1}, 𝑄={(𝑥,𝑦)∶𝑥+𝑦=1/2} and 𝐴=⎛⎜⎜⎝201012⎞⎟⎟⎠.(3.6) Then 𝐴∶ℝ²→ℝ² is a bounded linear operator. Obviously, 𝑆_𝛼={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥≥0,𝑦≥0} and it contains more than one element.

Proposition 3.2. For any 𝛼>0, 𝑥_𝛼∈𝑆_𝛼 if and only if there exists some 𝜉∈𝜕(‖𝑥‖₁) satisfying the following inequality: 𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝛼+𝛼𝜉,𝑣−𝑥_𝛼≥0,∀𝑣∈𝐶.(3.7)

Proof. Let 𝑓(𝑥)=12‖‖𝐼−𝑃^𝑄𝐴𝑥‖‖²,(3.8) then 𝑓_𝛼(𝑥)=𝑓(𝑥)+𝛼‖𝑥‖₁.(3.9) Since 𝑓 is convex and differentiable with gradient ∇𝑓(𝑥)=𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,(3.10)𝑓_𝛼 is convex, coercive, and subdifferentiable with the subdifferential 𝜕𝑓_𝛼(𝑥)=𝜕𝑓(𝑥)+𝛼𝜕‖𝑥‖₁;(3.11) that is, 𝜕𝑓_𝛼(𝑥)=𝐴^∗𝐼−𝑃_𝑄𝐴𝑥+𝛼𝜕‖𝑥‖₁.(3.12) By Corollary 2.3 and Lemma 2.4, the proof is finished.

Theorem 3.3. Denote by 𝑥_𝛼 an arbitrary element of 𝑆_𝛼, then the following assertions hold:(i)‖𝑥_𝛼‖₁ is decreasing for 𝛼∈(0,∞);(ii)‖(𝐼−𝑃_𝑄)𝐴𝑥_𝛼‖ is increasing for 𝛼∈(0,∞).

Proof. Let 𝛼>𝛽>0, for any 𝑥_𝛼∈𝑆_𝛼, 𝑥_𝛽∈𝑆_𝛽. We immediately obtain 12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛼‖‖²+𝛼‖‖𝑥_𝛼‖‖₁≤12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛽1‖‖²+𝛼‖‖𝑥_𝛽‖‖₁,(3.13)2‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛽‖‖²+𝛽‖‖𝑥_𝛽‖‖₁≤12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛼‖‖²+𝛽‖‖𝑥_𝛼‖‖₁.(3.14) Adding up (3.13) and (3.14) yields 𝛼‖‖𝑥_𝛼‖‖₁+𝛽‖‖𝑥_𝛽‖‖₁≤𝛼‖‖𝑥_𝛽‖‖₁+𝛽‖‖𝑥_𝛼‖‖₁,(3.15) which implies ‖𝑥_𝛼‖₁≤‖𝑥_𝛽‖₁. Hence (i) holds.
Using (3.14) again, we have 12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛽‖‖²≤12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛼‖‖²+𝛽‖‖𝑥_𝛼‖‖₁−‖‖𝑥_𝛽‖‖₁,(3.16) which together with (i) implies ‖‖𝐼−𝑃_𝑄𝐴𝑥_𝛽‖‖²≤‖‖𝐼−𝑃_𝑄𝐴𝑥_𝛼‖‖²,(3.17) and hence (ii) holds.

Let ℱ=𝐶∩𝐴⁻¹(𝑄), where 𝐴⁻¹(𝑄)={𝑥∈ℝ^𝑁∶𝐴𝑥∈𝑄}. In what follows, we assume that ℱ≠∅; that is, the solution set of SFP (1.1) is nonempty. The fact that ℱ is nonempty closed convex set thus allows us to introduce the concept of minimum-norm solution of SFP (1.1) in the sense of norm ‖⋅‖ (induced by the inner product).

Definition 3.4 (see [14]). An element 𝑥^†∈ℱ is said to be the minimum-norm solution of SFP (1.1) in the sense of norm ‖⋅‖ if ‖𝑥^†‖=inf_𝑥∈ℱ‖𝑥‖. In other words, 𝑥^† is the projection of the origin onto the solution set ℱ of SFP (1.1). Thus there exists only one minimum-norm solution of SFP (1.1) in the sense of norm ‖⋅‖, which is always denoted by 𝑥^†.
We can also give the concept of minimum-norm solution of SFP (1.1) in other senses.

Definition 3.5. An element 𝑥̃∈ℱ is said to be a minimum-norm solution of SFP (1.1) in the sense 𝑜𝑓 1-norm if ‖𝑥̃‖₁=inf_𝑥∈ℱ‖𝑥‖₁. We use ℱ₁ to stand for all minimum-norm solutions of SFP (1.1) in the sense of 1-norm and ℱ₁ is called the minimum-norm solution set of SFP (1.1) in the sense of 1-norm.
Obviously, ℱ₁ is a closed convex subset of ℱ. Moreover, it is easy to see that ℱ₁≠∅. Indeed, taking a sequence {𝑥_𝑛}⊂ℱ such that ‖𝑥_𝑛‖₁→inf_𝑥∈ℱ‖𝑥‖₁ as 𝑛→∞, then {𝑥_𝑛} is bounded. There exists a convergent subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛}. Set 𝑥⁻=lim_𝑘→∞𝑥_𝑛𝑘, then 𝑥⁻∈ℱ since ℱ is closed. On the other hand, using lower semicontinuity of the norm, we have ‖‖𝑥⁻‖‖≤lim_𝑘→∞‖‖𝑥_𝑛𝑘‖‖=inf_𝑥∈ℱ‖𝑥‖₁,(3.18) and this implies that 𝑥⁻∈ℱ₁.
However, ℱ₁ may contain more than one elements, in general (see Example 3.1, ℱ₁={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥,𝑦≥0}).

Theorem 3.6. Let 𝛼>0 and 𝑥_𝛼∈𝑆_𝛼. Then 𝜔(𝑥_𝛼)⊂ℱ₁, where 𝜔(𝑥_𝛼)={𝑥∶∃{𝑥_𝛼𝑘}⊂{𝑥_𝛼},𝑥_𝛼𝑘→𝑥𝑤𝑒𝑎𝑘𝑙𝑦}.

Proof. Taking 𝑥̃∈ℱ₁ arbitrarily, for any 𝛼∈(0,∞), we always have 12‖‖𝐼−𝑃^𝑄𝐴𝑥_𝛼‖‖²+𝛼‖‖𝑥_𝛼‖‖₁≤12‖‖𝐼−𝑃^𝑄𝐴𝑥̃‖‖²+𝛼‖𝑥̃‖₁.(3.19) Since 𝑥̃ is a solution of SFP (1.1), ‖(𝐼−𝑃_𝑄)𝐴𝑥̃‖=0. This implies that 12‖‖(𝐼−𝑃^𝑄)𝐴𝑥_𝛼‖‖²+𝛼‖‖𝑥_𝛼‖‖₁≤𝛼‖𝑥̃‖₁,(3.20) then, ‖‖𝑥_𝛼‖‖₁≤‖𝑥̃‖₁;(3.21) thus {𝑥_𝛼} is bounded.
Take 𝜔∈𝜔(𝑥_𝛼) arbitrarily, then there exists a sequence {𝛼_𝑛} such that 𝛼_𝑛→0 and 𝑥_𝛼𝑛→𝜔 as 𝑛→∞. Put 𝑥_𝛼𝑛=𝑥_𝑛. By Proposition 3.2, we deduce that there exists some 𝜉_𝑛∈𝜕(‖𝑥_𝑛‖₁) such that 𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+𝛼_𝑛𝜉_𝑛,𝑥̃−𝑥_𝑛≥0.(3.22) This implies that 𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝐴𝑥̃−𝑥_𝑛≥𝛼_𝑛⟨𝜉_𝑛,𝑥_𝑛−𝑥̃⟩.(3.23) Since 𝐴𝑥̃∈𝑄, the characterizing inequality (2.2) gives 𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝐴𝑥̃−𝑃_𝑄𝐴𝑥_𝑛≤0,(3.24) then, ‖‖𝐼−𝑃_𝑄𝐴𝑥_𝑛‖‖²≤𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝐴𝑥_𝑛−𝑥̃.(3.25) Combining (3.23) and (3.25), we have ‖‖𝐼−𝑃_𝑄𝐴𝑥_𝑛‖‖²≤≤𝛼𝛼_𝑛𝑛⟨𝜉_𝑛,𝑥̃−𝑥_𝑛⟩‖‖𝜉_𝑛‖‖_∞‖‖𝑥̃−𝑥_𝑛‖‖₁≤2𝛼_𝑛‖𝑥̃‖₁.(3.26) Consequently, we get lim_𝑛→∞‖‖𝐼−𝑃_𝑄𝐴𝑥_𝑛‖‖=0.(3.27) Furthermore, noting the fact that 𝑥_𝑛→𝜔 and 𝐼−𝑃_𝑄 and 𝐴 are all continuous operators, we have (𝐼−𝑃_𝑄)𝐴𝜔=0; that is, 𝐴𝜔∈𝑄; thus, 𝜔∈ℱ. Since 𝑥̃ is a minimum-norm solution of SFP (1.1) in the sense of 1-norm, using (3.21) again, we get ‖𝜔‖₁≤liminf_𝑛→∞‖‖𝑥_𝑛‖‖₁≤‖𝑥̃‖₁=min‖𝑥‖₁∶𝑥∈ℱ.(3.28) Thus we can assert that 𝜔∈ℱ₁ and this completes the proof.

Corollary 3.7. If ℱ₁ contains only one element 𝑥̃, then 𝑥_𝛼→𝑥̃, (𝛼→0).

Remark 3.8. It is worth noting that the minimum-norm solution of SFP (1.1) in the sense of norm ‖⋅‖ is very different from the minimum-norm solution of SFP (1.1) in the sense of 1-norm. In fact, 𝑥^† may not belong to ℱ₁! The following simple example shows this fact.

Example 3.9. Let 𝐶={(𝑥,𝑦)∶𝑥+2𝑦≥2,𝑥≥0,𝑦≥0},  𝑄={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥≥0,𝑦≥0}, and 𝐴=⎛⎜⎜⎝12010⎞⎟⎟⎠.(3.29) It is not hard to see that 𝐴∶ℝ²→ℝ² is a bounded linear operator and 𝐴(𝑥,𝑦)^𝑇=((1/2)𝑥,𝑦)^𝑇, for  all (𝑥,𝑦)∈𝐶. Obviously, ℱ={(𝑥,𝑦)∶𝑥+2𝑦=2,𝑥≥0,𝑦≥0}, 𝑥^†=(2/5,4/5), but ℱ₁={(0,1)}. Hence, 𝑥^†∈ℱ⧵ℱ₁.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (ZXH2012K001) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing. W. Zhu was also supported by the Postgraduate Science and Technology Innovation Funds (YJSCX12-22).