A Review of Atom Recognition Strategies for Greedy Class Reconstruction Algorithms Based on Compressed Sensing
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摘要: 在压缩感知(CS)重构算法中，贪婪类算法因其硬件实现的简易性与良好的恢复精度得到了广泛研究，但算法多样化的同时出现了算法选择困难的问题。原子识别策略作为贪婪类算法的核心，其差异往往决定了算法重构性能的优劣。该文以贪婪类算法最关键的一环原子识别作为研究对象，对贪婪类重构算法的原子识别策略进行了提取与分类。根据不同策略的适用阶段和特点归纳提炼出3种一步式原子识别策略、8种进阶式原子识别策略以及3种稀疏度自适应原子识别策略。最后对原子识别策略所对应原始算法的重构性能进行了分类仿真对比。整理后的策略方便于实际应用中对算法的选择，同时为贪婪类重构算法的进一步优化提供了参考。Abstract: Among the Compressed Sensing(CS) reconstruction algorithms, greedy algorithm has been widely studied for its simple hardware implementation and excellent recovery accuracy. However, the diversity of algorithms also presents the problem of difficult algorithm selection. As the core of greedy algorithms, the difference of atomic recognition strategy often determines its recovery performance. In this paper, atomic recognition strategy, which is the most important part of greedy algorithm, is taken as the research object. Three one-step atom recognition strategies, eight advanced atom recognition strategies and three sparsity adaptive atom recognition strategies are summarized according to the applicable stages and characteristics. Finally, the recovery performance of the original algorithms corresponding to the atomic recognition strategies are simulated and compared. The sorted strategies are convenient for the selection of algorithms in practical application, and they provide references for the further optimization of greedy algorithms.
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算法1 模糊阈值策略[37] 输入：内积向量${\boldsymbol{P}}$，潜在支撑集${{\boldsymbol{C}}_t}$ 执行： (1) 计算模糊阈值：${\boldsymbol{w}} = {\boldsymbol{\alpha }} \times ({P_1} + {P_2}) \times \tau$ (2) 模糊阈值缩减：${\boldsymbol{\varGamma} } = \{ j|\left| { {{\boldsymbol{P}}_{ {C_t} } }(j)} \right| > w\}$ (3) 缩减后的潜在支撑集：${{\boldsymbol{\varLambda}} _t} = {{\boldsymbol{C}}_t} \cap {\boldsymbol{\varGamma}}$ 输出：${{\boldsymbol{\varLambda}} _t}$ 
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算法2 增量最小二乘投影回溯策略[41] 输入：${\boldsymbol{A}}$, ${\boldsymbol{y}}$，开始标志${\rm{TR}}$，当前迭代次数$ t $，候选支撑集${{\boldsymbol{S}}_t}$ (1) 投影：${\hat {\boldsymbol{\theta} } _{ {{\boldsymbol{S}}_t} } } = {\boldsymbol{A} }_{_{ {{\boldsymbol{S}}_t} } }^\dagger {\boldsymbol{y} }$ (2) 判断：如果$t > {\rm{TR}}$并且$ t\% 2 = 1 $，则回溯: 选择出${\hat {\boldsymbol{\theta} } _{ {{\boldsymbol{S}}_t} } }$的绝对值的最小值所对应的索引：
${ {\boldsymbol{D} }_t} = \arg \min (\left| { { {\hat {\boldsymbol{\theta} } }_{ {{\boldsymbol{S}}_t} } } } \right|)$(3) 剔除错误索引并更新索引集：${{\boldsymbol{I}}_t} = {{\boldsymbol{S}}_t}/{{\boldsymbol{D}}_t}$ 输出：${{\boldsymbol{I}}_t}$
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算法3 基于投影的原子选择策略[17] 输入：${\boldsymbol{A}}$,${\boldsymbol{y}}$，最终支撑集${{\boldsymbol{I}}_{t - 1} }$，潜在支撑集${{\boldsymbol{C}}_t}$ 假设：${{\boldsymbol{C}}_t} \cap {{\boldsymbol{I}}_{t - 1} } = \varnothing$ 初始化：$\hat {\boldsymbol{\theta}} = 0 \in {\mathbb{R}^{M \times 1} }$ 执行： (1) 候选支撑集：${{\boldsymbol{S}}_t} = {{\boldsymbol{I}}_{t - 1} } \cup {{\boldsymbol{C}}_t}$ (2) 正交投影：${\hat {\boldsymbol{\theta} } _{ { {\boldsymbol{S} }_t} } } = {\boldsymbol{A} }_{_{ {{\boldsymbol{S}}_t} } }^\dagger {\boldsymbol{y} }$ (3) 部分置0：${\hat {\boldsymbol{\theta} } _{ { {\bar {\boldsymbol{C}}}_t} } } = 0$ (4) 索引集选择：${ {\boldsymbol{\varOmega} } _t} = {\rm{supp} }\_{\rm{max} }\_L({\hat {\boldsymbol{\theta} } _{ {{\boldsymbol{S}}_t} } },L)$ 输出：${{\boldsymbol{\varOmega}} _t}$
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算法4 压缩采样原子选择策略 输入：${\boldsymbol{A}}$, $ {\boldsymbol{y}} $, $ K $，最终支撑集${{\boldsymbol{I}}_{t - 1} }$, 潜在支撑集${{\boldsymbol{C}}_t}$ 初始化：$\hat {\boldsymbol{\theta}} = 0 \in {\mathbb{R}^{M \times 1} }$ 执行： (1)$ t = t + 1 $,${ {\boldsymbol{C} }_t} = {\rm{supp} }\_{\rm{max} }\_L{\text{(} }|\left\langle {{\boldsymbol{A}},{{\boldsymbol{r}}_{t - 1} } } \right\rangle |,2K{\text{)} }$ (2) 候选支撑集：${ {\boldsymbol{S} }_t} = { {\boldsymbol{I} }_{t - 1} } \cup {{\boldsymbol{C}}_t}$ (3) 正交投影：${\hat {\boldsymbol{\theta} } _{ {{\boldsymbol{S}}_t} } } = {\boldsymbol{A} }_{_{ {{\boldsymbol{S}}_t} } }^\dagger {\boldsymbol{y} }$ (4) 更新支撑集：${ {\boldsymbol{I} }_t} = {\rm{supp} }\_{\rm{max} }\_L{\text{(} }{\hat {\boldsymbol{\theta} } _{ {{\boldsymbol{S}}_t} } },K{\text{)} }$ (5) 残差更新：${{\boldsymbol{r}}_t} = {\boldsymbol{y}} - {{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} } }{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} }^\dagger {\boldsymbol{y}}$ 输出：${{\boldsymbol{I}}_t}$
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算法5 展望策略[17, 29] 输入：${\boldsymbol{A}}$,${\boldsymbol{y}}$,$ K $,${{\boldsymbol{I}}_{t - 1} }$，潜在索引$i(i \notin {{\boldsymbol{I}}_{t - 1} })$ 初始化： 迭代次数: $t = {\rm{length}}({{\boldsymbol{I}}_{t - 1} } \cup i)$ 更新支撑集：${{\boldsymbol{I}}_t} = {{\boldsymbol{I}}_{t - 1} } \cup i$ 更新残差：${ {\boldsymbol{r} }_t} = {\boldsymbol{y} } - { {\boldsymbol{A} }_{ { {\boldsymbol{I} }_t} } }{\boldsymbol{A} }_{ { {\boldsymbol{I} }_t} }^\dagger {\boldsymbol{y}}$ 迭代： (1) 迭代次数增加：$ t = t + 1 $ (2) 原子识别${i_t} = {\rm{supp}}\_{\rm{max}}\_L{\text{(} }|\left\langle {{\boldsymbol{A}},{{\boldsymbol{r}}_{t - 1} } } \right\rangle |,1{\text{)} }$ (3) 支撑集扩充：${{\boldsymbol{I}}_t} = {{\boldsymbol{I}}_{t - 1} } \cup {i_t}$ (4) 残差更新：${{\boldsymbol{r}}_t} = {\boldsymbol{y}} - {{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} } }{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} }^\dagger {\boldsymbol{y}}$ (5) 判断是否停止：如果${\left\| { {{\boldsymbol{r}}_t} } \right\|_2} > {\left\| { {{\boldsymbol{r}}_{t - 1} } } \right\|_2}$或者$ t > K $则
$ t = t - 1 $并停止迭代，否则跳转到步骤1继续迭代输出：${\boldsymbol{r}} = {\boldsymbol{y}} - {{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} } }{\boldsymbol{A}}_{ {{\boldsymbol{I}}_t} }^\dagger {\boldsymbol{y}}$, ${{\boldsymbol{I}}_t}$
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算法6 并行展望策略[30] 输入：${\boldsymbol{A}}$,${\boldsymbol{y}}$,$ K $，最终支撑集${{\boldsymbol{I}}_{t - 1} }$，潜在索引集
${ {\boldsymbol{C} }_t}({ {\boldsymbol{C} }_t} \notin {{\boldsymbol{I}}_{t - 1} })$, ${\rm{length}}({{\boldsymbol{C}}_t}) = L$执行： (1) for j = 1 to L $[{\boldsymbol{R} },{ {\boldsymbol{\varLambda} } _j}] = {\rm{Look}}\_{\rm{Ahead}}\_{\rm{Resi}}({\boldsymbol{A}},{\boldsymbol{y}},{\boldsymbol{K}},{{\boldsymbol{I}}_{t - 1} },{{\boldsymbol{C}}_t})$ ${{\boldsymbol{n}}_j} = {\left\| {\boldsymbol{R}} \right\|_2}$ end for (2) 选择出残差$ {l_2} $范数最小值所对应的一组索引：
${ {\boldsymbol{\varLambda} } _t} = \mathop {\arg \min }\limits_j ({{\boldsymbol{n}}_j})$(3) 取交集：${{\boldsymbol{\varGamma}} _t} = {{\boldsymbol{\varLambda}} _t} \cap {{\boldsymbol{C}}_t}$ 输出：${{\boldsymbol{\varGamma}} _t}$
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算法7 正交最小二乘一步展望策略[17] 输入：${\boldsymbol{A}}$,${\boldsymbol{y}}$,${{\boldsymbol{I}}_{t - 1} }$，潜在索引$i(i \notin {{\boldsymbol{I}}_{t - 1} })$ 执行： (1) 候选支撑集扩充：${{\boldsymbol{S}}_t} = {{\boldsymbol{I}}_{t - 1} } \cup i$ (2) 残差更新：${{\boldsymbol{r}}_t} = {\boldsymbol{y}} - {{\boldsymbol{A}}_{ {{\boldsymbol{S}}_t} } }{\boldsymbol{A}}_{ {{\boldsymbol{S}}_t} }^\dagger {\boldsymbol{y}}$ (3) 计算残差$ {l_2} $范数：${\boldsymbol{R} } = {\left\| { {{\boldsymbol{r}}_t} } \right\|_2}$ 输出：${\boldsymbol{R}}$
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算法8 固定步长稀疏度自适应策略[12,26] 输入：当前迭代过程中得到的残差${\boldsymbol{R}}$，上一次迭代后得到的残差
${{\boldsymbol{r}}_{t - 1} }$，最终支撑集${\boldsymbol{ F}}$，当前迭代次数$ t $，步长$ s $，阶段数$ j $判断：if ${\left\| {\boldsymbol{R}} \right\|_2} \le \varepsilon$ 退出主函数大循环 else if ${\left\| {\boldsymbol{R} } \right\|_2} \ge {\left\| { {{\boldsymbol{r}}_{t - 1} } } \right\|_2}$ 阶段数增加：$ j = j + 1 $ 支撑集长度更新：$ L = j \times s $ else 更新最终支撑集：${{\boldsymbol{I}}_t} = {\boldsymbol{F}}$ 更新最终残差：${{\boldsymbol{r}}_t} = {\boldsymbol{R}}$ 迭代次数增加：$ t = t + 1 $ end if 输出： $ L $, ${{\boldsymbol{I}}_t}$, ${{\boldsymbol{r}}_t}$, $ t $
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算法9 两阈值变步长稀疏度自适应策略[16] 输入：当前迭代得到的残差${\boldsymbol{R}}$，上一次迭代后得到的残差
${{\boldsymbol{r}}_{t - 1} }$,$\varepsilon_1$, $\varepsilon_2$，当前稀疏向量估计值$\hat {\boldsymbol{z}}$，上一次迭代后得到
的稀疏向量估计值${\hat {\boldsymbol{x}}_{t - 1} }$，最终支撑集${\boldsymbol{F}}$，当前迭代次数
$ t $，步长$ s $，阶段数$ j $判断：if ${\left\| {\hat {\boldsymbol{z} } - { {\hat {\boldsymbol{x}}}_{t - 1} } } \right\|_2} \ge \varepsilon_1$ if ${\left\| {\boldsymbol{R} } \right\|_2} \ge {\left\| { {{\boldsymbol{r}}_{t - 1} } } \right\|_2}$ 阶段数增加：$ j = j + 1 $ 支撑集长度更新：$ L = L + s $ else 更新最终支撑集：${{\boldsymbol{I}}_t} = {\boldsymbol{F}}$ 更新最终残差：${{\boldsymbol{r}}_t} = {\boldsymbol{R}}$ 迭代次数增加：$ t = t + 1 $ end if else if $\varepsilon_2 \le {\left\| {\hat {\boldsymbol{z} } - { {\hat {\boldsymbol{x} } }_{t - 1} } } \right\|_2} \le \varepsilon_1$ if ${\left\| {\boldsymbol{R} } \right\|_2} \ge {\left\| { {{\boldsymbol{r}}_{t - 1} } } \right\|_2}$ 阶段数增加：$ j = j + 1 $ 步长变化：$ s = \left\lceil {0.5 \times s} \right\rceil $ 支撑集长度更新：$ L = L + s $ else 更新最终支撑集：$ L = L + s $ 更新最终残差：${{\boldsymbol{r}}_t} = {\boldsymbol{R}}$ 迭代次数增加：$ t = t + 1 $ end if else if ${\left\| {\hat {\boldsymbol{z} } - { {\hat {\boldsymbol{x} } }_{t - 1} } } \right\|_2} \le \varepsilon_2$ 退出主函数大循环 end if 输出：$ L $,${{\boldsymbol{I}}_t}$,${{\boldsymbol{r}}_t}$,$ t $
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算法10 基于能量的变步长稀疏度自适应策略[12,33] 输入：当前迭代得到的残差${\boldsymbol{R}}$，上一次迭代后得到的残差
${{\boldsymbol{r}}_{t - 1} }$,${\boldsymbol{y}}$，当前稀疏向量估计值$\hat {\boldsymbol{z}}$，最终支撑集${\boldsymbol{F}}$，当前迭
代次数$ t $，步长$ s $，阶段数$ j $判断： if ${\left\| {\boldsymbol{R}} \right\|_2} \le \varepsilon$ 退出主函数大循环 else if ${\left\| {\boldsymbol{R}} \right\|_2} \ge {\left\| { {{\boldsymbol{r}}_{t - 1} } } \right\|_2}$ if $\dfrac{ {\left\| {\boldsymbol{y}} \right\|_2^2} }{ {\left\| {\hat {\boldsymbol{z}}} \right\|_2^2} } \ge \rho$ 状态数增加：$ j = j + 1 $ 支撑集长度更新：$ L = j \times s $ else 状态数增加：$ j = j + 1 $ (1)支撑集长度更新：$ L = L + \left\lceil {0.5 \times s} \right\rceil $[15] (2)步长变化：$ s = \left\lceil {0.5 \times s} \right\rceil $[42] 支撑集长度更新：$ L = L + s $ end if else 更新最终支撑集：${{\boldsymbol{I}}_t} = {\boldsymbol{F}}$ 更新最终残差：${{\boldsymbol{r}}_t} = {\boldsymbol{R}}$ 迭代次数增加：$ t = t + 1 $ end if end if 输出： $ L $,${{\boldsymbol{I}}_t}$,${{\boldsymbol{r}}_t}$,$ t $
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