﻿ 基于l1-al2最小化的部分支集已知的信号重建 Signal Reconstruction with Known Partial Support Based on l1-al2 Minimization

Vol. 11  No. 08 ( 2022 ), Article ID: 55197 , 14 pages
10.12677/AAM.2022.118634

Signal Reconstruction with Known Partial Support Based on ${\mathcal{l}}_{1}-\alpha {\mathcal{l}}_{2}$ Minimization

Siqi Wu*#, Ruying Song

Department of Mathematics, Taiyuan Normal University, Jinzhong Shanxi

Received: Jul. 24th, 2022; accepted: Aug. 17th, 2022; published: Aug. 26th, 2022

ABSTRACT

The restricted isometry property of the measurement matrix in compressed sensing can ensure the reconstruction of sparse signals under certain conditions. In this paper, the sufficient conditions for signal recovery under three kinds of noise (Gaussian noise, impulse noise and uniform noise) are studied according to the known prior support information of the signal and the restricted isometry property of the measurement matrix under ${\mathcal{l}}_{1}-\alpha {\mathcal{l}}_{2}$ $\left(0<\alpha \le 1\right)$ minimization model. These conditions intuitively reveal the close relationship between the restricted isometry property of the measurement matrix and signal recovery.

Keywords:Compressed Sensing, Signal Recovery, ${\mathcal{l}}_{1}-\alpha {\mathcal{l}}_{2}$ Minimization, Restricted Isometry Property, Partial Support Is Known

1. 引言

$\underset{x\in {ℝ}^{n}}{\mathrm{min}}{‖x‖}_{0}$ s.t. $b-Αx\in B,$ (1)

$\underset{x\in {ℝ}^{n}}{\mathrm{min}}{‖x‖}_{1}$ s.t. $b-Αx\in B,$ (2)

$\underset{x\in {ℝ}^{n}}{\mathrm{min}}{‖x‖}_{1}-\alpha {‖x‖}_{2}$ s.t. $b-Αx\in B,$ (3)

$\underset{x\in {ℝ}^{n}}{\mathrm{min}}{‖{x}_{{T}^{c}}‖}_{1}-\alpha {‖{x}_{{T}^{c}}‖}_{2}$ s.t. $b-Αx\in B,$ (4)

$\underset{x\in {ℝ}^{n}}{\mathrm{min}}{‖{x}_{{T}^{c}}‖}_{1}-\alpha {‖{x}_{{T}^{c}}‖}_{2}$ s.t. $b=Αx,$ (5)

2. 预备知识

$\left(1-{\delta }_{s}^{lb}\right){‖x‖}_{2}^{p}\le {‖Αx‖}_{p}^{p}\le \left(1+{\delta }_{s}^{ub}\right){‖x‖}_{2}^{p}$ (6)

$\left(1-{\delta }_{s}\right){‖x‖}_{2}^{2}\le {‖Αx‖}_{2}^{2}\le \left(1+{\delta }_{s}\right){‖x‖}_{2}^{2}$ (7)

$\mu \left(Α\right):=\underset{i\ne j}{\mathrm{max}}\frac{|〈{Α}_{i},{Α}_{j}〉|}{{‖{Α}_{i}‖}_{2}{‖{Α}_{j}‖}_{2}}.$

${‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\le {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2},$ (8)

${‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}-\alpha {‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}\le {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2},$ (9)

${‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\le {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+\alpha {‖h‖}_{2},$ (10)

${‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}-\alpha {‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}\le {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+\alpha {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2},$ (11)

1) 对 $0\le \alpha \le 1$，令 $T=\mathrm{supp}\left(x\right)$${‖x‖}_{0}=s$，则

$\left(s-\alpha \sqrt{s}\right)\underset{j\in T}{\mathrm{min}}|{x}_{j}|\le {‖x‖}_{1}-\alpha {‖x‖}_{2}\le \left(\sqrt{s}-\alpha \right){‖x‖}_{2}.$ (12)

2) 令 $S,{S}_{1},{S}_{2}\subseteq \left[n\right]$ 满足 $S={S}_{1}\cup {S}_{2}$${S}_{1}\cap {S}_{2}=\varnothing$，则

${‖{x}_{{S}_{1}}‖}_{1}-\alpha {‖{x}_{{S}_{1}}‖}_{2}+{‖{x}_{{S}_{2}}‖}_{1}-\alpha {‖{x}_{{S}_{2}}‖}_{2}\le {‖{x}_{S}‖}_{1}-\alpha {‖{x}_{S}‖}_{2}.$ (13)

${‖{h}_{{\stackrel{¯}{{T}_{01}}}^{c}}‖}_{2}\le \frac{1}{2\sqrt{t}}\left({‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}}{\sqrt{r+s}}\right),$ (14)

${‖h‖}_{2}\le \left(1+\frac{1}{2\sqrt{t}}\right){‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{t\left(r+s\right)}}+\frac{\alpha {‖h‖}_{2}}{2\sqrt{t\left(r+s\right)}},$ (15)

${‖Αh‖}_{1}\ge {\rho }_{k}{‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}-\frac{2\left(1+{\delta }_{k}^{ub}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{k}-\alpha },$ (16)

${\rho }_{k}=1-{\delta }_{r+s+k}^{lb}-\frac{1+{\delta }_{k}^{ub}}{a\left(r+s,k;\alpha \right)}$

$a\left(r+s,k;\alpha \right)=\frac{\sqrt{k}-\alpha }{\sqrt{r+s}+\alpha }$

$v=\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{u}^{\left(i\right)},$

$0<{\lambda }_{i}\le 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}=1,$ (17)

$\mathrm{supp}\left({u}^{\left(i\right)}\right)\subseteq \mathrm{supp}\left(v\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖{u}^{\left(i\right)}‖}_{0}\le s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖{u}^{\left(i\right)}‖}_{\infty }\le \left(1+\frac{\sqrt{2}}{2}\right)\theta ,$

$\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\le \left({\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(s-\sqrt{s}\right)+1\right){\theta }^{2}.$ (18)

$\text{χ}=\frac{\sqrt{r+s}+\alpha }{\sqrt{r+s}-1}\frac{{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{\sqrt{r+s}}+\frac{2}{r+s-\sqrt{r+s}}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1},$ (19)

${\text{W}}_{1}=\left\{i:|{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}\left(i\right)|>\frac{\text{χ}}{t-1}\right\},$ (20)

${\text{W}}_{2}=\left\{i:|{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}\left(i\right)|\le \frac{\text{χ}}{t-1}\right\},$ (21)

${h}_{{\text{W}}_{2}}=\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{u}^{\left(i\right)},$ (22)

$\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\le \frac{{\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1}{{\left(t-1\right)}^{2}}{\text{χ}}^{2}.$ (23)

3. 主要结论

${B}^{{\mathcal{l}}_{2}}\left(\epsilon \right)=\left\{e:{‖e‖}_{2}\le {\epsilon }_{1}\right\};$ (24)

${B}^{{\mathcal{l}}_{1}}\left(\epsilon \right)=\left\{e:{‖e‖}_{1}\le {\epsilon }_{1}\right\};$ (25)

${B}^{DS}\left(\epsilon \right)=\left\{e:{‖{Α}^{Τ}e‖}_{\infty }\le {\epsilon }_{1}\right\}.$ (26)

${\delta }_{t\left(r+s\right)}<\frac{1}{\sqrt{1+\frac{{\left(\sqrt{r+s}+\alpha \right)}^{2}\left({\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1\right)}{\left(r+s\right){\left(t-1\right)}^{2}{\left(\sqrt{r+s}-1\right)}^{2}}}},$ (27)

$\begin{array}{c}{‖{\stackrel{^}{x}}^{{\mathcal{l}}_{2}}-x‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)\\ \text{\hspace{0.17em}}+\left(\left(\sqrt{r+s}+\sqrt{\frac{{\alpha }^{2}}{4}+\alpha \sqrt{r+s}+\left(r+s\right)}+\frac{\alpha +\sqrt{2}}{2}\right)\left(\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right)\left(\sqrt{r+s}+\alpha \right)}\\ \text{\hspace{0.17em}}+\sqrt{\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right){\left(\sqrt{r+s}+\alpha \right)}^{2}}}\right)+\frac{\sqrt{2\left(r+s\right)}}{2}\right)\frac{{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}},\end{array}$ (28)

${‖Αh‖}_{2}\le {‖Α{\stackrel{^}{x}}^{{\mathcal{l}}_{2}}-y‖}_{2}+{‖y-Αx‖}_{2}\le {\epsilon }_{1}+{\epsilon }_{2}.$ (29)

${h}_{{w}_{2}}=\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{u}^{\left(i\right)},$ (30)

$\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\le \frac{{\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1}{{\left(t-1\right)}^{2}}{\text{χ}}^{2},$ (31)

$\begin{array}{c}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\le \frac{{\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1}{{\left(t-1\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(\frac{{\left(\sqrt{r+s}+\alpha \right)}^{2}}{\left(r+s\right){\left(\sqrt{r+s}-1\right)}^{2}}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}^{2}+\frac{4\left(\sqrt{r+s}+\alpha \right)}{\left(r+s\right){\left(\sqrt{r+s}-1\right)}^{2}}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\frac{4}{\left(r+s\right){\left(\sqrt{r+s}-1\right)}^{2}}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}^{2}\right)\\ =\frac{1-2\mu }{{\mu }^{2}}\left({‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}^{2}+\frac{4{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}+\alpha }+\frac{4{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}^{2}}{{\left(\sqrt{r+s}+\alpha \right)}^{2}}\right),\end{array}$ (32)

$\underset{i=1}{\overset{N}{\sum }}\frac{{\lambda }_{i}}{4}{‖Α\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}+\mu {u}^{\left(i\right)}\right)‖}_{2}^{2}=\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}+\mu h\right)‖}_{2}^{2}.$ (34)

$\begin{array}{c}\underset{i=1}{\overset{N}{\sum }}\frac{{\lambda }_{i}}{4}{‖Α\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}+\mu {u}^{\left(i\right)}\right)‖}_{2}^{2}\ge \left(1-{\delta }_{t\left(r+s\right)}\right)\underset{i=1}{\overset{N}{\sum }}\frac{{\lambda }_{i}}{4}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}+\mu {u}^{\left(i\right)}‖}_{2}^{2}\\ =\frac{1-{\delta }_{t\left(r+s\right)}}{4}\left({‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}^{2}+{\mu }^{2}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\right),\end{array}$ (35)

$\begin{array}{l}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}+\mu h\right)‖}_{2}^{2}\\ =\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}\right)‖}_{2}^{2}+2\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}〈Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}\right),\mu Αh〉+{\mu }^{2}{‖Αh‖}_{2}^{2}\\ =\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}\right)‖}_{2}^{2}+\left(1-\mu \right)\mu 〈Α\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right),Αh〉\\ \le \left(1+{\delta }_{t\left(r+s\right)}\right)\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}‖}_{2}^{2}+\left(1-\mu \right)\mu {‖Α\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)‖}_{2}{‖Αh‖}_{2}\\ \le \left(1+{\delta }_{t\left(r+s\right)}\right)\left({\left(\frac{1}{2}-\mu \right)}^{2}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}^{2}+\frac{{\mu }^{2}}{4}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\right)+\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\left({\epsilon }_{1}+{\epsilon }_{2}\right).\end{array}$ (36)

$\begin{array}{l}\left(\left(1+{\delta }_{t\left(r+s\right)}\right){\left(\frac{1}{2}-\mu \right)}^{2}-\frac{1-{\delta }_{t\left(r+s\right)}}{4}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}^{2}+\frac{{\mu }^{2}{\delta }_{t\left(r+s\right)}}{2}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\\ +\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\left({\epsilon }_{1}+{\epsilon }_{2}\right)\ge 0.\end{array}$

$\begin{array}{l}\left({\mu }^{2}-\mu +{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}^{2}+\left(\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)+\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\sqrt{r+s}+\alpha }{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\\ +\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{{\left(\sqrt{r+s}+\alpha \right)}^{2}}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}^{2}\ge 0.\end{array}$ (37)

${\delta }_{t\left(r+s\right)}<\frac{\mu }{1-\mu }=\frac{1}{\sqrt{1+\frac{{\left(\sqrt{r+s}+\alpha \right)}^{2}\left({\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1\right)}{\left(r+s\right){\left(t-1\right)}^{2}{\left(\sqrt{r+s}-1\right)}^{2}}}}.$

$\begin{array}{l}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\\ \le \frac{\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)+\left(\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\sqrt{r+s}+\alpha }+\sqrt{\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{{\left(\sqrt{r+s}+\alpha \right)}^{2}}\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right)}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}.\end{array}$ (38)

$\begin{array}{c}{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}^{2}\le {‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{\infty }\\ \le \left(\alpha {‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}+\left(\sqrt{r+s}+\alpha \right){‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}+2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\right)\frac{{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{\sqrt{r+s}}\\ =\frac{\alpha }{\sqrt{r+s}}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}+\frac{\sqrt{r+s}+\alpha }{\sqrt{r+s}}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}^{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{\sqrt{r+s}}.\end{array}$

${\left({‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}-\frac{\alpha {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{2\sqrt{r+s}}\right)}^{2}\le \left(\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}^{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{\sqrt{r+s}}.$

$2|a||b|\le \frac{{\left(|a|+|b|\right)}^{2}}{2}$，可以得到

$\begin{array}{c}{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{2}\le \left(\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha }{2\sqrt{r+s}}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}+\sqrt{\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}}{\sqrt{r+s}}}\\ \le \left(\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}+\frac{\sqrt{2}}{2}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}.\end{array}$ (39)

$\begin{array}{c}{‖h‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right){‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{2}+\frac{\sqrt{2}}{2}{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\\ \le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)\\ \text{\hspace{0.17em}}+\left(\left(\sqrt{r+s}+\sqrt{\frac{{\alpha }^{2}}{4}+\alpha \sqrt{r+s}+\left(r+s\right)}+\frac{\alpha +\sqrt{2}}{2}\right)\left(\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right)\left(\sqrt{r+s}+\alpha \right)}\\ \text{\hspace{0.17em}}+\sqrt{\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right){\left(\sqrt{r+s}+\alpha \right)}^{2}}}\right)+\frac{\sqrt{2\left(r+s\right)}}{2}\right)\frac{{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}},\end{array}$

${‖{\stackrel{^}{x}}^{{\mathcal{l}}_{2}}-x‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{1+{\delta }_{t\left(r+s\right)}}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right),$

${\delta }_{t\left(r+s\right)}^{ub}+a\left(r+s,k;\alpha \right){\delta }_{r+s+k}^{lb} (40)

$\begin{array}{c}{‖x-\stackrel{^}{x}‖}_{2}\le \frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{2\sqrt{t\left(r+s\right)}-\alpha }\left(\frac{1}{{\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)+\frac{2\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{{\rho }_{t\left(r+s\right)}\left(\sqrt{t\left(r+s\right)}-\alpha \right)}\right)+\frac{2}{2\sqrt{t\left(r+s\right)}-\alpha }{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\\ =\frac{2}{2\sqrt{k}-\alpha }\left(1+\frac{\left(2\sqrt{t}+1\right)\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right)\sqrt{r+s}}{{\rho }_{t\left(r+s\right)}\left(\sqrt{k}-\alpha \right)}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{\left(2\sqrt{k}-\alpha \right){\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right).\end{array}$ (41)

${‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\le {‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}.$ (42)

${‖e‖}_{1}={‖b-Αx‖}_{1}\le {\epsilon }_{2}$${‖b-Α{\stackrel{^}{x}}^{{\mathcal{l}}_{1}}‖}_{1}\le {\epsilon }_{1}$，可以得到

${‖Αh‖}_{1}={‖Α{\stackrel{^}{x}}^{{\mathcal{l}}_{1}}-Αx‖}_{1}\le {‖Α{\stackrel{^}{x}}^{{\mathcal{l}}_{1}}-b‖}_{1}+{‖b-Αx‖}_{1}\le {\epsilon }_{1}+{\epsilon }_{2}.$ (43)

${T}_{0}$ 是h中 $s\in {ℤ}_{+}$ 个最大绝对值项组成的指标集， ${T}_{1}$${h}_{{T}_{0}{}^{c}}$$k=t\left(r+s\right)\in {ℤ}_{+}$ 个最大绝对值项组成的指标集且 ${T}_{\stackrel{¯}{01}}=T\cup {T}_{0}\cup {T}_{1}$。然后，通过引理4的式(16)和式(43)，得到

$\left(1-{\delta }_{r+s+k}^{lb}-\frac{1+{\delta }_{t\left(r+s\right)}^{ub}}{a\left(r+s,k;\alpha \right)}\right){‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}-\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right)\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{k}-\alpha }\le {\epsilon }_{1}+{\epsilon }_{2},$ (44)

${\rho }_{t\left(r+s\right)}=1-{\delta }_{r+s+k}^{lb}-\frac{1+{\delta }_{t\left(r+s\right)}^{ub}}{a\left(r+s,k;\alpha \right)}>0.$

${‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}\le \frac{1}{{\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)+\frac{\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right)\sqrt{r+s}}{{\rho }_{t\left(r+s\right)}\left(\sqrt{k}-\alpha \right)}\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}}.$ (45)

$\begin{array}{c}{‖{h}_{{\stackrel{¯}{{T}_{01}}}^{c}}‖}_{2}\le \sqrt{{‖{h}_{{\stackrel{¯}{{T}_{01}}}^{c}}‖}_{1}{‖{h}_{{\stackrel{¯}{{T}_{01}}}^{c}}‖}_{\infty }}\le \sqrt{\left({‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}-\underset{j\in {T}_{1}}{\sum }|{h}_{j}|\right)|{h}_{r+s+k}|}\\ \le \sqrt{\left({‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}-k|{h}_{r+s+k}|\right)|{h}_{r+s+k}|}=\sqrt{-k{\left(|{h}_{r+s+k}|-\frac{{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{2k}\right)}^{2}+\frac{{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}^{2}}{4k}}\le \frac{{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{2\sqrt{k}}.\end{array}$ (46)

$\begin{array}{c}\frac{{‖{h}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{2\sqrt{k}}\le \frac{{‖{h}_{\stackrel{¯}{{T}_{0}}}‖}_{1}+2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}}{2\sqrt{k}}\\ \le \frac{1}{2}\sqrt{\frac{r+s}{k}}\left({‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}}{\sqrt{r+s}}\right)\\ =\frac{1}{2\sqrt{t}}\left({‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}}{\sqrt{r+s}}\right).\end{array}$ (47)

$\begin{array}{c}{‖h‖}_{2}\le \sqrt{{‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}^{2}+\frac{1}{4t}{\left({‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\alpha {‖h‖}_{2}}{\sqrt{r+s}}\right)}^{2}}\\ \le \left(1+\frac{1}{2\sqrt{t}}\right){‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{1}{2\sqrt{t}}\frac{2{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}}+\frac{1}{2\sqrt{t}}\frac{\alpha {‖h‖}_{2}}{\sqrt{r+s}},\end{array}$ (48)

${‖h‖}_{2}\le \frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{2\sqrt{t\left(r+s\right)}-\alpha }{‖{h}_{\stackrel{¯}{{T}_{01}}}‖}_{2}+\frac{2}{2\sqrt{t\left(r+s\right)}-\alpha }{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}.$ (49)

$\begin{array}{c}{‖h‖}_{2}\le \frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{2\sqrt{t\left(r+s\right)}-\alpha }\left(\frac{1}{{\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)+\frac{2\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{{\rho }_{t\left(r+s\right)}\left(\sqrt{t\left(r+s\right)}-\alpha \right)}\right)+\frac{2}{2\sqrt{t\left(r+s\right)}-\alpha }{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}\\ =\frac{2}{2\sqrt{t\left(r+s\right)}-\alpha }\left(1+\frac{\left(2\sqrt{t}+1\right)\left(1+{\delta }_{t\left(r+s\right)}^{ub}\right)\sqrt{r+s}}{{\rho }_{t\left(r+s\right)}\left(\sqrt{t\left(r+s\right)}-\alpha \right)}\right){‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}+\frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{\left(2\sqrt{t\left(r+s\right)}-\alpha \right){\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right).\end{array}$

${‖{\stackrel{^}{x}}^{{\mathcal{l}}_{1}}-x‖}_{2}\le \frac{\left(2\sqrt{t}+1\right)\sqrt{r+s}}{\left(2\sqrt{t\left(r+s\right)}-\alpha \right){\rho }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right),$

${\delta }_{t\left(r+s\right)}<\frac{1}{\sqrt{1+\frac{{\left(\sqrt{r+s}+\alpha \right)}^{2}\left({\left(1+\frac{\sqrt{2}}{2}\right)}^{2}\left(t\left(r+s\right)-\left(r+s\right)-\sqrt{t\left(r+s\right)-\left(r+s\right)}\right)+1\right)}{\left(r+s\right){\left(t-1\right)}^{2}{\left(\sqrt{r+s}-1\right)}^{2}}}},$ (50)

$\begin{array}{c}{‖{\stackrel{^}{x}}^{DS}-x‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{t\left(r+s\right)}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)\\ \text{\hspace{0.17em}}+\left(\left(\sqrt{r+s}+\sqrt{\frac{{\alpha }^{2}}{4}+\alpha \sqrt{r+s}+\left(r+s\right)}+\frac{\alpha +\sqrt{2}}{2}\right)\left(\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right)\left(\sqrt{r+s}+\alpha \right)}\\ \text{\hspace{0.17em}}+\sqrt{\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right){\left(\sqrt{r+s}+\alpha \right)}^{2}}}\right)+\frac{\sqrt{2\left(r+s\right)}}{2}\right)\frac{{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}}.\end{array}$ (51)

${‖{Α}^{Τ}Αh‖}_{\infty }\le {‖{Α}^{Τ}\left(Α{\stackrel{^}{x}}^{DS}-y\right)‖}_{\infty }+{‖{Α}^{Τ}\left(y-Αx\right)‖}_{\infty }\le {\epsilon }_{1}+{\epsilon }_{2}$ (52)

$〈Α\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right),Αh〉=〈{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}},{Α}^{Τ}Αh〉\le {‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{1}{‖{Α}^{Τ}Αh‖}_{\infty }\le \sqrt{t\left(r+s\right)}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\left({\epsilon }_{1}+{\epsilon }_{2}\right).$ (53)

$\begin{array}{l}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖Α\left(\left(\frac{1}{2}-\mu \right)\left({h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}\right)-\frac{1}{2}\mu {u}^{\left(i\right)}+\mu h\right)‖}_{2}^{2}\\ \le \left(1+{\delta }_{t\left(r+s\right)}\right)\left({\left(\frac{1}{2}-\mu \right)}^{2}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}^{2}+\frac{{\mu }^{2}}{4}\underset{i=1}{\overset{N}{\sum }}{\lambda }_{i}{‖{u}^{\left(i\right)}‖}_{2}^{2}\right)+\left(1-\mu \right)\mu \sqrt{t\left(r+s\right)}{‖{h}_{\stackrel{¯}{{T}_{0}}}+{h}_{{\text{W}}_{1}}‖}_{2}\left({\epsilon }_{1}+{\epsilon }_{2}\right).\end{array}$

$\begin{array}{c}{‖h‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{t\left(r+s\right)}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right)\\ \text{\hspace{0.17em}}+\left(\left(\sqrt{r+s}+\sqrt{\frac{{\alpha }^{2}}{4}+\alpha \sqrt{r+s}+\left(r+s\right)}+\frac{\alpha +\sqrt{2}}{2}\right)\left(\frac{2{\delta }_{t\left(r+s\right)}\left(1-2\mu \right)}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right)\left(\sqrt{r+s}+\alpha \right)}\\ \text{\hspace{0.17em}}+\sqrt{\frac{2\left(1-2\mu \right){\delta }_{t\left(r+s\right)}}{\left(\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}\right){\left(\sqrt{r+s}+\alpha \right)}^{2}}}\right)+\frac{\sqrt{2\left(r+s\right)}}{2}\right)\frac{{‖{x}_{{\stackrel{¯}{{T}_{0}}}^{c}}‖}_{1}}{\sqrt{r+s}}.\end{array}$

${‖{\stackrel{^}{x}}^{DS}-x‖}_{2}\le \left(1+\sqrt{\frac{{\alpha }^{2}}{4\left(r+s\right)}+\frac{\alpha +\sqrt{r+s}}{\sqrt{r+s}}}+\frac{\alpha +\sqrt{2}}{2\sqrt{r+s}}\right)\frac{\left(1-\mu \right)\mu \sqrt{t\left(r+s\right)}}{\mu -{\mu }^{2}-{\left(\mu -1\right)}^{2}{\delta }_{t\left(r+s\right)}}\left({\epsilon }_{1}+{\epsilon }_{2}\right),$

4. 结论

Signal Reconstruction with Known Partial Support Based on l1-al2 Minimization[J]. 应用数学进展, 2022, 11(08): 6015-6028. https://doi.org/10.12677/AAM.2022.118634

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24. NOTES

*第一作者。

#通讯作者。