Appendix
1.1 Proof of Theorem 1
Before proving Theorem 1, we will introduce the following Lemmas.
Lemma 1
[13, 14] For disjoint support\({{{\mathcal{T}}}}_{1}\),\({{{\mathcal{T}}}}_{2}\)where\(\left| {{{{\mathcal{T}}}}_{1} } \right| \le S_{1}\),\(\left| {{{{\mathcal{T}}}}_{2} } \right| \le S_{2}\), \({{{\mathcal{T}}}}_{1} \cap {{{\mathcal{T}}}}_{2} = \emptyset\), the chunk sparse matrix\({\tilde{\mathbf{H}}}\)with support set\({{{\mathcal{T}}}}_{1}\)has the following inequality condition
$$\text{If}\;S_{1} \le S_{2} ,\;\text{then},\;\delta_{{S_{1} }} \le \delta_{{S_{2} }} .$$
(40)
$$\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }}^{H} {\mathbf{\tilde{\Psi }\tilde{H}}}} \right\|_{F} = \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }}^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} \le \delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right| + \left| {{{{\mathcal{T}}}}_{2} } \right|}} \left\| {{\tilde{\mathbf{H}}}} \right\|_{F} .$$
(41)
$$\left( {1 - \frac{{\delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right| + \left| {{{{\mathcal{T}}}}_{2} } \right|}} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{2} } \right|}} } \right)} }}} \right)\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} \le \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} \le \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} .$$
(42)
$$\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{1} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} \le {{\delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right| + \left| {{{{\mathcal{T}}}}_{2} } \right|}} } \mathord{\left/ {\vphantom {{\delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right| + \left| {{{{\mathcal{T}}}}_{2} } \right|}} } {\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{2} } \right|}} } \right)} }}} \right. \kern-0pt} {\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{1} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}_{2} } \right|}} } \right)} }}\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{2} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}_{2} }} } \right\|_{F} .$$
(43)
Lemma 2
[14] Suppose that\(\sigma_{\hbox{min} } \left( {\mathbf{A}} \right)\)and\(\sigma_{\hbox{max} } \left( {\mathbf{A}} \right)\)represents the maximum and minimum singular values of\({\mathbf{A}}\), respectively, we have
$$\sigma_{\hbox{min} } \left( {\mathbf{A}} \right)\left\| {\mathbf{B}} \right\|_{F} \le \left\| {{\mathbf{AB}}} \right\|_{F} \le \sigma_{\hbox{max} } \left( {\mathbf{A}} \right)\left\| {\mathbf{B}} \right\|_{F}$$
(44)
The proof of Theorem 1: According to the orthogonal projection theorem, we have
$$\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} = {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} = 0$$
(45)
Using the above formula, the unequal relationship between \(\left\| {{\mathbf{R}}^{k} } \right\|_{F}\) and \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F}\) is expressed as
$$\begin{aligned} \left\| {{\mathbf{R}}^{k} } \right\|_{F} &= \left\| {{\mathbf{Y}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} {\mathbf{\bar{\tilde{H}}}}_{{{{{\mathcal{T}}}}^{k} }}^{k} } \right\|_{F} = \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right)} \right.\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}}} + {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right) + \left. {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right){\mathbf{N}}} \right\|_{F} \\ &\le \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right){\mathbf{N}}} \right\|_{F} \\ &\le \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\mathbf{N}} \right\|_{F} \\ &\le \sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} } \right|}} } \left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\mathbf{N}} \right\|_{F} \\ &\le \sqrt {1 + \delta_{S} } \left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(46)
In the above formula, (45) is used in the first inequality and the projection theorem \(\left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k} }} } \right)^{\dag } } \right){\mathbf{X}}} \right\|_{F} \le \left\| {\mathbf{X}} \right\|_{F}\) and Lemma 1 (40) are used in the second inequality and the last inequality, respectively. Similarly, we can obtain the unequal relationship between \(\left\| {{\mathbf{R}}^{k - 1} } \right\|_{F}\) and \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F}\).
$$\begin{aligned} \left\| {{\mathbf{R}}^{k - 1} } \right\|_{F} &= \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right)\left( {{\tilde{\mathbf{\Psi }}}_{{{{\mathcal{T}}}}} {\tilde{\mathbf{H}}} + {\mathbf{N}}} \right)} \right\|_{F} \\ &\ge \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\varPsi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} - \left\| {\left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\mathbf{N}}} \right\|_{F} \\ &\ge \left( {1 - \frac{{\delta_{S} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}^{k - 1} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} } \right|}} } \right)} }}} \right)\left\| {{\tilde{\mathbf{\varPsi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} - \left\| {\mathbf{N}} \right\|_{F} \\ &\ge \frac{{1 - 2\delta_{S} }}{{\sqrt {1 - \delta_{S} } }}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} - \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(47)
Lemma 1 (40) and (45) are adopted in the second and the last inequality. From (46) and (47), we know that the relationship between \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F}\) and \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F}\) must be get for the proof of (31) which will be demonstrated below.
In the step 11, the estimated error \(\Delta {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}^{k} }}\) can be expressed as equality (48) when the effective support is \({{{\mathcal{T}}}}^{k}\) in the \(k\)th iteration.
$$\begin{aligned} \Delta {\mathbf{\tilde{H}}}_{{{{\mathcal{T} }}^{k} }} & = {\mathbf{\bar{\tilde{H}}}}^{k} - {\mathbf{\tilde{H}}}_{{{{\mathcal{T}}}^{k} {{ }}}} \\ & = \left( {{\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T}}}^{k} }} } \right)^{\dag } {\mathbf{Y}} - {\mathbf{\tilde{H}}}_{{{{\mathcal{T}}}^{k} {{}}}} \\ & = \left( {{\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T}}}^{k} }} } \right)^{\dag } \left( {{\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T} }}^{k} }} {\mathbf{\tilde{H}}}_{{{{\mathcal{T} }}^{k} }} + {\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T}}}\begin{array}{*{20}c} {{{\backslash \mathcal{T} }}^{k} } \\ \end{array} }} {\mathbf{\tilde{H}}}_{{{{\mathcal{T}}}\begin{array}{*{20}c} {{{\backslash \mathcal{T} }}^{k} } \\ \end{array} }} + {\mathbf{N}}} \right) - {\mathbf{\tilde{H}}}_{{{{\mathcal{T}}}^{k} {{ }}}} \\ & = \left( {{\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T} }}^{k} }} } \right)^{\dag } {\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T}}}\begin{array}{*{20}c} {{{\backslash \mathcal{T}}}^{k} {{ }}} \\ \end{array} }} {\mathbf{\tilde{H}}}_{{{{\mathcal{T}}}\begin{array}{*{20}c} {{{\backslash \mathcal{T}}}^{k} {{}}} \\ \end{array} }} + \left( {{\mathbf{\tilde{\Psi }}}_{{{{\mathcal{T} }}^{k} }} } \right)^{\dag } {\mathbf{N}} \\ \end{aligned}$$
(48)
Furthermore, the F norm of the correlation matrix of \({\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }}\) and \(\Delta {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}^{k} }}\) is obtained
$$\begin{aligned} \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \Delta {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}^{k} }} } \right\|_{F} &= \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \left( {{\mathbf{\bar{\tilde{H}}}}^{k} - {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}^{k} }} } \right)} \right\|_{F} \\ &= \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} + {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} } \right)^{\dag } {\mathbf{N}}} \right\|_{F} \\ &\ge \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} } \right\|_{F} - \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} } \right)^{\dag } {\mathbf{N}}} \right\|_{F} \\ &\ge \frac{{\delta_{S} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} } \right\|_{F} - \left\| {\mathbf{N}} \right\|_{F} \\ &\ge \frac{{\delta_{S} \left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)}}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} } \right\|_{F} - \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(49)
The establishment of the second inequality in the above formula takes advantage of the formula (43) in Lemma 1. Similarly, in the step 8, the F norm of the correlation matrix of estimated error \(\Delta {\mathbf{Z}}_{{{{{\mathcal{T}} }}_{b}^{k} }}\) and \({\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }}\) satisfies inequality (50) when the effective support is \({{{\mathcal{T}}}}^{k}\) in the \(k\)th iteration.
$$\begin{aligned} \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \Delta {\mathbf{Z}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right\|_{F} &= \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \left( {{\mathbf{Z}}_{{{{{\mathcal{T}} }}_{b}^{k} }} - {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right)} \right\|_{F} \\ &= \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} + {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right)^{\dag } {\mathbf{N}}} \right\|_{F} \\ &\le \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right)^{\dag } {\mathbf{N}}} \right\|_{F} \\ &\le \frac{{\delta_{S} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}_{b}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)} }}\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\mathbf{N}} \right\|_{F} \\ &\le \frac{{\delta_{S} \left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)}}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}_{b}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)} }}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F} + \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(50)
According to step 7 and 10, we have \({{{\mathcal{T}}}}_{b}^{k} \cap {{{\mathcal{T}}}} \subset {{{\mathcal{T}}}}^{k} \cap {{{\mathcal{T}}}}\). Besides, in the light of the definition of the least squares residual, the measurement residuals of support set \({{{\mathcal{T}}}}_{b}^{k}\) and \({{{\mathcal{T}}}}^{k}\) satisfy
$$\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}_{b}^{k} }} \Delta {\mathbf{Z}}_{{{{{\mathcal{T}} }}_{b}^{k} }} } \right\|_{F} \ge \left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}} }}^{k} }} \Delta {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}} }}^{k} }} } \right\|_{F}$$
(51)
Thus, combining (49), (50) and (51), we have the relationship between \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} } \right\|_{F}\) and \(\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F}\).
$$\begin{aligned} \left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}}}}^{k} } \\ \end{array} }} } \right\|_{F} &\le \frac{{\left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}_{b}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)} \left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)}}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F} + \frac{{2\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}{{\delta_{S} \left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)}}\left\| {\mathbf{N}} \right\|_{F} \hfill \\ &= a\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} }} } \right\|_{F} + b\left\| {\mathbf{N}} \right\|_{F} \hfill \\ \end{aligned}$$
(52)
According to steps 6 and 7, we have the following relationship
$$\begin{aligned} \left\| {\left( {{\varvec{\Psi}}_{{{{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| = \left\| {\left( {{\varvec{\Psi}}_{{{{{\mathcal{T}}}}_{a}^{k} \cup {{{\mathcal{T}}}}^{k} }} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| \\ \ge \left\| {\left( {{\varvec{\Psi}}_{{{{{\mathcal{T}}}}_{a}^{k} }} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| \\ \ge \left\| {\left( {{\varvec{\Psi}}_{{{{\mathcal{T}}}}} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| \\ \end{aligned} ,$$
(53)
which means
$$\left\| {\left( {{\varvec{\Psi}}_{{{{{\mathcal{T}}}}_{b}^{k} \begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}} \\ \end{array} }} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| \ge \left\| {\left( {{\varvec{\Psi}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\|$$
(54)
On the other hand, for the F norm of the correlation matrix \(\left( {{\tilde{\mathbf{\varPsi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\mathbf{R}}^{k - 1}\) based on support \({{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}\) obtained in step 6, we have
$$\begin{aligned} \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\|_{F} &= \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} \left( {{\mathbf{Y}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } {\mathbf{Y}}} \right)} \right\|_{F} \\ &\le \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} \left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{\mathcal{T}}}}} {\mathbf{H}}} \right\|_{F} + \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} \left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\mathbf{N}}} \right\|_{F} \\ \end{aligned}$$
(55)
For convenience, we make the following simplification
$$\begin{aligned} \left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\tilde{\mathbf{\Psi }}}_{{{{\mathcal{T}}}}} {\tilde{\mathbf{H}}} &= \left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right)\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} + {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right) \\ &= {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cap {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} \\ &= \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} ,} & {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \\ { - \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \\ \end{array} } \right] \\ &= {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} \\ \end{aligned}$$
(56)
Submitting (56) into (55), so (55) can be simplified as
$$\begin{aligned} &\left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\mathbf{R}}^{k - 1} } \right\| \le \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} + \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} \left( {{\mathbf{I}} - {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} \left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } } \right){\mathbf{N}}} \right\|_{F} \\ &\quad \le \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} + \sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right|}} } \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(57)
Similarly,
$$\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} \backslash \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\mathbf{R}}^{k - 1} \ge \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} \backslash \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} - \sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} \backslash \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \left\| {\mathbf{N}} \right\|_{F}$$
(58)
Thus, we have
$$\left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} \ge \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} \backslash \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} - 2\sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right|}} } \left\| {\mathbf{N}} \right\|_{F}$$
(59)
For \(\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1}\), the following inequality is hold.
$$\begin{aligned} &\left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} \\ &\quad= \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} \left( {{\tilde{\mathbf{\Psi }}}_{{\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} } \right\}{{\backslash }}\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} } \right\}}} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}_{{\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} } \right\}{{\backslash }}\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} } \right\}}}^{k} + {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} }}^{k} } \right)} \right\|_{F} \\&\quad \ge \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} - \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} } \right\}{{\backslash }}\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} } \right\}}} {\tilde{\mathbf{H}}}_{{\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} } \right\}{{\backslash }}\left\{ {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{\backslash }} \\ \end{array} {{{\mathcal{T}}}}_{b}^{k} } \right\}}}^{k} } \right\|_{F} \\&\quad \ge \sigma_{\hbox{min} } \left( {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} }} } \right)^{H} } \right)\left\| {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} - \delta_{3S} \left\| {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} \\ &\quad\ge \left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} - \delta_{3S} \left\| {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} \\ \end{aligned}$$
(60)
Due to \(\left\{ {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right\} \cap \left\{ {{{{\mathcal{T}}}} \cup {{{\mathcal{T}}}}^{k - 1} } \right\} = \emptyset\), by making use of Lemma 1 we can get
$$\left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}}} } \right)^{H} {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} { \cup {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} \le \delta_{3S} \left\| {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F}$$
(61)
$$\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} \le 2\delta_{3S} \left\| {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} + 2\sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right|}} } \left\| {\mathbf{N}} \right\|_{F}$$
(62)
Besides, it is obvious that \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1}\) satisfies the following inequality condition
$$\begin{aligned} \left\| {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} }}^{k - 1} } \right\|_{F} & \le \left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} + \left\| {\left( {{\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}^{k - 1} }} } \right)^{\dag } {\tilde{\mathbf{\Psi }}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} {\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} \\ & \le \frac{{1 - \delta_{S} + \delta_{2S} }}{{1 - \delta_{S} }}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} \\ \end{aligned}$$
(63)
Combining (61) and (62), the following inequality is obtained
$$\begin{aligned} \left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} \le & \frac{{2\delta_{3S} \left( {1 - \delta_{S} + \delta_{2S} } \right)}}{{\left( {1 - \delta_{S} } \right)}}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} \\ & + \;2\sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right|}} } \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(64)
Thus, we have
$$\begin{aligned} \left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}_{b}^{k} } \\ \end{array} }} } \right\|_{F} & \le \frac{{2\delta_{3S} \left( {1 - \delta_{S} + \delta_{2S} } \right)}}{{\left( {1 - \delta_{S} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)}}\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} + \frac{{2\sqrt {1 + \delta_{{\left| {{{{\mathcal{T}}}}_{b}^{k} \backslash {{{\mathcal{T}}}}} \right|}} } }}{{\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)}}\left\| {\mathbf{N}} \right\|_{F} \\ & = c\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} + d\left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(65)
Substituting Eq. (64) into (52), we can get
$$\left\| {{\tilde{\mathbf{H}}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} }} } \right\|_{F} \le ac\left\| {{\mathbf{H}}_{{{{{\mathcal{T}}}}\begin{array}{*{20}c} {\backslash {{{\mathcal{T}}}}^{k - 1} } \\ \end{array} }} } \right\|_{F} + e\left\| {\mathbf{N}} \right\|_{F}$$
(66)
Combining (46), (47) and (65) we can get
$$\begin{aligned} \left\| {{\mathbf{R}}^{k} } \right\|_{F} & \le \frac{{ac\sqrt {\left( {1 - \delta_{S}^{2} } \right)} }}{{1 - 2\delta_{S} }}\left\| {{\mathbf{R}}^{k - 1} } \right\|_{F} + C_{2} \left\| {\mathbf{N}} \right\|_{F} \\ & = C_{1} \left\| {{\mathbf{R}}^{k - 1} } \right\|_{F} + C_{2} \left\| {\mathbf{N}} \right\|_{F} \\ \end{aligned}$$
(67)
$$\left\| {{\tilde{\mathbf{H}}} - {\mathbf{\hat{\tilde{H}}}}^{k} } \right\|_{F} \le C_{3} \left\| {\mathbf{N}} \right\|_{F}$$
(68)
Below, we will prove that there exists \(C_{1}\) satisfies \(C_{1} < 1\).
$$\begin{aligned} C_{1} & = \frac{{\left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}{{\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}_{b}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)} \left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)}}\frac{{2\delta_{3S} \left( {1 - \delta_{S} + \delta_{2S} } \right)}}{{\left( {1 - \delta_{S} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)}}\frac{{\sqrt {\left( {1 - \delta_{S}^{2} } \right)} }}{{1 - 2\delta_{S} }} \\ & \le \frac{{\left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}_{b}^{k} } \\ \end{array} } \right|}} } \right)\sqrt {\left( {1 - \delta_{{\left| {{{{\mathcal{T}} }}^{k} } \right|}} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)} }}{{\left( {1 - \delta_{S} } \right)\left( {1 + \delta_{{\left| {{{{\mathcal{T}}}}\begin{array}{*{20}c} {{{\backslash {\mathcal{T}} }}^{k} } \\ \end{array} } \right|}} } \right)}}\frac{{2\delta_{3S} \left( {1 - \delta_{S} + \delta_{2S} } \right)}}{{\left( {1 - \delta_{S} } \right)\left( {1 - \delta_{{\left| {{{{\mathcal{T}}}}\backslash {{{\mathcal{T}}}}_{b}^{k} } \right|}} } \right)}}\frac{{\sqrt {\left( {1 - \delta_{S}^{2} } \right)} }}{{1 - 2\delta_{S} }} \\ & \le \frac{{2\delta_{3S} \left( {1 + \delta_{S} } \right)\left( {1 - \delta_{S} + \delta_{2S} } \right)\sqrt {\left( {1 - \delta_{S}^{2} } \right)} }}{{\left( {1 - \delta_{S} } \right)^{3} \left( {1 - 2\delta_{S} } \right)}} \\ \end{aligned}$$
(69)
By numerical calculation we find that when \(\delta_{3S}\) satisfies \(0 \le \delta_{S} \le \delta_{3S} \le 0.17\), we have \(C_{1} < 1\).
