SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} = \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} [ (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} - (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}) ^ {- 1} ] \\ + \left[ H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left(H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \\ + \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \\ + \boldsymbol {H} _ {\mathcal {S} _ {C} ^ {*} \mathcal {S} _ {r}} \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \\ \end{array}
$$
$$
\begin{array}{l} \left\| H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \leqslant \left\| H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \right\| _ {B, 1, \infty} \\ + \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \| _ {B, 1, \infty} \\ + \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \| _ {B, 1, \infty} \\ + \| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}) ^ {- 1} \| _ {B, 1, \infty}. \\ \end{array}
$$
By Hoeffding's inequality,
$$
\operatorname {P r o b} \left(\left| \left(\tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}}\right) _ {i j} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}}\right) _ {i j} \right| \geqslant t\right) \leqslant 2 \exp \left(- \frac {m t ^ {2}}{2}\right),
$$
for  $t > 0$ . Taking  $t = \frac{t}{\rho_i|\mathcal{S}_r|}$  and applying the union bound over  $i \in \mathbf{C}\mathbf{o}_r$ , we observe that
$$
\begin{array}{l} \operatorname {P r o b} \left(\| \tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \| _ {B, 1, \infty} \geqslant t\right) \leqslant \sum_ {i \in \mathbf {C o} _ {r}} 2 \rho_ {i} | \mathcal {S} _ {r} | \exp \left(- \frac {m t ^ {2}}{2 \rho_ {i} ^ {2} | \mathcal {S} _ {r} | ^ {2}}\right) \\ \leqslant 2 \left| \mathcal {S} _ {r} ^ {c} \right| \left| \mathcal {S} _ {r} \right| \exp \left(- \frac {m t ^ {2}}{2 \rho_ {\max } ^ {2} \left| \mathcal {S} _ {r} \right| ^ {2}}\right). \\ \end{array}
$$
Similarly, taking  $t = \frac{t}{|\mathcal{S}_r|}$
$$
\begin{array}{l} \operatorname {P r o b} \left(\| \tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \| _ {\infty , \infty} \geqslant t\right) \leqslant \sum_ {i \in \mathcal {S} _ {r}} \sum_ {j \in \mathcal {S} _ {r}} 2 \exp \left(- \frac {m t ^ {2}}{2 | \mathcal {S} _ {r} | ^ {2}}\right) \\ = 2 \left| \mathcal {S} _ {r} \right| ^ {2} \exp \left(- \frac {m t ^ {2}}{2 \left| \mathcal {S} _ {r} \right| ^ {2}}\right). \\ \end{array}
$$
In order to bound  $\| H_{\mathcal{S}_r^c\mathcal{S}_r}^{\mathbb{Q}} - H_{\mathcal{S}_r^c\mathcal{S}_r}\|_{B,1,\infty}$ , for  $\mathbb{Q}\neq \tilde{\mathbb{P}}$ , consider
$$
\begin{array}{l} \left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right\| _ {B, 1, \infty} \leqslant \left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right\| _ {B, 1, \infty} + \left\| \tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right\| _ {B, 1, \infty} \\ \leqslant \mathbb {E} _ {\mathbb {Q}} \| \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {\bar {r}}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B, 1, \infty} + \mathbb {E} _ {\tilde {\mathbb {P}} _ {m}} \| \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {\bar {r}}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B, 1, \infty} \\ = \sup  _ {\tilde {\mathbb {P}} _ {m} ^ {\prime}, \mathbb {Q} ^ {\prime} \in \mathcal {A} _ {\varepsilon} ^ {W _ {p}} (\tilde {\mathbb {P}} _ {m} ^ {\prime})} | \mathbb {E} _ {\mathbb {Q} ^ {\prime}} \xi_ {1} \| \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r} ^ {c}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B, 1, \infty} - \mathbb {E} _ {\tilde {\mathbb {P}} _ {m} ^ {\prime}} \xi_ {2} \| \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r} ^ {c}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B, 1, \infty} |, \\ \end{array}
$$
where  $\mathbb{Q}'$  and  $\tilde{\mathbb{P}}_m'$  are probability measures on  $\mathcal{X} \times \Xi$  with  $\Xi = \{-1, +1\}$  and identical marginals as  $\mathbb{Q}$  and  $\tilde{\mathbb{P}}_m$  respectively. We assume that  $\mathbb{Q} \neq \tilde{\mathbb{P}}$  because otherwise  $\| H_{\mathcal{S}_r^c\mathcal{S}_r}^\mathbb{Q} - \tilde{H}_{\mathcal{S}_r^c\mathcal{S}_r}\|_{B,1,\infty} = 0$  holds trivially. In this way, the equality is always achieved by some  $\mathbb{Q}', \tilde{\mathbb{P}}_m'$ , i.e., setting  $\mathbb{Q}'(\mathcal{X},\xi = 1) = 1$  and  $\tilde{\mathbb{P}}_m'(\mathcal{X},\xi = -1) = 1$ .
Define the transport cost function in the ambiguity set  $\mathcal{A}_{\varepsilon}^{W_p}(\tilde{\mathbb{P}}_m')$  to be  $c'((X_1, \xi_1), (X_2, \xi_2)) := \| \mathcal{E}(X_1) - \mathcal{E}(X_2) \|_1$  with zero cost for  $\xi$ . Let  $g(\pmb{X}, \xi) := \xi_1 \| \mathcal{E}(\pmb{X}_{\bar{r}})_{\mathcal{S}_r^c} \mathcal{E}(\pmb{X}_{\bar{r}})_{\mathcal{S}_r}^\intercal \|_{B, 1, \infty}$ . Consider the Lipschitz constants of  $g$ :
$$
\begin{array}{l} \operatorname {l i p} _ {c ^ {\prime}} (g) \leqslant \sup  _ {\boldsymbol {X}, \xi , \boldsymbol {X} ^ {\prime}, \xi^ {\prime}} \frac {| g (\boldsymbol {X} , \xi) - g \left(\boldsymbol {X} ^ {\prime} , \xi^ {\prime}\right) |}{c ^ {\prime} \left((\boldsymbol {X} , \xi) , \left(\boldsymbol {X} ^ {\prime} , \xi^ {\prime}\right)\right)} \\ \leqslant \sup  _ {\boldsymbol {X}, \boldsymbol {X} ^ {\prime}} \frac {\| \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r} ^ {c}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B , 1 , \infty} + \| \mathcal {E} (\boldsymbol {X} _ {\bar {r}} ^ {\prime}) _ {\mathcal {S} _ {r} ^ {c}} \mathcal {E} (\boldsymbol {X} _ {\bar {r}} ^ {\prime}) _ {\mathcal {S} _ {r}} ^ {\intercal} \| _ {B , 1 , \infty}}{\| \mathcal {E} (\boldsymbol {X}) - \mathcal {E} (\boldsymbol {X} ^ {\prime}) \| _ {1}} \\ \leqslant 2 \rho_ {\max } | \mathcal {S} _ {r} |. \tag {9} \\ \end{array}
$$
Therefore, by the Kantorovich-Rubinstein theorem [Kantorovich and Rubinshtein, 1958],
$$
\begin{array}{l} \| \boldsymbol {H}^{\mathbb{Q}}_{\mathcal{S}_{r}^{c}\mathcal{S}_{r}} - \tilde{\boldsymbol{H}}_{\mathcal{S}_{r}^{c}\mathcal{S}_{r}}\|_{B,1,\infty}\leqslant \sup_{\substack{\tilde{\mathbb{P}}^{\prime}_{m},\mathbb{Q}^{\prime}\in \mathcal{A}_{\varepsilon}^{W_{p}}(\tilde{\mathbb{P}}^{\prime}_{m})}}|\mathbb{E}_{\mathbb{Q}^{\prime}}g(\boldsymbol {X},\xi) - \mathbb{E}_{\tilde{\mathbb{P}}^{\prime}_{m}}g(\boldsymbol {X},\xi)| \\ \leqslant \sup  _ {\tilde {\mathbb {P}} _ {m} ^ {\prime}, \mathbb {Q} ^ {\prime} \in \mathcal {A} _ {\varepsilon} ^ {W _ {p}} (\tilde {\mathbb {P}} _ {m} ^ {\prime})} \operatorname {l i p} _ {c ^ {\prime}} (g) | \mathbb {E} _ {\mathbb {Q} ^ {\prime}} g (\boldsymbol {X}, \xi) / \operatorname {l i p} _ {c ^ {\prime}} (g) - \mathbb {E} _ {\tilde {\mathbb {P}} _ {m} ^ {\prime}} g (\boldsymbol {X}, \xi) / \operatorname {l i p} _ {c ^ {\prime}} (g) | \\ \leqslant \sup  _ {\tilde {\mathbb {P}} _ {m} ^ {\prime}, \mathbb {Q} ^ {\prime} \in \mathcal {A} _ {\varepsilon} ^ {W _ {p}} (\tilde {\mathbb {P}} _ {m} ^ {\prime})} \operatorname {l i p} _ {c ^ {\prime}} (g) W _ {1} (\mathbb {Q} ^ {\prime}, \tilde {\mathbb {P}} _ {m} ^ {\prime}) \\ \leqslant \operatorname {l i p} _ {c ^ {\prime}} (g) \varepsilon \\ \leqslant 2 \varepsilon \rho_ {\max } | \mathcal {S} _ {r} |. \\ \end{array}
$$
$$
\left\| \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \tilde {\boldsymbol {H}} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \right\| _ {\infty , \infty} \leqslant 2 \varepsilon | \mathcal {S} _ {r} |.
$$
Based on the above two inequalities, we find that
$$
\begin{array}{l} \left\| H _ {S _ {r} ^ {c}} ^ {\mathbb {Q}} \right. _ {S _ {r}} - H _ {S _ {r} ^ {c}} S _ {r} \| _ {B, 1, \infty} \leqslant \left\| H _ {S _ {r} ^ {c}} ^ {\mathbb {Q}} \right. _ {S _ {r}} - \tilde {H} _ {S _ {r} ^ {c}} S _ {r} \| _ {B, 1, \infty} + \left\| \tilde {H} _ {S _ {r} ^ {c}} S _ {r} - H _ {S _ {r} ^ {c}} S _ {r} \right\| _ {B, 1, \infty} \\ \leqslant 2 \varepsilon \rho_ {\max } \left| S _ {r} \right| + t, \tag {10} \\ \end{array}
$$
with probability at least  $1 - 2|\mathcal{S}_r^c ||\mathcal{S}_r|\exp \left(-\frac{mt^2}{2\rho_{\max}^2|\mathcal{S}_r|^2}\right)$ , and
$$
\left\| \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \right\| _ {\infty , \infty} \leqslant 2 \varepsilon | \mathcal {S} _ {r} | + t, \tag {11}
$$
with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2}{2|\mathcal{S}_r|^2}\right)$ .
Based on Equation (8), we also have
$$
\left\| \left[ H _ {\mathcal {S} _ {r}} \mathcal {S} _ {r} - H _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r} \right] \right\| _ {F} \leqslant 2 \varepsilon | \mathcal {S} _ {r} | + t, \tag {12}
$$
Next we look at the upper bound on the difference between the inverses of  $H_{S_r S_r}^{\mathbb{Q}}$  and  $H_{S_r S_r}$ . Observe that
$$
\begin{array}{l} \| \big (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \big) ^ {- 1} - \big (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \big) ^ {- 1} \| _ {\infty , \infty} = \| \big (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} \big) ^ {- 1} [ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} ] (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} \| _ {\infty , \infty} \\ \leqslant \sqrt {\left| \mathcal {S} _ {r} \right|} \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \| _ {2, 2} \\ \leqslant \sqrt {\left| \mathcal {S} _ {r} \right|} \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \| _ {2, 2} \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right] \| _ {2, 2} \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \| _ {2, 2} \\ \leqslant \sqrt {\frac {| \mathcal {S} _ {r} |}{\Lambda_ {\min} (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}})}} \| [ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} ] \| _ {2, 2} \| (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} \| _ {2, 2}. \\ \end{array}
$$
According to Lemma 7, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2}{2|\mathcal{S}_r|^2}\right)$ , we have
Let  $t = \frac{1}{2}\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})$  and  $\varepsilon \leqslant \frac{\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}$ . We get that, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$ ,
$$
\begin{array}{l} \Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) \geqslant \frac {1}{4} \Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) \\ \Longrightarrow \left\| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \right\| _ {2, 2} \leqslant \sqrt {\frac {4}{\Lambda_ {\operatorname* {m i n}} \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right)}}. \tag {13} \\ \end{array}
$$
Set  $t = \frac{t\Lambda_{\min}(H_{S_r S_r})}{4\sqrt{|S_r|}}$  and  $\varepsilon \leqslant \frac{t\Lambda_{\min}(H_{S_r S_r})}{8|\mathcal{S}_r|\sqrt{|\mathcal{S}_r|}}$  in Equation (12), we get that, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2(\Lambda_{\min}(\mathcal{H}_{S_r S_r}))^2}{32|\mathcal{S}_r|^3}\right)$
$$
\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right] \| _ {2, 2} \leqslant \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - \boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right] \| _ {F} \leqslant \frac {t \Lambda_ {\min } \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right)}{2 \sqrt {| \mathcal {S} _ {r} |}}.
$$
Therefore, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp \left(-\frac{mt^2(\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{32|\mathcal{S}_r|^3}\right) - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$  and  $\varepsilon \leqslant \min \left(\frac{t\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{8|\mathcal{S}_r|\sqrt{|\mathcal{S}_r|}},\frac{\Lambda_{\min}(\boldsymbol{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}\right)$ ,
$$
\left\| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {\infty , \infty} \leqslant t. \tag {14}
$$
Now we are ready to obtain upper bounds for the four terms recapitulated here:
$$
\begin{array}{l} \| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} \| _ {B, 1, \infty} \leqslant \| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} [ (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} - (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}) ^ {- 1} ] \| _ {B, 1, \infty} \\ + \left\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \\ + \left\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c}} ^ {\mathbb {Q}} \mathcal {S} _ {r} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c}} \mathcal {S} _ {r} \right] \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} \mathcal {S} _ {r}\right) ^ {- 1} \right] \right\| _ {B, 1, \infty} \\ + \| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} (\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}) ^ {- 1} \| _ {B, 1, \infty}. \\ \end{array}
$$
We derive the bounds separately.
For the first term, based on Assumption 4, consider
$$
\begin{array}{l} \left\| \mathbf {H} _ {\mathcal {S} _ {r} ^ {c}} \mathcal {S} _ {r} \left[ \left(\mathbf {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\mathbf {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \right\| _ {B, 1, \infty} \\ = \left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c}} \mathcal {S} _ {r} \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} \mathcal {S} _ {r}\right) ^ {- 1} \left[ \boldsymbol {H} _ {\mathcal {S} _ {r}} \mathcal {S} _ {r} - \boldsymbol {H} _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r}\right) ^ {- 1} \right\| _ {B, 1, \infty} \\ \leqslant \left\| H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \left(H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \left\| H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \right\| _ {\infty , \infty} \left\| \left(H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} \right\| _ {\infty , \infty} \\ \leqslant (1 - \alpha) \| H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} - H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}} \| _ {\infty , \infty} \sqrt {| \mathcal {S} _ {r} |} \| (H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}) ^ {- 1} \| _ {2, 2}. \\ \end{array}
$$
Taking  $t = \frac{\alpha}{24(1 - \alpha)}\sqrt{\frac{\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{|\mathcal{S}_r|}}$  and  $\varepsilon \leqslant \frac{\alpha}{48(1 - \alpha)|\mathcal{S}_r|}\sqrt{\frac{\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{|\mathcal{S}_r|}}$  in Equation (11) and adopting Equation (13), we conclude that, with probability at least  $1 - 2|\mathcal{S}_r|^2\exp (-\frac{m\alpha^2\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{1152(1 - \alpha)^2|\mathcal{S}_r|^3}) - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$  and  $\varepsilon \leqslant \min \left(\frac{\alpha}{48(1 - \alpha)|\mathcal{S}_r|}\sqrt{\frac{\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{|\mathcal{S}_r|}},\frac{\Lambda_{\min}(\pmb{H}_{\mathcal{S}_r\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}\right)$ ,
$$
\left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c}} \mathcal {S} _ {r} \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r}} \mathcal {S} _ {r}\right) ^ {- 1} \right] \right\| _ {B, 1, \infty} \leqslant \frac {\alpha}{6}.
$$
For the second term, rewrite it as
$$
\begin{array}{l} \left\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \\ \leqslant \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \| _ {B, 1, \infty} \| \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \| _ {\infty , \infty} \\ \leqslant \left\| \left[ H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - H _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \right\| _ {B, 1, \infty} \sqrt {| \mathcal {S} _ {r} |} \| \left(H _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \| _ {2, 2} \\ \leqslant \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \| _ {B, 1, \infty} \sqrt {\frac {\left| \mathcal {S} _ {r} \right|}{\Lambda_ {\min} \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right)}}. \\ \end{array}
$$
Using Equation (10) by setting  $t = \frac{\alpha}{12}\sqrt{\frac{\Lambda_{\min}(\boldsymbol{H}_{\boldsymbol{S}_r\boldsymbol{S}_r})}{|\boldsymbol{S}_r|}}$  and  $\varepsilon \leqslant \frac{\alpha}{24\rho_{\max}|\mathcal{S}_r|}\sqrt{\frac{\Lambda_{\min}(\boldsymbol{H}_{\boldsymbol{S}_r\boldsymbol{S}_r})}{|\mathcal{S}_r|}}$ , we have, with probability at least  $1 - 2|\mathcal{S}_r^c ||\mathcal{S}_r|\exp \left(-\frac{m\alpha^2\Lambda_{\min}(\boldsymbol{H}_{\boldsymbol{S}_r\boldsymbol{S}_r})}{288\rho_{\max}^2|\mathcal{S}_r|^3}\right)$  and  $\varepsilon \leqslant \frac{\alpha}{24\rho_{\max}|\mathcal{S}_r|}\sqrt{\frac{\Lambda_{\min}(\boldsymbol{H}_{\boldsymbol{S}_r\boldsymbol{S}_r})}{|\mathcal{S}_r|}}$ ,
$$
\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \| _ {B, 1, \infty} \leqslant \frac {\alpha}{6}.
$$
For the third term, we obtain the upper bound
$$
\begin{array}{l} \left\| \left[ H _ {\mathcal {S} _ {r} ^ {c}} ^ {\mathbb {Q}} \mathcal {S} _ {r} - H _ {\mathcal {S} _ {r} ^ {c}} \mathcal {S} _ {r} \right] \left[ \left(H _ {\mathcal {S} _ {r}} ^ {\mathbb {Q}} \mathcal {S} _ {r}\right) ^ {- 1} - \left(H _ {\mathcal {S} _ {r}} \mathcal {S} _ {r}\right) ^ {- 1} \right] \right\| _ {B, 1, \infty} \\ \leqslant \| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \| _ {B, 1, \infty} \| \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \| | _ {\infty , \infty}. \\ \end{array}
$$
Taking  $t = \sqrt{\frac{\alpha}{6}}$  in Equation (14). Taking  $t = \frac{1}{2}\sqrt{\frac{\alpha}{6}}$  and  $2\varepsilon \rho_{\max}|\mathcal{S}_r| \leqslant \frac{1}{2}\sqrt{\frac{\alpha}{6}}$  in Equation (10). We establish the upper bound that, with probability at least  $1 - 2|\mathcal{S}_r^c||\mathcal{S}_r|\exp \left(-\frac{m\alpha}{48\rho_{\max}^2|\mathcal{S}_r|^2}\right) - 2|\mathcal{S}_r|^2\exp \left(-\frac{m\alpha(\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r,\mathcal{S}_r}))^2}{192|\mathcal{S}_r|^3}\right) - 2|\mathcal{S}_r|^2\exp \left(-\frac{m(\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r,\mathcal{S}_r}))^2}{8|\mathcal{S}_r|^2}\right)$  and  $\varepsilon \leqslant \min \left(\frac{1}{4\rho_{\max}|\mathcal{S}_r|}\sqrt{\frac{\alpha}{6}},\frac{\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r,\mathcal{S}_r})}{8|\mathcal{S}_r|}\sqrt{\frac{\alpha}{6|\mathcal{S}_r|},\frac{\Lambda_{\min}(\mathbf{H}_{\mathcal{S}_r,\mathcal{S}_r})}{16|\mathcal{S}_r|^{\frac{1}{2}}}\right)$ ,
$$
\| \left[ \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} ^ {\mathbb {Q}} - \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \right] \left[ \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}} ^ {\mathbb {Q}}\right) ^ {- 1} - \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right] \| _ {B, 1, \infty} \leqslant \frac {\alpha}{6}.
$$
For the fourth term, in accordance with Assumption 4,
$$
\left\| \boldsymbol {H} _ {\mathcal {S} _ {r} ^ {c} \mathcal {S} _ {r}} \left(\boldsymbol {H} _ {\mathcal {S} _ {r} \mathcal {S} _ {r}}\right) ^ {- 1} \right\| _ {B, 1, \infty} \leqslant 1 - \alpha .
$$