SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| (\mathbb {P} _ {\tilde {\mathbf {U}}, \tilde {\mathbf {V}}} ^ {\perp} - \mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp}) \boldsymbol {\Xi} \right\| _ {F} = \left\| (\mathbb {P} _ {\tilde {\mathbf {U}}, \tilde {\mathbf {V}}} - \mathbb {P} _ {\mathbf {U}, \mathbf {V}}) \boldsymbol {\Xi} \right\| _ {F} \leq \left\| (\mathbb {P} _ {\tilde {\mathbf {U}}, \tilde {\mathbf {V}}} - \mathbb {P} _ {\mathbf {U} _ {\star}, \mathbf {V} _ {\star}}) \boldsymbol {\Xi} \right\| _ {F} + \left\| (\mathbb {P} _ {\mathbf {U} _ {\star}, \mathbf {V} _ {\star}} - \mathbb {P} _ {\mathbf {U}, \mathbf {V}}) \boldsymbol {\Xi} \right\| _ {F} \\ \leq 8 \frac {\left\| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \right\|}{(1 - 1 / 4 8) \sigma_ {r} \left(\mathbf {X} _ {\star}\right)} \| \boldsymbol {\Xi} \| _ {F}. \tag {53} \\ \end{array}
$$
To estimate the fourth summand in (51), we argue analogously that
$$
\begin{array}{l} \left\| \mathbb {P} _ {S ^ {c}} \boldsymbol {\Xi} \left(\widetilde {\mathbf {V}} \widetilde {\mathbf {V}} ^ {*} - \mathbf {V} \mathbf {V} ^ {*}\right) \right\| _ {F} \leq \left\| \boldsymbol {\Xi} \left(\widetilde {\mathbf {V}} \widetilde {\mathbf {V}} ^ {*} - \mathbf {V} \mathbf {V} ^ {*}\right) \right\| _ {F} \leq \left(\| \widetilde {\mathbf {V}} \widetilde {\mathbf {V}} ^ {*} - \mathbf {V} _ {\star} \mathbf {V} _ {\star} ^ {*} \| + \| \mathbf {V} _ {\star} \mathbf {V} _ {\star} ^ {*} - \mathbf {V} \mathbf {V} ^ {*} \|\right) \| \boldsymbol {\Xi} \| _ {F} \\ \leq 2 \left(\| \mathbf {V} _ {\star} \widetilde {\mathbf {V}} _ {\perp} ^ {*} \| + \| \mathbf {V} _ {\star} \mathbf {V} _ {\perp} ^ {*} \|\right) \| \boldsymbol {\Xi} \| _ {F} \\ \leq 2 \left(\frac {\| \mathbf {H} _ {s} (\mathbf {X} ^ {(k)}) - \mathbf {X} _ {\star} \|}{\sigma_ {r} (\mathbf {X} _ {\star}) - \sigma_ {r + 1} (\mathbf {H} _ {s} (\mathbf {X} ^ {(k)}))} + \frac {\| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \|}{\sigma_ {r} (\mathbf {X} _ {\star}) - \sigma_ {r + 1} (\mathbf {X} ^ {(k)})}\right) \| \boldsymbol {\Xi} \| _ {F} \\ \leq 4 \frac {\left\| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \right\|}{\sigma_ {r} \left(\mathbf {X} _ {\star}\right) - \sigma_ {r + 1} \left(\mathbf {X} ^ {(k)}\right)} \| \boldsymbol {\Xi} \| _ {F} \leq 4 \frac {\left\| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \right\|}{(1 - 1 / 4 8) \sigma_ {r} \left(\mathbf {X} _ {\star}\right)} \| \boldsymbol {\Xi} \| _ {F}, \tag {54} \\ \end{array}
$$
using again that  $\sigma_{r + 1}(\mathbf{X}^{(k)})\leq \| \mathbf{X}_{\star} - \mathbf{X}^{(k)}\| \leq \frac{1}{48}\sigma_r(\mathbf{X}_\star)$  due to (25).
Let now  $\Xi^{(k + 1)} = \mathbf{X}^{(k + 1)} - \mathbf{X}_{\star}$ . Combining (24) and (51)-(52) we can proceed to estimate that
$$
\begin{array}{l} \| \Xi^ {(k + 1)} \| _ {F} ^ {2} \\ \leq c ^ {2} \left\| \mathbb {P} _ {T _ {k} ^ {\perp}} (\boldsymbol {\Xi} ^ {(k + 1)}) \right\| _ {F} ^ {2} \\ \leq 2 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} \boldsymbol {\Xi} ^ {(k + 1)} \rangle \\ + 2 \left[ \left(\frac {8}{4 7 / 4 8}\right) ^ {2} + \left(\frac {4}{4 7 / 4 8}\right) ^ {2} \right] c ^ {2} \| \boldsymbol {\Xi} ^ {(k + 1)} \| _ {F} ^ {2} \frac {\| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \| ^ {2}}{\sigma_ {r} ^ {2} (\mathbf {X} _ {\star})} \\ \leq 2 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} \left(\mathbf {X} ^ {(k)}\right)}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} \left(\mathbf {X} ^ {(k)}\right)}{\delta_ {k}} \right\} ^ {2} \left\langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} \left(\boldsymbol {\Xi} ^ {(k + 1)}\right) \right\rangle + 1 6 7 c ^ {2} \| \boldsymbol {\Xi} ^ {(k + 1)} \| _ {F} ^ {2} \frac {1}{(1 9 c) ^ {2}} \\ \leq 2 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} \left(\mathbf {X} ^ {(k)}\right)}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} \left(\mathbf {X} ^ {(k)}\right)}{\delta_ {k}} \right\} ^ {2} \left\langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} \left(\boldsymbol {\Xi} ^ {(k + 1)}\right) \right\rangle + \frac {1}{2} \| \boldsymbol {\Xi} ^ {(k + 1)} \| _ {F} ^ {2}, \tag {55} \\ \end{array}
$$
where the third inequality follows from (8) and (25). Hence, rearranging (55) yields
$$
\| \boldsymbol {\Xi} ^ {(k + 1)} \| _ {F} ^ {2} \leq 4 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} (\boldsymbol {\Xi} ^ {(k + 1)}) \rangle .
$$
By Lemma B.5, we know that  $\mathbf{X}^{(k + 1)}$  fulfills
$$
0 = \left\langle \Xi^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} (\mathbf {X} ^ {(k + 1)}) \right\rangle = \left\langle \Xi^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} (\Xi^ {(k + 1)}) \right\rangle + \left\langle \Xi^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} (\mathbf {X} _ {\star}) \right\rangle
$$
such that we conclude that
$$
\begin{array}{l} \left\| \boldsymbol {\Xi} ^ {(k + 1)} \right\| ^ {2} \leq \left\| \boldsymbol {\Xi} ^ {(k + 1)} \right\| _ {F} ^ {2} \\ \leq 4 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} \left(\mathbf {X} ^ {(k)}\right)}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} \left(\mathbf {X} ^ {(k)}\right)}{\delta_ {k}} \right\} ^ {2} \langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} \left(\boldsymbol {\Xi} ^ {(k + 1)}\right) \rangle \\ = - 4 c ^ {2} \min \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}, \delta_ {k}} (\mathbf {X} _ {\star}) \rangle \\ = - 4 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \left(\left\langle \boldsymbol {\Xi} ^ {(k + 1)}, W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r} (\mathbf {X} _ {\star}) \right\rangle + \left\langle \boldsymbol {\Xi} ^ {(k + 1)}, \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \cdot \mathbf {X} _ {\star} \right\rangle\right) \\ \leq 4 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \left(\left\| W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r} (\mathbf {X} _ {\star}) \right\| _ {*} \left\| \boldsymbol {\Xi} ^ {(k + 1)} \right\| + \left\| \boldsymbol {\Xi} ^ {(k + 1)} \right\| \left\| \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \cdot \mathbf {X} _ {\star} \right\| _ {1, 2}\right) \\ = 4 c ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \left(\left\| W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r} (\mathbf {X} _ {\star}) \right\| _ {*} + \left\| \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \cdot \mathbf {X} _ {\star} \right\| _ {1, 2}\right) \left\| \boldsymbol {\Xi} ^ {(k + 1)} \right\|, \\ \end{array}
$$
which completes the proof. We used in the penultimate line Hölder's inequality and that
$$
| \langle \mathbf {A}, \mathbf {B} \rangle_ {F} | = \left| \sum_ {i, j} A _ {i, j} B _ {i, j} \right| \leq \sum_ {i} \| \mathbf {A} _ {i,:} \| _ {2} \| \mathbf {B} _ {i,:} \| _ {2} \leq \left(\max  _ {i} \| \mathbf {A} _ {i,:} \| _ {2}\right) \cdot \sum_ {i} \| \mathbf {B} _ {i,:} \| _ {2} \leq \| \mathbf {A} \| \| \mathbf {B} \| _ {1, 2},
$$
for all matrices  $\mathbf{A},\mathbf{B}$
C.5 Proof of Lemma B.9
Note that by Lemma B.2 and (27),  $S \coloneqq \operatorname{supp}(\mathsf{H}_s(\mathbf{X}^{(k)})) = \operatorname{supp}(\mathbf{X}_{\star})$ . Since by assumption  $\delta_k \leq \rho_s(\mathbf{X}^{(k)})$  we have by definition of  $\mathbf{W}_{\mathbf{X}^{(k)},\delta_k}^{sp}$  that  $\mathbf{Z} \coloneqq \mathbf{W}_{\mathbf{X}^{(k)},\delta_k}^{sp} \cdot \mathbf{X}_{\star}$  is a matrix with row-support  $S$  and rows
$$
\mathbf {Z} _ {i,:} = \min  \left\{\frac {\delta_ {k} ^ {2}}{\| (\mathbf {X} ^ {(k)}) _ {i,:} \| _ {2} ^ {2}}, 1 \right\} (\mathbf {X} _ {\star}) _ {i,:}, = \frac {\delta_ {k} ^ {2}}{\| (\mathbf {X} ^ {(k)}) _ {i,:} \| _ {2} ^ {2}} (\mathbf {X} _ {\star}) _ {i,:}
$$
for  $i\in S$  . Now note that if (27) holds, then
$$
\| \mathbf {Z} _ {i,:} \| _ {2} = \frac {\delta_ {k} ^ {2} \| (\mathbf {X} _ {\star}) _ {i , :} \| _ {2}}{\| (\mathbf {X} ^ {(k)}) _ {i , :} \| _ {2} ^ {2}} \leq \frac {\delta_ {k} ^ {2}}{(1 - \zeta) ^ {2} \rho_ {s} (\mathbf {X} _ {\star})},
$$
where we used in the last estimate that with (27) and  $\| (\mathbf{X}_{\star})_{i,:}\| _2\geq \rho_s(\mathbf{X}_{\star})$  , for  $i\in S$  , we have
$$
\begin{array}{l} \left\| \left(\mathbf {X} ^ {(k)}\right) _ {i,:} \right\| _ {2} ^ {2} \geq \left(\left\| \left(\mathbf {X} _ {\star}\right) _ {i,:} \right\| _ {2} - \left\| \left(\mathbf {X} ^ {(k)}\right) _ {i,:} - \left(\mathbf {X} _ {\star}\right) _ {i,:} \right\| _ {2}\right) ^ {2} \geq \left(\left\| \left(\mathbf {X} _ {\star}\right) _ {i,:} \right\| _ {2} - \zeta \rho_ {s} \left(\mathbf {X} _ {\star}\right)\right) ^ {2} \\ \geq \left((1 - \zeta) \rho_ {s} \left(\mathbf {X} _ {\star}\right)\right) \left((1 - \zeta) \| \left(\mathbf {X} _ {\star}\right) _ {i,:} \| _ {2}\right), \\ \end{array}
$$
for all  $i \in S$ . The claim easily follows since  $\mathbf{Z}$  has only  $s$  non-zero rows.
C.6 Proof of Lemma B.10
In the proof of Lemma B.10, we use a simple technical observation.
Lemma C.3. Let  $\mathbf{X}_{\star} \in \mathcal{M}_{r,s}$ , let  $\mathbf{X}^{(k)}$  be the  $k$ -th iterate of Algorithm 1, and abbreviate  $\Xi^{(k)} = \mathbf{X}^{(k)} - \mathbf{X}_{\star}$ . Then the following two statements hold true:
1. If  $\varepsilon_{k} \leq \sigma_{r + 1}(\mathbf{X}^{(k)})$ , then  $\varepsilon_{k} \leq \| \boldsymbol{\Xi}^{(k)}\|$ .
2. If  $\delta_{k} \leq \rho_{s+1}(\mathbf{X}^{(k)})$ , then  $\delta_{k} \leq \| \boldsymbol{\Xi}^{(k)} \|_{\infty,2}$ .
Proof: By defining  $[\mathbf{X}^{(k)}]_r$  to be the best rank-  $r$  approximation of  $\mathbf{X}^{(k)}$  in any unitarily invariant norm, we bound
$$
\varepsilon_ {k} \leq \sigma_ {r + 1} (\mathbf {X} ^ {(k)}) = \| \mathbf {X} ^ {(k)} - [ \mathbf {X} ^ {(k)} ] _ {r} \| \leq \| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \| = \| \boldsymbol {\Xi} ^ {(k)} \|,
$$
where the inequality follows the fact that  $\mathbf{X}_{\star}$  is a rank-  $r$  matrix.
Similarly, for the second statement, we have that
$$
\delta_ {k} \leq \rho_ {s + 1} (\mathbf {X} ^ {(k)}) = \| \mathbf {X} ^ {(k)} - \mathsf {H} _ {s} (\mathbf {X} ^ {(k)}) \| _ {\infty , 2} \leq \| \mathbf {X} ^ {(k)} - \mathbf {X} _ {\star} \| _ {\infty , 2} = \| \boldsymbol {\Xi} ^ {(k)} \| _ {\infty , 2},
$$
using that  $\mathbf{X}_{\star}$  is  $s$  -row sparse.
Proof of Lemma B.10: First, we note that, using Lemma B.3, the observation in Remark B.7 yields
$$
\left\| \boldsymbol {\Xi} \right\| _ {F} ^ {2} \leq 4 c _ {\| \mathcal {A} \| _ {2 \rightarrow 2}} ^ {2} \min  \left\{\frac {\sigma_ {r + 1} \left(\mathbf {X} ^ {(k)}\right)}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} \left(\mathbf {X} ^ {(k)}\right)}{\delta_ {k}} \right\} ^ {2} \tag {56}
$$
$$
\cdot \left\langle \boldsymbol {\Xi}, \left(\mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r} \mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} + \mathbb {P} _ {S ^ {c}} \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \mathbb {P} _ {S ^ {c}}\right) \boldsymbol {\Xi} \right\rangle
$$
for all  $\Xi \in \mathbb{R}^{n_1\times n_2}$  as the assumption (28) implies  $\| \mathbf{X}^{(k)} - \mathbf{X}_{\star}\| \leq \min \left\{\frac{1}{48},\frac{1}{19c}\right\} \sigma_r(\mathbf{X}_{\star})$ . Thus, this holds in particular also for  $\Xi^{(k)} = \mathbf{X}^{(k)} - \mathbf{X}_{\star}$ . (Recall that  $\mathbf{U}$  and  $\mathbf{V}$  contain the leading singular vectors of  $\mathbf{X}^{(k)}$ , see Definition 2.1.) We estimate that
$$
\begin{array}{l} \sqrt {\left\langle \mathbb {P} _ {\mathbf {U} , \mathbf {V}} ^ {\perp} \boldsymbol {\Xi} ^ {(k)} , W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r} \mathbb {P} _ {\mathbf {U} , \mathbf {V}} ^ {\perp} \boldsymbol {\Xi} ^ {(k)} \right\rangle} = \left\| \left(W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) ^ {1 / 2} \left(\mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} \left(\boldsymbol {\Xi} ^ {(k)}\right)\right) \right\| _ {F} \\ \leq \left\| \left(W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) ^ {1 / 2} \left(\mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} \left(\mathbf {X} _ {\star}\right)\right) \right\| _ {F} + \left\| \left(W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) ^ {1 / 2} \left(\mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} \left(\mathbf {X} ^ {(k)}\right)\right) \right\| _ {F} \\ \leq \left\| (W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}) ^ {1 / 2} (\mathbf {X} _ {\star}) \right\| _ {F} + \sqrt {\sum_ {i = r + 1} ^ {\min (n _ {1} , n _ {2})} \frac {\sigma_ {i} ^ {2}}{\max \left(\frac {\sigma_ {i} ^ {2}}{\varepsilon_ {k} ^ {2}} , 1\right)}} \leq \left\| (W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}) ^ {1 / 2} (\mathbf {X} _ {\star}) \right\| _ {F} + \sqrt {n - r} \varepsilon_ {k} \\ \end{array}
$$
Furthermore, since  $\max (\varepsilon_k,\| \Xi^{(k)}\|)\leq \frac{1}{48}\sigma_r(\mathbf{X}_\star)$  by assumption and  $\varepsilon_{k}\leq \| \Xi^{(k)}\|$  by Lemma C.3, we can use a variant of Lemma B.8 to obtain
$$
\begin{array}{l} \left\| \left(W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) ^ {1 / 2} \left(\mathbf {X} _ {\star}\right) \right\| _ {F} \leq \frac {4 8}{4 7} \left(\sqrt {r} \varepsilon_ {k} + 2 \varepsilon_ {k} \frac {\| \boldsymbol {\Xi} ^ {(k)} \| _ {F}}{\sigma_ {r} \left(\mathbf {X} _ {\star}\right)} + 2 \frac {\| \boldsymbol {\Xi} ^ {(k)} \| \| \boldsymbol {\Xi} ^ {(k)} \| _ {F}}{\sigma_ {r} \left(\mathbf {X} _ {\star}\right)}\right) \\ \leq 1. 0 4 \sqrt {r} \varepsilon_ {k} + \frac {4 . 1 6 \| \boldsymbol {\Xi} ^ {(k)} \|}{\sigma_ {r} (\mathbf {X} _ {\star})} \| \boldsymbol {\Xi} ^ {(k)} \| _ {F}. \\ \end{array}
$$
On the other hand, we note that  $\Xi^{(k)}$  restricted to  $S^c$  coincides with the restriction of  $\mathbf{X}^{(k)}$  to  $S^c$  under assumption (28), cf. Lemma B.2, and therefore
$$
\begin{array}{l} \left\langle \mathbb {P} _ {S ^ {c}} \boldsymbol {\Xi} ^ {(k)}, \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \mathbb {P} _ {S ^ {c}} \boldsymbol {\Xi} ^ {(k)} \right\rangle = \left\langle \mathbb {P} _ {S ^ {c}} \mathbf {X} ^ {(k)}, \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \mathbb {P} _ {S ^ {c}} \mathbf {X} ^ {(k)} \right\rangle \\ = \sum_ {i = s + 1} ^ {n _ {1}} \frac {\| (\mathbf {X} ^ {(k)}) _ {i , :} \| _ {2} ^ {2}}{\max \{\| (\mathbf {X} ^ {(k)}) _ {i , :} \| _ {2} ^ {2} / \delta_ {k} ^ {2} , 1 \}} \leq (n _ {1} - s) \delta_ {k} ^ {2}. \\ \end{array}
$$
With the estimate of above, this implies that
$$
\begin{array}{l} \left\langle \right. \boldsymbol {\Xi}, \left(\mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) \mathbb {P} _ {\mathbf {U}, \mathbf {V}} ^ {\perp} + \mathbb {P} _ {S ^ {c}} \mathbf {W} _ {\mathbf {X} ^ {(k)}, \delta_ {k}} ^ {s p} \mathbb {P} _ {S ^ {c}}\left. \right) \boldsymbol {\Xi} \left. \right\rangle \\ \leq \left(\left\| \left(W _ {\mathbf {X} ^ {(k)}, \varepsilon_ {k}} ^ {l r}\right) ^ {1 / 2} (\mathbf {X} _ {\star}) \right\| _ {F} + \sqrt {n - r} \varepsilon_ {k}\right) ^ {2} + (n _ {1} - s) \delta_ {k} ^ {2} \\ \leq \frac {1 3}{4} r \varepsilon_ {k} ^ {2} + \frac {5 2 \| \boldsymbol {\Xi} ^ {(k)} \| ^ {2}}{\sigma_ {r} ^ {2} (\mathbf {X} _ {\star})} \| \boldsymbol {\Xi} ^ {(k)} \| _ {F} ^ {2} + 3 (n - r) \varepsilon_ {k} ^ {2} + (n _ {1} - s) \delta_ {k} ^ {2}. \\ \end{array}
$$
Inserting these estimates into (56), we obtain
$$
\| \boldsymbol {\Xi} ^ {(k)} \| _ {F} ^ {2} \leq 4 c _ {\| \mathcal {A} \| _ {2 \rightarrow 2}} ^ {2} \min  \left\{\frac {\sigma_ {r + 1} (\mathbf {X} ^ {(k)})}{\varepsilon_ {k}}, \frac {\rho_ {s + 1} (\mathbf {X} ^ {(k)})}{\delta_ {k}} \right\} ^ {2} \left(\frac {1 3}{4} n \varepsilon_ {k} ^ {2} + n _ {1} \delta_ {k} ^ {2} + \frac {5 2 \| \boldsymbol {\Xi} ^ {(k)} \| ^ {2}}{\sigma_ {r} ^ {2} (\mathbf {X} _ {\star})} \| \boldsymbol {\Xi} ^ {(k)} \| _ {F} ^ {2}\right).
$$
If now either one of the two equations  $\varepsilon_{k} = \sigma_{r + 1}(\mathbf{X}^{(k)})$  or  $\delta_{k} = \rho_{s + 1}(\mathbf{X}^{(k)})$  is true, it follows that
$$
\begin{array}{l} \left\| \boldsymbol {\Xi} ^ {(k)} \right\| _ {F} ^ {2} \leq c _ {\left\| \boldsymbol {\mathcal {A}} \right\| _ {2 \rightarrow 2}} ^ {2} \left(1 3 n \varepsilon_ {k} ^ {2} + 4 n _ {1} \delta_ {k} ^ {2}\right) + c _ {\left\| \boldsymbol {\mathcal {A}} \right\| _ {2 \rightarrow 2}} ^ {2} \frac {2 0 8 \left\| \boldsymbol {\Xi} ^ {(k)} \right\| ^ {2}}{\sigma_ {r} ^ {2} (\mathbf {X} _ {\star})} \| \boldsymbol {\Xi} ^ {(k)} \| _ {F} ^ {2} \\ \leq c _ {\| \mathcal {A} \| _ {2 \rightarrow 2}} ^ {2} \left(1 3 n \varepsilon_ {k} ^ {2} + 4 n _ {1} \delta_ {k} ^ {2}\right) + \frac {1}{2} \| \boldsymbol {\Xi} ^ {(k)} \| _ {F} ^ {2} \\ \end{array}
$$
if the proximity condition  $\| \Xi^{(k)}\| = \| \mathbf{X}^{(k)} - \mathbf{X}_{\star}\| \leq \frac{1}{21c_{\|\mathcal{A}\|_{2\to 2}}}\sigma_r(\mathbf{X}_{\star})$  is satisfied. Rearranging the latter inequality yields the conclusion of Lemma B.10.