SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\left\| \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \right\| _ {2} \leq \frac {9}{2} \max  _ {w \in S _ {\varepsilon} ^ {d - 1}} \left| w ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} w \right|.
$$
Now,  $S_{1/3}^{d-1}$  can be selected so that its cardinality can be upper bounded by  $|\mathcal{S}_{1/3}^{d-1}| \leq 18^d$  (see, for example, Pollard [70]). For a fixed  $w \in S_{1/3}^{d-1}$ , we let  $z = \bar{\mathbf{Q}} w$  and note that since  $S_{1/3}^{d-1} \subset S^{d-1}$ , that  $\|z\|_2 \leq 1$ , and
$$
\left| w ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} w \right| = \left| \sum_ {i = 1} ^ {n (M + 1)} \sum_ {j = 1} ^ {n (M + 1)} \bar {e} _ {i j} z _ {i} z _ {j} \right| = 2 \left| \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n M} e _ {i j} z _ {i} z _ {n + j} \right|
$$
Now, over the event that entries  $e_{ij} \lesssim M \log n$ , for all  $i, j$ , which occurs which overwhelming probability by Lemma 3, Hoeffding's inequality and a union bound over  $w \in S_{1/3}^{d-1}$  gives
$$
\begin{array}{l} \mathbb{P}\left\{\left\| \bar{\mathbf{Q}}^{\top}\bar{\mathbf{E}}\bar{\mathbf{Q}}\right\|_{2} > t\right\} \leq \sum_{w\in \mathcal{S}_{1 / 3}^{d - 1}}\mathbb{P}\left(\left|w^{\top}\bar{\mathbf{Q}}^{\top}\bar{\mathbf{E}}\bar{\mathbf{Q}} w\right| > \frac{2t}{9}\right) \\ = \sum_ {w \in \mathcal {S} _ {1 / 3} ^ {d - 1}} \mathbb {P} \left\{\left| \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n M} e _ {i j} z _ {i} z _ {n + j} \right| > \frac {t}{9} \right\} \\ \leq 2 \cdot 1 8 ^ {d} \exp \left\{- \frac {2 t ^ {2}}{(9 c M \log n) ^ {2}} \right\} \\ = 2 \cdot \exp \left\{d \log (1 8) - \frac {2 t ^ {2}}{(9 c M \log n) ^ {2}} \right\} \\ \end{array}
$$
Setting  $t = M \log^{3/2} n$  gives
$$
\left\| \mathbf {Q} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \tilde {\boldsymbol {\Lambda}}\right) \mathbf {R} \right\| _ {2} = \left\| \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \right\| _ {2} \stackrel {{\mathbb {P}}} {{\lesssim}} M \log^ {3 / 2} n,
$$
G.4 Proof of Proposition 5
Denote the singular value decomposition of  $\bar{\mathbf{U}}^{\top}\hat{\mathbf{U}}$  by  $\Omega_1\Xi \Omega_2^\top$ , where  $\Xi = \mathrm{diag}(\xi_1,\dots ,\xi_d)$ , and let  $\mathbf{W} := \Omega_1\Omega_2^\top$ . The principal angles  $\{\theta_i\}_{i = 1}^d$  between the column spaces of  $\bar{\mathbf{U}}$  and  $\hat{\mathbf{U}}$  are defined by  $\xi_{i} = \cos (\theta_{i})$ , and by the Wedin sin $\Theta$  theorem, we have
$$
\begin{array}{l} \left\| \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \mathbf {W} \right\| _ {2} = \| \boldsymbol {\Xi} - \mathbf {I} \| _ {2} = \max _ {i \in [ d ]} | 1 - \xi_ {i} | = \max _ {i \in [ d ]} | 1 - \cos \theta_ {i} | \leq \max _ {i \in [ d ]} | 1 - \cos^ {2} \theta_ {i} | \\ = \max  _ {i \in [ d ]} \sin^ {2} \theta_ {i} \lesssim \frac {\left\| \widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} ^ {2}}{\left(\sigma_ {d} (\bar {\boldsymbol {\Lambda}}) - \sigma_ {d + 1} (\bar {\boldsymbol {\Lambda}})\right) ^ {2}} \stackrel {\mathbb {P}} {\lesssim} \frac {M n \lambda_ {\operatorname* {m a x}}}{M \delta^ {2}} = \frac {n \lambda_ {\operatorname* {m a x}}}{\delta^ {2}} \lesssim \frac {\sqrt {n \lambda_ {\operatorname* {m a x}}}}{\delta}. \tag {21} \\ \end{array}
$$
We apply the Wedin  $\sin \Theta$  theorem again to obtain a bound which we will require later:
$$
\begin{array}{l} \left\| \widehat {\mathbf {U}} \widehat {\mathbf {U}} ^ {\top} - \bar {\mathbf {U}} \bar {\mathbf {U}} ^ {\top} \right\| _ {2} \vee \left\| \widehat {\mathbf {V}} \widehat {\mathbf {V}} ^ {\top} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top} \right\| _ {2} = \left\| \sin \Theta (\widehat {\mathbf {U}}, \bar {\mathbf {U}}) \right\| _ {2} \vee \left\| \sin \Theta (\widehat {\mathbf {V}}, \bar {\mathbf {V}}) \right\| _ {2} \\ \lesssim \frac {\left\| \widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2}}{\sigma_ {d} (\bar {\boldsymbol {\Lambda}}) - \sigma_ {d + 1} (\bar {\boldsymbol {\Lambda}})} \\ \stackrel {\mathbb {P}} {\lesssim} \frac {\sqrt {n \lambda_ {\max}}}{\delta}. \\ \end{array}
$$
We now establish a bound on  $\left\| \bar{\mathbf{U}}^{\top}\widehat{\mathbf{U}} -\bar{\mathbf{V}}^{\top}\widehat{\mathbf{V}}\right\|_{2}$ . We start by showing that
$$
\begin{array}{l} \left\| \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} \right\| _ {2} = \underset {x: | | x | | _ {2} \leq 1} {\arg \max } x ^ {\top} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) x \\ = \arg \max  _ {x: \| x \| _ {2} \leq 1} \sum_ {i, j = 1} ^ {d} x _ {i} x _ {j} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) _ {i j} \\ \leq \underset {x: \| x \| _ {2} \leq 1} {\arg \max } \sum_ {i, j = 1} ^ {d} (1 + \bar {s} _ {i}) x _ {i} (1 + \bar {s} _ {j} ^ {- 1}) x _ {j} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) _ {i j} \\ = \underset {x: \| x \| _ {2} \leq 1} {\arg \max } \sum_ {i, j = 1} ^ {d} x ^ {\top} \left[ \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) + \bar {\mathbf {S}} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) \hat {\mathbf {S}} ^ {- 1} \right] x \\ = \left\| \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) + \bar {\mathbf {S}} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) \hat {\mathbf {S}} ^ {- 1} \right\| _ {2}, \\ \end{array}
$$
and then we employ the decomposition
$$
\begin{array}{l} \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} + \bar {\mathbf {S}} \left(\bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}}\right) \hat {\mathbf {S}} ^ {- 1} \\ = \left[ \bar {\mathbf {U}} ^ {\top} \widehat {\mathbf {U}} \widehat {\mathbf {S}} - \bar {\mathbf {S}} \bar {\mathbf {V}} ^ {\top} \widehat {\mathbf {V}} + \bar {\mathbf {S}} \bar {\mathbf {U}} ^ {\top} \widehat {\mathbf {U}} - \bar {\mathbf {V}} \widehat {\mathbf {V}} \right] \widehat {\mathbf {S}} ^ {- 1} \\ = \left[ \bar {\mathbf {U}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \widehat {\mathbf {V}} + \bar {\mathbf {V}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \widehat {\mathbf {U}} \right] \widehat {\mathbf {S}} ^ {- 1} \\ = \bar {\mathbf {U}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \left(\widehat {\mathbf {V}} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top} \widehat {\mathbf {V}}\right) \widehat {\mathbf {S}} ^ {- 1} + \bar {\mathbf {U}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top} \widehat {\mathbf {V}} \widehat {\mathbf {S}} ^ {- 1} \\ + \bar {\mathbf {V}} ^ {\top} \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \left(\hat {\mathbf {U}} - \bar {\mathbf {U}} \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}}\right) \hat {\mathbf {S}} ^ {- 1} + \bar {\mathbf {V}} ^ {\top} \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \bar {\mathbf {U}} \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} \hat {\mathbf {S}} ^ {- 1}. \\ \end{array}
$$
$$
\begin{array}{l} \left\| \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} \right\| _ {2} \leq \left\| \hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} \left(\left\| \hat {\mathbf {V}} \hat {\mathbf {V}} ^ {\top} - \bar {\mathbf {V}} \bar {\mathbf {V}} \right\| _ {2} + \left\| \hat {\mathbf {U}} \hat {\mathbf {U}} ^ {\top} - \bar {\mathbf {U}} \bar {\mathbf {U}} \right\| _ {2}\right) \left\| \hat {\mathbf {S}} ^ {- 1} \right\| _ {2} \\ \left\| \bar {\mathbf {U}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \bar {\mathbf {V}} \right\| _ {2} \left\| \widehat {\mathbf {S}} ^ {- 1} \right\| _ {2} + \left\| \bar {\mathbf {V}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \bar {\mathbf {U}} \right\| _ {2} \left\| \widehat {\mathbf {S}} ^ {- 1} \right\| _ {2} \\ \stackrel {\mathbb {P}} {\lesssim} \sqrt {M n \lambda_ {\max}} \cdot \frac {\sqrt {n \lambda_ {\max}}}{\delta} \cdot \frac {1}{\sigma_ {d} (\bar {\Lambda})} + \frac {M \log^ {3 / 2} n}{\sigma_ {d} (\bar {\Lambda})} \\ = \frac {n \lambda_ {\max}}{\delta^ {2}} + \frac {\sqrt {M} \log^ {3 / 2} n}{\delta} \\ \lesssim \frac {\sqrt {n \lambda_ {\mathrm {m a x}}}}{\delta}. \\ \end{array}
$$
Combining this with (21), we have
$$
\left\| \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} - \mathbf {W} \right\| _ {2} \leq \left\| \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} - \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} \right\| _ {2} + \left\| \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} - \mathbf {W} \right\| _ {2} \stackrel {\mathbb {P}} {\lesssim} \frac {\sqrt {n \lambda_ {\max}}}{\delta}.
$$
G.5 Proof of Proposition 6
We begin by decomposing  $\mathbf{W}\widehat{\mathbf{S}} -\mathbf{S}\mathbf{W}$  as
$$
\begin{array}{l} \mathbf {W} \widehat {\mathbf {S}} - \bar {\mathbf {S}} \mathbf {W} = (\mathbf {W} - \bar {\mathbf {U}} ^ {\top} \widehat {\mathbf {U}}) \widehat {\mathbf {S}} + \bar {\mathbf {S}} (\mathbf {V} ^ {\top} \widehat {\mathbf {V}} - \mathbf {W}) + \bar {\mathbf {U}} ^ {\top} \widehat {\mathbf {U}} \widehat {\mathbf {S}} - \bar {\mathbf {S}} \bar {\mathbf {V}} ^ {\top} \widehat {\mathbf {V}} \\ = \left(\mathbf {W} - \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}}\right) \hat {\mathbf {S}} + \bar {\mathbf {S}} \left(\mathbf {V} ^ {\top} \hat {\mathbf {V}} - \mathbf {W}\right) + \bar {\mathbf {U}} ^ {\top} \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \hat {\mathbf {V}} \\ = \left(\mathbf {W} - \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}}\right) \hat {\mathbf {S}} + \bar {\mathbf {S}} \left(\mathbf {V} ^ {\top} \hat {\mathbf {V}} - \mathbf {W}\right) + \bar {\mathbf {U}} ^ {\top} \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \left(\hat {\mathbf {V}} \hat {\mathbf {V}} ^ {\top} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}\right) \hat {\mathbf {V}} \\ + \bar {\mathbf {U}} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top} \widehat {\mathbf {V}}, \\ \end{array}
$$
and therefore we have that
$$
\begin{array}{l} \left\| \mathbf {W} \hat {\mathbf {S}} - \bar {\mathbf {S}} \mathbf {W} \right\| _ {2} \leq \left\| \mathbf {W} - \bar {\mathbf {U}} ^ {\top} \hat {\mathbf {U}} \right\| _ {2} \left\| \hat {\boldsymbol {\Lambda}} \right\| _ {2} + \left\| \mathbf {W} - \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} \right\| _ {2} \| \bar {\boldsymbol {\Lambda}} \| _ {2} \\ + \left\| \hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} \left\| \hat {\mathbf {V}} \hat {\mathbf {V}} ^ {\top} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top} \right\| _ {2} + \left\| \bar {\mathbf {U}} ^ {\top} \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \bar {\mathbf {V}} \right\| _ {2} \\ \stackrel {\mathbb {P}} {\lesssim} \frac {\sqrt {n \lambda_ {\operatorname* {m a x}}} \kappa}{\delta} + \frac {\sqrt {M} n \lambda_ {\operatorname* {m a x}}}{\delta} + M \log^ {3 / 2} n \\ \lesssim M \log^ {3 / 2} n, \\ \end{array}
$$
G.6 Proof of Proposition 7
A key tool in proving Proposition 7 is a theorem due [11], providing entrywise eigenvector bounds for random matrices. The original statement is given for the eigenvectors of symmetric random matrices
with row and column-wise independence. We state a generalisation for the singular vectors of rectangular matrices with block-wise independence structure. The extension to block-wise independence structure has been handled in [12] (see Proposition 2.1(b) of that paper), although the exposition of the results in this paper is more complicated. For this reason, we choose to state the result due to [11] with this generalisation, which can be seen by following through the relevant parts of their proof.
Lemma 8 (A slight generalisation of Theorem 2.1 of [11]). Let  $\mathbf{M}_0$  be an  $n_1 \times n_2$  real-valued random matrix. Define  $n_0 = n_1 + n_2$  and let  $\pi_1 = \sqrt{2n_1 / n_0}$  and  $\pi_2 = \sqrt{2n_2 / n_0}$ . Define  $\kappa_0 := \sigma_1(\mathbb{EM}_0) / \sigma_d(\mathbb{EM}_0)$ ,  $\delta_0 = \sigma_d(\mathbb{EM}_0) - \sigma_{d+1}(\mathbb{EM}_0)$ . Suppose there exists some  $\gamma > 0$  and a function  $\varphi: \mathbb{R}_+ \to \mathbb{R}_+$  which is continuous and non-decreasing on  $\mathbb{R}_+$ , with  $\varphi(0) = 0$  and  $\varphi(x) / x$  non-increasing on  $\mathbb{R}_+$ , such that the following conditions hold:
B1 (Incoherence).  $\| \mathbb{E}\mathbf{M}_0\|_{2,\infty}\vee \left\| \mathbb{E}\mathbf{M}_0^\top \right\|_{2,\infty}\leq \gamma \delta_0$
B2 (Block-wise independence). Assume that for any  $k \in [n_1], \ell \in [n_2]$ , there exists  $\mathcal{N}_k^1 \subset [n_1]$  and  $\mathcal{N}_{\ell}^{2} \subset [n_{2}]$ , such that the  $k$ th row of  $\mathbf{M}_0$  is independent of the columns  $\{j:j\notin \mathcal{N}_k^1\}$ , and the  $\ell$ th column of  $\mathbf{M}_0$  is independent of the rows  $\{i:i\notin \mathcal{N}_{\ell}^{2}\}$ . Let  $m_0 = \max_{k,\ell}\left\{\left|\mathcal{N}_k^1\right| \vee \left|\mathcal{N}_{\ell}^{2}\right|\right\}$  and assume  $m_0 \lesssim \delta_0$ .
B3 (Spectral norm concentration).  $\kappa_0\max \{\gamma ,\varphi (\gamma)\} \lesssim 1$  and  $\mathbb{P}(|\mathbf{M}_0 - \mathbb{EM}_0||_2 > \gamma \Delta)\leq \eta_0$  for some  $\eta_0\in (0,1)$
B4. [Row and column concentration] There exists some  $\eta_1 \in (0, 1)$  such that for any matrices  $\mathbf{Q} \in \mathbb{R}^{n_1 \times d}$ ,  $\mathbf{R} \in \mathbb{R}^{n_2 \times d}$  and  $i \in [n_1], j \in [n_2]$ ,
$$
\mathbb {P} \left\{\left\| (\mathbf {M} _ {0} - \mathbb {E} \mathbf {M} _ {0}) _ {., i} \mathbf {R} \right\| _ {2} \leq \delta_ {0} b _ {\infty} \varphi \left(\frac {b _ {\mathrm {F}}}{\sqrt {n _ {0}} b _ {\infty}}\right) \right\} \geq 1 - \frac {\eta_ {1}}{n _ {0}},
$$
$$
\mathbb {P} \left\{\left\| (\mathbf {M} _ {0} - \mathbb {E} \mathbf {M} _ {0}) _ {j, \cdot} \mathbf {Q} \right\| _ {2} \leq \delta_ {0} b _ {\infty} \varphi \left(\frac {b _ {\mathrm {F}}}{\sqrt {n _ {0}} b _ {\infty}}\right) \right\} \geq 1 - \frac {\eta_ {1}}{n _ {0}}
$$
where  $b_{\infty} \coloneqq \pi_1\| \mathbf{Q}\|_{2,\infty} \vee \pi_2\| \mathbf{R}\|_{2,\infty}$ , and  $b_{\mathrm{F}} \coloneqq \left(\pi_1\| \mathbf{Q}\|_{\mathrm{F}}^2 + \pi_2\| \mathbf{R}\|_{\mathrm{F}}^2\right)^{1/2}$ .
Let  $\widehat{\mathbf{U}}_0, \mathbf{U}_0$  (respectively  $\widehat{\mathbf{V}}_0, \mathbf{V}_0$ ) be the matrices containing the left (respectively, right) singular vectors corresponding to the  $d$  leading singular values of  $\mathbf{M}_0$  and  $\mathbb{E}\mathbf{M}_0$ . Then, with probability at least  $1 - \eta_0 - 2\eta_1$ , we have
$$
\begin{array}{l} \pi_ {1} \| \widehat {\mathbf {U}} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| \widehat {\mathbf {V}} _ {0} \| _ {2, \infty} \lesssim \left\{\kappa_ {0} + \varphi (1) \right\} \left(\pi_ {1} \| \mathbf {U} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| \mathbf {V} _ {0} \| _ {2, \infty}\right) \\ + \gamma (\pi_ {1} \| \mathbb {E} \mathbf {M} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| (\mathbb {E} \mathbf {M} _ {0}) ^ {\top} \| _ {2, \infty}) / \delta_ {0}; \\ \end{array}
$$
$$
\begin{array}{l} \pi_ {1} \| \widehat {\mathbf {U}} _ {0} \mathbf {O} - \mathbf {U} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| \widehat {\mathbf {V}} _ {0} \mathbf {O} - \mathbf {V} _ {0} \| _ {2, \infty} \lesssim [ \kappa_ {0} \left\{\kappa_ {0} + \varphi (1) \right\} \left\{\gamma + \varphi (\gamma) \right\} + \varphi (1) ] (\pi_ {1} \| \mathbf {U} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| \mathbf {V} _ {0} \| _ {2, \infty}) \\ + \gamma (\pi_ {1} \| \mathbb {E} \mathbf {M} _ {0} \| _ {2, \infty} \vee \pi_ {2} \| (\mathbb {E} \mathbf {M} _ {0}) ^ {\top} \| _ {2, \infty}) / \delta_ {0}. \\ \end{array}
$$
The following is an adaptation of Lemma D.2 of [66] (see also Lemma 7 of [11]) who showed an analogous result for Bernoulli random variables.
Lemma 9. Let  $Y_{i} \sim \mathrm{Poisson}(\lambda_{i})$  independently for all  $i = 1, \dots, n$ , and suppose  $\mathbf{Q}$  is a deterministic matrix. The  $Q_{i}$  denote the  $i$ th row of  $\mathbf{Q}$ , and set  $\lambda_{\max} := \max_{i \in [n]} \lambda_{i}$ . Then for any  $\alpha > 0$ ,
$$
\mathbb {P} \left\{\left\| \sum_ {i = 1} ^ {n} \left(Y _ {i} - \lambda_ {i}\right) Q _ {i} \right\| > \frac {(2 + \alpha) n \lambda_ {\max } \| \mathbf {Q} \| _ {2 , \infty}}{1 \vee \log \left(\sqrt {n} \| \mathbf {Q} \| _ {2 , \infty} / \| \mathbf {Q} \| _ {\mathrm {F}}\right)} \right\} \leq 2 d e ^ {- \alpha n \lambda_ {\max }}.
$$
We omit the proof of Lemma 9, which is identical to the proof of Lemma D.2 of [66] with the Bernoulli moment generating function with the Poisson moment generating function.
With these tools to hand, we begin by obtaining a bound on  $\| \widehat{\mathbf{U}}\|_{2,\infty}$  using Lemma 8, with  $\mathbf{M}_0\coloneqq \widehat{\boldsymbol{\Lambda}}$  We set  $\gamma \coloneqq \sqrt{n\lambda_{\mathrm{max}}}\big{/}\delta$  and
$$
\varphi (x) := \frac {n \lambda_ {\operatorname* {m a x}}}{\delta \left\{1 \vee \log (1 / x) \right\}}.
$$
First observe that  $n_0 = n + nM \asymp nM$  and  $\pi_1 \asymp M^{-1/2}$  and  $\pi_2 \asymp 1$ , and that  $\kappa_0 = \kappa$  and  $\delta_0 = \sqrt{M}\delta$ . B1 holds since  $\| \bar{\Lambda} \|_{2,\infty} \leq \sqrt{n}\lambda_{\max} \lesssim \sqrt{n\lambda_{\max}}$  since  $\lambda_{\max} \lesssim 1$  by Assumption 1. Using Assumptions 3 and 4, we have
$$
M \lesssim \frac {n \lambda_ {\operatorname* {m a x}}}{\log^ {3} n} \lesssim \frac {\delta \log (\delta / \sqrt {n \lambda_ {\operatorname* {m a x}}})}{\kappa \log^ {3} n} \lesssim \frac {\delta \log n}{\kappa \log^ {3} n} \lesssim \delta ,
$$
and therefore B2 holds. B3 holds from Proposition 2, and observing that by Assumption 3,  $\kappa_0\max \{\gamma ,\phi (\gamma)\} \lesssim 1$
To see that B4 holds, note that each row and column of  $M^{-1}(\widehat{\mathbf{A}} - \bar{\mathbf{A}})$  contains independent Poisson random variables with means not exceeding  $n\lambda_{\max} / M$ . Then for  $\mathbf{Q} \in \mathbb{R}^{n \times d}$ ,  $\mathbf{R} \in \mathbb{R}^{nM \times d}$ , setting  $\alpha = \log n / n\lambda_{\max}$  in Lemma 9 implies that
$$
\begin{array}{l} \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {i, \cdot} \mathbf {R} \right\| _ {2, \infty} = M \left\| \frac {1}{M} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {i, \cdot} \mathbf {R} \right\| _ {2, \infty} \\ \stackrel {\mathbb {P}} {\lesssim} M \cdot \frac {(n \lambda_ {\max } / M + \log n) \| \mathbf {R} \| _ {2 , \infty}}{1 \vee \log \left(\frac {\sqrt {n _ {0}} \| \mathbf {R} \| _ {2 , \infty}}{\| \mathbf {R} \| _ {\mathrm {F}}}\right)} \\ \lesssim \frac {n \lambda_ {\operatorname* {m a x}} \| \mathbf {R} \| _ {2 , \infty}}{1 \vee \log \left(\frac {\sqrt {n _ {0}} \| \mathbf {R} \| _ {2 , \infty}}{\| \mathbf {R} \| _ {\mathrm {F}}}\right)} \\ = \delta \| \mathbf {R} \| _ {2, \infty} \varphi \left(\frac {\| \mathbf {R} \| _ {\mathrm {F}}}{\sqrt {n _ {0}} \| \mathbf {R} \| _ {2 , \infty}}\right) \\ \leq \delta_ {0} b _ {\infty} \varphi \left(\frac {b _ {\mathrm {F}}}{\sqrt {n _ {0}} b _ {\infty}}\right). \\ \end{array}
$$
Similarly, setting  $\alpha = M\log n / n\lambda_{\mathrm{max}}$  we have
$$
\left\| \left(\tilde {\mathbf {A}} - \tilde {\boldsymbol {\Lambda}}\right) _ {., i} ^ {\top} \mathbf {Q} \right\| _ {2, \infty} \lesssim \frac {\mathbb {P}}{1 \vee \log \left(\sqrt {n _ {0}} \| \mathbf {Q} \| _ {2 , \infty} / \| \mathbf {Q} \| _ {\mathrm {F}}\right)} \leq \delta_ {0} b _ {\infty} \varphi \left(\frac {\sqrt {n} b _ {\infty}}{b _ {\mathrm {F}}}\right),
$$
which establishes B4. Having established B1-B4, we are ready to apply Lemma 8:
$$
\left\| \widehat {\mathbf {U}} \right\| _ {2, \infty} \stackrel {\mathbb {P}} {\lesssim} \sqrt {M} \left\{\kappa_ {0} + \varphi (1) \right\} (\pi_ {1} \| \mathbf {U} \| _ {2, \infty} \vee \pi_ {2} \| \mathbf {V} \| _ {2, \infty}) + \sqrt {M} \gamma \left(\pi_ {1} \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} \vee \pi_ {2} \left\| \bar {\boldsymbol {\Lambda}} ^ {\top} \right\| _ {2, \infty}\right) / \delta_ {0}
$$
$$
\begin{array}{l} \left\| \bar {\mathbf {V}} \right\| _ {2, \infty} = \left\| \bar {\boldsymbol {\Lambda}} \bar {\mathbf {U}} \bar {\mathbf {S}} ^ {- 1} \right\| _ {2, \infty} \leq \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {\infty} \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \left\| \bar {\mathbf {S}} \right\| _ {2} ^ {- 1} \lesssim \frac {n \lambda_ {\max } \mu \sqrt {d / n}}{\sigma_ {d} ^ {1 / 2} (\boldsymbol {\Sigma})} \tag {22} \\ \lesssim \frac {n \lambda_ {\operatorname* {m a x}} \mu \sqrt {d / n}}{\sqrt {M \delta}} \lesssim \sqrt {\frac {d}{n M}} \mu \log n. \\ \end{array}
$$
where we used Assumption 3 in the final inequality. Therefore
$$
\pi_ {1} \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \vee \pi_ {2} \left\| \bar {\mathbf {V}} \right\| _ {2, \infty} \leq \sqrt {\frac {d}{n M}} \mu \log n,
$$
$$
\kappa = \frac {\sigma_ {1} ^ {1 / 2} (\boldsymbol {\Sigma})}{\sigma_ {d} ^ {1 / 2} (\boldsymbol {\Sigma})} \lesssim \frac {n \lambda_ {\max}}{\delta} = \varphi (1)
$$
and so the first term satisfies
$$
\begin{array}{l} \sqrt {M} \left\{\kappa_ {0} + \varphi (1) \right\} \left(\pi_ {1} \| \mathbf {U} \| _ {2, \infty} \vee \pi_ {2} \| \mathbf {V} \| _ {2, \infty}\right) \lesssim \sqrt {M} \varphi (1) \left(\pi_ {1} \| \mathbf {U} \| _ {2, \infty} \vee \pi_ {2} \| \mathbf {V} \| _ {2, \infty}\right) \\ \lesssim \sqrt {M} \cdot \frac {n \lambda_ {\operatorname* {m a x}}}{\delta} \cdot \sqrt {\frac {d}{n M}} \mu \log n \\ = \frac {\mu \lambda_ {\max } \sqrt {n d} \log n}{\delta}. \\ \end{array}
$$
To control the second term, we first observe that
$$
\begin{array}{l} \pi_ {1} \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} \leq M ^ {- 1 / 2} \left\| \bar {\mathbf {U}} \bar {\mathbf {U}} ^ {\top} \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} + M ^ {- 1 / 2} \left\| (\mathbf {I} - \bar {\mathbf {U}} \bar {\mathbf {U}}) \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} \\ \lesssim M ^ {- 1 / 2} \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {2} + M ^ {- 1 / 2} \max  _ {i \in [ n ]} \sup  _ {t \in (0, 1 ]} r _ {i} (t) \\ \lesssim \sqrt {\frac {d}{M n}} \mu \cdot \sqrt {M} \kappa \delta + \mu \sqrt {d \lambda_ {\mathrm {m a x}}} \log^ {5 / 2} n \\ \lesssim \mu \delta \sqrt {d \lambda_ {\max}} \\ \end{array}
$$
where the final inequality follows from  $\kappa \lesssim \log n \leq \sqrt{n}$  and  $\lambda_{\max} \lesssim 1$ . Similarly we obtain  $\pi_2\| \bar{\Lambda}^\top \|_{2,\infty} \lesssim \mu \delta \sqrt{d\lambda_{\max}} \log n$  using (22), and therefore
$$
\pi_ {1} \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} \vee \pi_ {2} \left\| \bar {\boldsymbol {\Lambda}} ^ {\top} \right\| _ {2, \infty} \lesssim \mu \delta \sqrt {d \lambda_ {\max }} \log n.
$$
$$
\begin{array}{l} \sqrt {M} \gamma \left(\pi_ {1} \left\| \bar {\boldsymbol {\Lambda}} \right\| _ {2, \infty} \vee \pi_ {2} \left\| \bar {\boldsymbol {\Lambda}} ^ {\top} \right\| _ {2, \infty}\right) / \delta_ {0} \lesssim \sqrt {M} \cdot \frac {\sqrt {n \lambda_ {m a x}}}{\delta} \cdot \mu \delta \sqrt {d \lambda_ {\max}} \log n \cdot \frac {1}{\sqrt {M} \delta} \\ \lesssim \frac {\mu \sqrt {n d} \lambda_ {\max} \log n}{\delta}. \\ \end{array}
$$
Combining these bounds, we obtain
$$
\left\| \widehat {\mathbf {U}} \right\| _ {2, \infty} \stackrel {{\mathbb {P}}} {{\lesssim}} \frac {\mu \sqrt {n d} \lambda_ {\max } \log n}{\delta}.
$$