SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\left\| \widehat {\mathbf {U}} ^ {(m)} \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} - \widehat {\mathbf {U}} \widehat {\mathbf {U}} ^ {\top} \right\| _ {2} \lesssim \frac {\mu \lambda_ {\max } ^ {3 / 2} \sqrt {d} n \log^ {3 / 2} n}{\delta^ {2}} \tag {16}
$$
The proof of Proposition 8 requires a delicate "leave-one-out"-style argument.
F.5 Bounding terms (9)-(12)
Firstly observe that
$$
\left\| \bar {\mathbf {V}} \right\| _ {2, \infty} = \left\| \bar {\boldsymbol {\Lambda}} ^ {\top} \bar {\mathbf {U}} \bar {\mathbf {S}} ^ {- 1} \right\| _ {2, \infty} \leq \left\| \bar {\boldsymbol {\Lambda}} ^ {\top} \right\| _ {\infty} \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \left\| \bar {\mathbf {S}} ^ {- 1} \right\| _ {2} \leq \frac {\sqrt {n d} \lambda_ {\max} \mu}{\sqrt {M \sigma_ {d} (\boldsymbol {\Sigma})}}
$$
and therefore term (9) can be bounded as
$$
\begin{array}{l} \left\| \bar {\mathbf {V}} (\bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} \hat {\mathbf {S}} - \bar {\mathbf {S}} \mathbf {W}) \right\| _ {2, \infty} \leq \| \bar {\mathbf {V}} \| _ {2, \infty} \left(\left\| \bar {\mathbf {V}} ^ {\top} \hat {\mathbf {V}} - \mathbf {W} \right\| _ {2} \left\| \hat {\mathbf {S}} \right\| + \left\| \mathbf {W} \hat {\mathbf {S}} - \bar {\mathbf {S}} \mathbf {W} \right\| _ {2}\right) \\ \stackrel {\mathbb {P}} {\lesssim} \frac {\sqrt {n d} \lambda_ {\operatorname* {m a x}} \mu}{\sqrt {M \sigma_ {d} (\pmb {\Sigma})}} \left(\frac {\sqrt {n \lambda_ {\operatorname* {m a x}}}}{\delta} \cdot \sqrt {M \sigma_ {1} (\pmb {\Sigma})} + M \log^ {3 / 2} n\right) \\ \lesssim \frac {n \sqrt {M d} \lambda_ {\mathrm {m a x}} ^ {3 / 2} \mu \kappa}{\delta} \\ \lesssim \mu \sqrt {M \lambda_ {\mathrm {m a x}} d} \log n. \\ \end{array}
$$
where the third inequality follows from Assumption 4 that  $\sqrt{M}\log^{3 / 2}n\lesssim n\lambda_{\mathrm{max}}$ , and the definition  $\kappa \coloneqq \sqrt{\sigma_1(\Sigma) / \sigma_d(\Sigma)}$ , and the fourth inequality follows from Assumption 3 that  $\delta \log n\geq \delta \log (\delta /\sqrt{n\lambda_{\mathrm{max}}})\gtrsim \kappa n\lambda_{\mathrm{max}}$
To bound (10), we first apply Wedin's  $\sin \Theta$  theorem to obtain
$$
\left\| \widehat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W} \right\| _ {2} = \left\| \sin \Theta (\widehat {\mathbf {U}}, \bar {\mathbf {U}}) \right\| _ {2} \leq \frac {\left\| \widehat {\mathbf {A}} - \bar {\mathbf {A}} \right\|}{\sigma_ {d} (\bar {\mathbf {A}}) - \sigma_ {d + 1} (\bar {\mathbf {A}})} \stackrel {\mathbb {P}} {\lesssim} \frac {\sqrt {M n \lambda_ {\max}}}{\sqrt {M} \delta} = \frac {\sqrt {n \lambda_ {\max}}}{\delta}
$$
Then, we use Assumption 2 to obtain the bound
$$
\big \| (\mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}) \bar {\boldsymbol {\Lambda}} ^ {\top} \big \| _ {2, \infty} = \big \| \bar {\boldsymbol {\Lambda}} ^ {\top} (\mathbf {I} - \bar {\mathbf {U}} \bar {\mathbf {U}}) \big \| _ {2, \infty} \lesssim \max _ {i \in [ n ]} \sup _ {t \in \mathcal {T}} r _ {i} (t) \lesssim \sqrt {\frac {d}{n}} \mu \delta \log^ {5 / 2} n.
$$
Putting these two bounds together, we bound (10) as
$$
\left\| (\mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}) \bar {\boldsymbol {\Lambda}} ^ {\top} (\hat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W}) \right\| _ {2, \infty} \leq \left\| (\mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}) \bar {\boldsymbol {\Lambda}} ^ {\top} \right\| _ {2, \infty} \left\| \hat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W} \right\| _ {2} \stackrel {{\mathbb {P}}} {{\lesssim}} \mu \sqrt {d \lambda_ {\max }} \log^ {5 / 2} n
$$
To bound term (11), we set  $\mathbf{E} = \widehat{\mathbf{\Lambda}} -\bar{\mathbf{\Lambda}}$  and note that each column of  $M^{-1}\mathbf{E}$  contains independent Poisson random variables with means no greater than  $M^{-1}\lambda_{\mathrm{max}}$ . We will use Lemma 4 to bound the rows  $\mathbf{E}\bar{\mathbf{U}}$  as
$$
\begin{array}{l} [ \mathbf {E} ^ {\top} \bar {\mathbf {U}} ] _ {i} = \sum_ {j = 1} ^ {n} e _ {j i} \bar {U} _ {j} \stackrel {\mathbb {P}} {\lesssim} M \log^ {2} n \| \bar {\mathbf {U}} \| _ {2, \infty} + \sqrt {M \lambda_ {\max } \log n} \| \bar {\mathbf {U}} \| _ {\mathrm {F}} \\ \lesssim \left(M \log^ {2} n + \sqrt {M n \lambda_ {\max} \log n}\right) \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \\ \lesssim \sqrt {M \lambda_ {\mathrm {m a x}} n \log n} \left\| \bar {\mathbf {U}} \right\| _ {2, \infty} \\ \lesssim \mu \sqrt {M \lambda_ {\mathrm {m a x}} d \log n}. \\ \end{array}
$$
where the third inequality uses Assumption 4 and a union bound over  $i = 1, \dots, n$ . Therefore, we have
$$
\left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \bar {\mathbf {U}} \right\| _ {2, \infty} \stackrel {\mathbb {P}} {\lesssim} \sqrt {M \lambda_ {\max } d \log n}. \tag {17}
$$
Noting that  $\left\| \mathbf{I} - \bar{\mathbf{V}}\bar{\mathbf{V}}^{\top}\right\|_{\infty}\leq \left\| \mathbf{I} - \bar{\mathbf{V}}\bar{\mathbf{V}}^{\top}\right\|_{2}\lesssim 1$  , we bound (11) as
$$
\left\| (\mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}) (\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}) ^ {\top} \bar {\mathbf {U}} \mathbf {W} \right\| _ {2, \infty} \lesssim \left\| \mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} \right\| _ {\infty} \left\| (\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}) ^ {\top} \bar {\mathbf {U}} \right\| _ {2, \infty} \stackrel {\mathbb {P}} {\lesssim} \mu \sqrt {M \lambda_ {\max} d \log n}.
$$
Finally, we bound term (12). Let  $\widehat{\Lambda}^{(1)},\ldots ,\widehat{\Lambda}^{(n)}$  denote the auxiliary matrices described in (14), and let  $\widehat{\mathbf{U}}^{(m)}$  denote the matrix of leading left singular values of  $\widehat{\Lambda}^{(m)}$ . We can then decompose the Euclidean norm of  $(\widehat{\Lambda} -\bar{\Lambda})_{\cdot ,m}^{\top}(\widehat{\mathbf{U}} -\mathbf{U}\mathbf{W})$  as
$$
\begin{array}{l} \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {., m} ^ {\top} \left(\widehat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W}\right) \right\| _ {2} \leq \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {., m} ^ {\top} \widehat {\mathbf {U}} \left(\mathbf {W} - \widehat {\mathbf {U}} ^ {\top} \bar {\mathbf {U}}\right) \right\| _ {2} (18) \\ + \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {., m} ^ {\top} \left(\widehat {\mathbf {U}} \widehat {\mathbf {U}} ^ {\top} \bar {\mathbf {U}} - \widehat {\mathbf {U}} ^ {(m)} \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} \bar {\mathbf {U}}\right) \right\| _ {2} (19) \\ + \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {., m} ^ {\top} \left(\widehat {\mathbf {U}} ^ {(m)} \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} \bar {\mathbf {U}} - \bar {\mathbf {U}}\right) \right\| _ {2}. (20) \\ \end{array}
$$
The first term (18) is bounded as
$$
\begin{array}{l} \left\| \left(\widehat {\mathbf {A}} - \bar {\mathbf {A}}\right) _ {., m} ^ {\top} \widehat {\mathbf {U}} \left(\mathbf {W} - \widehat {\mathbf {U}} ^ {\top} \bar {\mathbf {U}}\right) \right\| _ {2} \leq \left\| \widehat {\mathbf {A}} - \bar {\mathbf {A}} \right\| _ {1} \left\| \widehat {\mathbf {U}} \right\| _ {2, \infty} \left\| \mathbf {W} - \widehat {\mathbf {U}} ^ {\top} \bar {\mathbf {U}} \right\| _ {2} \\ \stackrel {\mathbb {P}} {\lesssim} \sqrt {M n \lambda_ {m a x} \log n} \cdot \frac {\mu \lambda_ {\max} \sqrt {d n} \log n}{\delta} \cdot \frac {\sqrt {n \lambda_ {\max}}}{\delta} \\ = \frac {\sqrt {M} n ^ {3 / 2} \lambda_ {\max } ^ {2} \mu \sqrt {d} \log^ {3 / 2} n}{\delta^ {2}} \\ \lesssim \mu \sqrt {M \lambda_ {\max} d} \log^ {5 / 2} n. \\ \end{array}
$$
To bound the second term (19), we employ Proposition 8 to obtain
$$
\begin{array}{l} \left\| \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) _ {, m} ^ {\top} \left(\widehat {\mathbf {U}} \widehat {\mathbf {U}} ^ {\top} \bar {\mathbf {U}} - \widehat {\mathbf {U}} ^ {(m)} \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} \bar {\mathbf {U}}\right) \right\| _ {2} \leq \left\| \widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} \left\| \widehat {\mathbf {U}} \widehat {\mathbf {U}} ^ {\top} - \widehat {\mathbf {U}} ^ {(m)} \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} \right\| _ {2} \\ \stackrel {\mathbb {P}} {\lesssim} \sqrt {M n \lambda_ {\max }} \cdot \frac {\mu \lambda_ {\max } ^ {3 / 2} \sqrt {d n} \log^ {3 / 2} n}{\delta^ {2}} \\ = \frac {\sqrt {M} \mu \lambda_ {\max } ^ {2} \sqrt {d n} ^ {3 / 2} \log^ {3 / 2} n}{\delta^ {2}} \\ \lesssim \mu \sqrt {M \lambda_ {\max} d} \log^ {5 / 2} n \\ \end{array}
$$
We now set about bounding the third term (20). Let  $\Omega_1 \Xi \Omega_2^\top$  denote a singular value decomposition of  $(\widehat{\mathbf{U}}^{(m)})^\top \bar{\mathbf{U}}$ , and set  $\mathbf{W}^{(m)} \coloneqq \Omega_1 \Omega_2^\top$ . Let  $\theta_i^{(m)}$  denote the principal angles between the column spaces of  $\widehat{\mathbf{U}}^{(m)}$  and  $\bar{\mathbf{U}}$  defined by  $\xi_i^{(m)} = \cos(\theta_i^{(m)})$ , where  $\xi_i^{(m)}$  are the singular values of  $(\widehat{\mathbf{U}}^{(m)})^\top \bar{\mathbf{U}}$ . We invoke Wedin's theorem to show that
$$
\begin{array}{l} \left\| \mathbf {W} ^ {(m)} - \left(\widehat {\mathbf {U}} ^ {(m)}\right) ^ {\top} \bar {\mathbf {U}} \right\| _ {2} = \| \mathbf {I} - \boldsymbol {\Xi} \| _ {2} = \max  _ {i \in [ d ]} (1 - \xi_ {i} ^ {(m)}) = \max  _ {i \in [ d ]} (1 - \cos \theta_ {i} ^ {(m)}) \\ \leq \max _ {i \in [ d ]} (1 - \cos^ {2} \theta_ {i} ^ {(m)}) = \max _ {i \in [ d ]} \sin^ {2} \theta_ {i} ^ {(m)} \lesssim \frac {\left\| \widehat {\boldsymbol {\Lambda}} ^ {(m)} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} ^ {2}}{(\sigma_ {d} (\bar {\boldsymbol {\Lambda}}) - \sigma_ {d + 1} (\widehat {\boldsymbol {\Lambda}})) ^ {2}} \stackrel {{\mathbb {P}}} {{\lesssim}} \frac {M n \lambda_ {\max}}{M \delta^ {2}} = \frac {n \lambda_ {\max}}{\delta^ {2}} \lesssim 1 \\ \end{array}
$$
We define  $\mathbf{H}^{(m)}\coloneqq \widehat{\mathbf{U}}^{(m)}(\widehat{\mathbf{U}}^{(m)})^{\top} - \mathbf{U}\mathbf{U}^{\top}$  and note that  $\mathbf{H}^{(m)}$  is independent of  $(\widehat{\pmb{\Lambda}} -\bar{\pmb{\Lambda}})_{m}$ , and that
$$
\begin{array}{l} \left\| \mathbf {H} ^ {(m)} \right\| _ {2, \infty} \leq \left\| \widehat {\mathbf {U}} ^ {(m)} \mathbf {W} ^ {(m)} - \bar {\mathbf {U}} \right\| _ {2, \infty} + \left\| \widehat {\mathbf {U}} ^ {(m)} \right\| _ {2, \infty} \left\| (\widehat {\mathbf {U}} ^ {(m)}) ^ {\top} \bar {\mathbf {U}} - \mathbf {W} ^ {(m)} \right\| _ {2} \\ \lesssim \left\| \widehat {\mathbf {U}} ^ {(m)} \mathbf {W} ^ {(m)} - \bar {\mathbf {U}} \right\| _ {2, \infty} + \left\| \widehat {\mathbf {U}} ^ {(m)} \right\| _ {2, \infty} \\ \stackrel {\mathbb {P}} {\lesssim} \frac {\mu \lambda_ {\max } \sqrt {d n} \log n}{\delta} \\ \end{array}
$$
Then, using Lemma 4 we have that
$$
\begin{array}{l} \left\| \left(\tilde {\mathbf {A}} - \tilde {\mathbf {A}}\right) _ {, m} ^ {\top} \mathbf {H} ^ {(m)} \right\| _ {2} \stackrel {{\mathbb {P}}} {{\lesssim}} M \log^ {2} n \left\| \mathbf {H} ^ {(m)} \right\| _ {2, \infty} + \sqrt {\lambda_ {\max } \log n} \left\| \mathbf {H} ^ {(m)} \right\| _ {\mathrm {F}} \\ \stackrel {\mathbb {P}} {\lesssim} \sqrt {M \log n \lambda_ {\max}} \left\| \mathbf {H} ^ {(m)} \right\| _ {2, \infty} + \sqrt {\lambda_ {\max} d \log n} \left\| \mathbf {H} ^ {(m)} \right\| _ {2} \\ \stackrel {\mathbb {P}} {\lesssim} \sqrt {M \log n \lambda_ {\max }} \cdot \frac {\mu \lambda_ {\max } \sqrt {d n} \log n}{\delta} + \sqrt {\lambda_ {\max } d \log n} \frac {n \lambda_ {\max }}{\delta} \\ \leq \frac {\sqrt {M} \mu n \lambda_ {\max } ^ {3 / 2} \sqrt {d} \log^ {3 / 2} n}{\delta} \\ \lesssim \mu \sqrt {M d \lambda_ {\max}} \log^ {5 / 2} n. \\ \end{array}
$$
Combining these bounds and taking a union bound over  $m \in [n]$ , we have
$$
\left\| \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \left(\hat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W}\right) \right\| _ {2, \infty} \stackrel {{\mathbb {P}}} {\lesssim} \mu \sqrt {M d \lambda_ {\max }} \log^ {5 / 2} n,
$$
and the term (12) is bounded as
$$
\begin{array}{l} \left\| \left(\mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} ^ {\top}\right) \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \left(\hat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W}\right) \right\| _ {2, \infty} \leq \left\| \mathbf {I} - \bar {\mathbf {V}} \bar {\mathbf {V}} \right\| _ {\infty} \left\| \left(\hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) ^ {\top} \left(\hat {\mathbf {U}} - \bar {\mathbf {U}} \mathbf {W}\right) \right\| _ {2, \infty} \\ \stackrel {\mathbb {P}} {\lesssim} \mu \sqrt {M \lambda_ {\max } d} \log^ {5 / 2} n. \\ \end{array}
$$
Combining the bounds on (9)-(12), we have
$$
\max  _ {i, j \in [ n ]} \sup  _ {t \in \mathcal {T}} \left\| \mathbf {W} _ {1} \widehat {Y} _ {i} (t) - \bar {Y} _ {i} (t) \right\| _ {2} = \left\| \widehat {\mathbf {V}} \widehat {\mathbf {S}} \mathbf {W} _ {1} ^ {\top} - \bar {\mathbf {V}} \bar {\mathbf {S}} \right\| _ {2, \infty} \stackrel {{\mathbb {P}}} {\lesssim} \mu \sqrt {M d \lambda_ {\max }} \log^ {5 / 2} n,
$$
F.6 Controlling the bias term
F.6.1 Edge level bias
We begin by studying the edge-level bias of the histogram intensity estimator. Let  $\rho_{ij}(t) = \int_0^t\lambda_{ij}(s)\mathrm{d}s$  denote the cumulative intensity of edge  $i,j$ . Now we have, for  $t\in B_{\ell}$ ,
$$
\begin{array}{l} \bar {\lambda} _ {i j} (t) = M \int_ {B _ {\ell}} \lambda_ {i j} (s) \mathrm {d} t \\ = M \left\{\rho_ {i j} \left(\frac {\ell}{M}\right) - \rho_ {i j} \left(\frac {\ell - 1}{M}\right) \right\} \\ = \frac {\rho_ {i j} \left(\frac {\ell}{M}\right) - \rho_ {i j} \left(\frac {\ell - 1}{M}\right)}{\frac {\ell}{M} - \frac {\ell - 1}{M}} \\ = \lambda_ {i j} (t ^ {\star}) \\ \end{array}
$$
for some  $t^\star \in B^\ell$ , which follows by an application of the mean value theorem, where  $\rho'(t) = \lambda_{ij}(t)$ . We then apply the  $L$ -Lipschitz continuity of  $\lambda_{ij}(t)$  to obtain
$$
\begin{array}{l} \bar {\lambda} _ {i j} (t) - \lambda_ {i j} (t) | = | \lambda_ {i j} \left(t ^ {*}\right) - \lambda_ {i j} (t) | \\ \leq L \cdot | t ^ {\star} - t | \\ \leq \frac {L}{M}. \\ \end{array}
$$
F.6.2 A subspace perturbation bound
Define the operator  $\mathcal{A}:(\mathcal{T}\to \mathbb{R}^n)\to \mathbb{R}^n$  by
$$
\mathcal {A} v (\cdot) = \int_ {\mathcal {T}} \boldsymbol {\Lambda} (t) v (t) d t
$$
and define the operator  $\mathcal{A}^{\star}:\mathbb{R}^{n}\to (\mathcal{T}\to \mathbb{R}^{n})$  by
$$
\mathcal {A} ^ {\star} u = \Lambda (\cdot) u.
$$
Then  $\Sigma \equiv \mathcal{A}\mathcal{A}^{\star}$  since
$$
\mathcal {A} \mathcal {A} ^ {\star} u = \mathcal {A} (\boldsymbol {\Lambda} (\cdot) u) = \int_ {\mathcal {T}} \boldsymbol {\Lambda} ^ {2} (t) u d t = \boldsymbol {\Sigma} u.
$$
Denote its eigenvalues  $\sigma_1^2, \ldots, \sigma_n^2$ , and its corresponding orthonormal eigenvectors  $u_1, \ldots, u_n$ , and define  $v_i(\cdot) = \Lambda(\cdot) u_i / \xi_i$  for all  $i = 1, \ldots, n$ . Then,  $\Lambda(\cdot)$  admits the (functional) singular value decomposition
$$
\boldsymbol {\Lambda} (\cdot) = \sum_ {i = 1} ^ {n} \sigma_ {i} u _ {i} v _ {i} (\cdot).
$$
Define  $\bar{\mathcal{A}}$  and its corresponding parameters analogously. By definition,
$$
\left\| \bar {\mathcal {A}} - \mathcal {A} \right\| _ {2} \leq \sup  _ {t \in \mathcal {T}} \left\| \bar {\boldsymbol {\Lambda}} (t) - \boldsymbol {\Lambda} (t) \right\| _ {2} \leq \sup  _ {t \in \mathcal {T}} \max  _ {i, j \in [ n ]} \left| \bar {\lambda} _ {i j} (t) - \lambda_ {i j} (t) \right| \leq \frac {n L}{M}.
$$
Therefore, by (a functional version of) Wedin's  $\sin \Theta$  theorem
$$
\left\| \bar {\mathbf {U}} \mathbf {W} _ {1} - \mathbf {U} \right\| _ {2} \lesssim \frac {\left\| \bar {\mathcal {A}} - \mathcal {A} \right\| _ {2}}{\sigma_ {d} - \sigma_ {d + 1}} \leq \frac {n L}{M \delta}.
$$
F.6.3 Controlling the bias term
Combining the above bounds, we have that uniformly for all  $i,j,t$
$$
\begin{array}{l} \left\| \bar {Y} _ {i} (t) \mathbf {W} _ {1} - Y _ {i} (t) \right\| _ {2} = \left\| \bar {\boldsymbol {\Lambda}} (t) \bar {\mathbf {U}} \mathbf {W} _ {1} - \boldsymbol {\Lambda} (t) \mathbf {U} \right\| _ {2} \\ \leq \left\| \bar {\mathbf {A}} (t) \right\| _ {2, \infty} \left\| \bar {\mathbf {U}} \mathbf {W} _ {2} - \mathbf {U} \right\| _ {2} + \left\| \bar {\mathbf {A}} (t) - \mathbf {A} (t) \right\| _ {2, \infty} \| \mathbf {U} \| _ {2} \\ \lesssim \frac {n ^ {3 / 2} \lambda_ {\max } L}{M \delta} + \frac {\sqrt {n} L}{M} \\ \leq \frac {n ^ {3 / 2} \lambda_ {\mathrm {m a x}} L}{M \delta}, \\ \end{array}
$$
where the final inequality follows from the fact that  $\delta \leq n\lambda_{\max}$ .
G Proofs of the technical propositions
G.1 Proof of Proposition 2
We have that  $M^{-1}$  times the lower-triangular elements of each block of  $\widehat{\Lambda}$  are independent Poisson random variables with mean given by  $M^{-1}$  times the lower-triangular elements of each block of  $\bar{\Lambda}$ . Define the matrices  $\widehat{\Lambda}^{\mathrm{L}}$  and  $\widehat{\Lambda}^{\mathrm{U}}$  with the upper and lower triangles, respectively, of each block set to zero, and the diagonals of each block halved, and define  $\bar{\Lambda}^{\mathrm{L}}$  and  $\bar{\Lambda}^{\mathrm{U}}$  similarly, so that  $M^{-1}\widehat{\Lambda}^{\mathrm{L}}$  (respectively  $M^{-1}\widehat{\Lambda}^{\mathrm{U}}$ ) has independent Poisson entries with means  $M^{-1}\bar{\Lambda}^{\mathrm{L}}$  (respectively  $M^{-1}\bar{\Lambda}^{\mathrm{U}}$ ), and  $\widehat{\Lambda} - \bar{\Lambda} = (\widehat{\Lambda}^{\mathrm{L}} - \bar{\Lambda}^{\mathrm{L}}) + (\widehat{\Lambda}^{\mathrm{U}} - \bar{\Lambda}^{\mathrm{U}})$ .
We condition on the event that  $(\widehat{\Lambda}^{\mathrm{L}} - \bar{\Lambda}^{\mathrm{L}})_{ij}\lesssim M\log n$  for all  $i,j$ , which occurs with overwhelming probability by Lemma 3 and a union bound. Now, we employ Lemma 5 with  $B\coloneqq M\log n$  and  $\nu \coloneqq Mn\lambda_{\max}$  to obtain
$$
\mathbb {P} \left(\left\| \widehat {\boldsymbol {\Lambda}} ^ {\mathrm {L}} - \bar {\boldsymbol {\Lambda}} ^ {\mathrm {L}} \right\| _ {2} \geq 4 \sqrt {M n \lambda_ {\max }} + t\right) \leq n \exp \left(- \frac {t ^ {2}}{c (M \log n) ^ {2}}\right).
$$
Setting  $t = M\log^{3 / 2}n$  we have that
$$
\left\| \hat {\boldsymbol {\Lambda}} ^ {\mathrm {L}} - \bar {\boldsymbol {\Lambda}} ^ {\mathrm {L}} \right\| _ {2} \stackrel {{\mathbb {P}}} {{\lesssim}} \sqrt {M n \lambda_ {\max }} + M \log^ {3 / 2} n \lesssim \sqrt {M n \lambda_ {\max }}
$$
where the final inequality follows from Assumption 4. We obtain an analogous bound for  $\left\| \widehat{\boldsymbol{\Lambda}}^{\mathrm{U}} - \bar{\boldsymbol{\Lambda}}^{\mathrm{U}}\right\|_{2}$  and combine the with the triangle inequality:
$$
\left\| \hat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} \leq \left\| \hat {\boldsymbol {\Lambda}} ^ {\mathrm {L}} - \bar {\boldsymbol {\Lambda}} ^ {\mathrm {L}} \right\| _ {2} + \left\| \hat {\boldsymbol {\Lambda}} ^ {\mathrm {U}} - \bar {\boldsymbol {\Lambda}} ^ {\mathrm {U}} \right\| _ {2} \stackrel {{\mathbb {P}}} {{\lesssim}} \sqrt {M n \lambda_ {\max }}.
$$
We now establish a bound on  $\left\| \widehat{\mathbf{\Lambda}} - \bar{\mathbf{\Lambda}} \right\|_1$ . We condition on the event  $|\widehat{\mathbf{\Lambda}}_{ij} - \bar{\mathbf{\Lambda}}_{ij}| \lesssim M \log n$  for all  $i, j$ , which occurs with overwhelming probability due to Lemma 3 and a union bound, and note that we have  $\sum_{j=1}^{n} \mathbb{E}(\widehat{\mathbf{\Lambda}}_{ji} - \bar{\mathbf{\Lambda}}_{ji})^2 \leq Mn\lambda_{\max}$ . Then, by the classical Bernstein inequality, we have for any  $t > 0$ ,
$$
\mathbb {P} \left\{\sum_ {j = 1} ^ {n} \left| \widehat {\boldsymbol {\Lambda}} _ {j i} - \bar {\boldsymbol {\Lambda}} _ {j i} \right| \geq t \right\} \leq 2 \exp \left\{\frac {- t ^ {2}}{2 (M n \lambda_ {\max } + t M \log n / 3)} \right\},
$$
and setting  $t = \sqrt{nM\lambda_{\mathrm{max}}\log n}$ , we obtain
$$
\sum_ {j = 1} ^ {n} \left| \widehat {\boldsymbol {\Lambda}} _ {j i} - \bar {\boldsymbol {\Lambda}} _ {j i} \right| \stackrel {\mathbb {P}} {\lesssim} \sqrt {n M \lambda_ {\max} \log n}.
$$
A union bound establishes that
$$
\left\| \widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {1} \stackrel {\mathbb {P}} {\lesssim} \sqrt {n M \lambda_ {\max} \log n},
$$
which establishes Proposition 2.
G.2 Proof of Proposition 3
Proposition 3 follows from an application of Weyl's inequality. We have
$$
\sigma_ {1} (\widehat {\mathbf {A}}) \leq \sigma_ {1} (\bar {\mathbf {A}}) + | \sigma_ {1} (\widehat {\mathbf {A}}) - \sigma_ {1} (\bar {\mathbf {A}}) | \leq \sigma_ {1} (\bar {\mathbf {A}}) + \left\| \widehat {\mathbf {A}} - \bar {\mathbf {A}} \right\| _ {2} \stackrel {\mathbb {P}} {\lesssim} \sqrt {M \sigma_ {1} (\mathbf {\Sigma})} + \sqrt {M n \lambda_ {\max}} \lesssim \sqrt {M \sigma_ {1} (\mathbf {\Sigma})}
$$
since  $\sigma_1(\bar{\Lambda}) = \sqrt{M\sigma_1(\Sigma)}\gtrsim \sqrt{M\delta}\gtrsim \sqrt{Mn\lambda_{\max}}$ . Similarly, we have
$$
\sigma_ {d} (\widehat {\boldsymbol {\Lambda}}) \geq \sigma_ {d} (\bar {\boldsymbol {\Lambda}}) - | \sigma_ {1} (\widehat {\boldsymbol {\Lambda}}) - \sigma_ {1} (\bar {\boldsymbol {\Lambda}}) | \geq \sigma_ {d} (\bar {\boldsymbol {\Lambda}}) - \left\| \widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}} \right\| _ {2} \stackrel {\mathbb {P}} {\gtrsim} \sqrt {M \sigma_ {d} (\boldsymbol {\Sigma})} - \sqrt {M n \lambda_ {\max}} \gtrsim \sqrt {M \sigma_ {d} (\boldsymbol {\Sigma})},
$$
which establishes the proposition.
G.3 Proof of Proposition 4
We begin by constructing matrices  $\bar{\mathbf{Q}}$  and  $\bar{\mathbf{E}}$ , via a symmetric dilation trick, such that the spectral norms of  $\mathbf{Q}^{\top}(\hat{\boldsymbol{\Lambda}} - \bar{\boldsymbol{\Lambda}})\mathbf{R}$  and  $\bar{\mathbf{Q}}^{\top}\bar{\mathbf{E}}\bar{\mathbf{Q}}$  coincide, and then apply a classical  $\varepsilon$ -net argument to the spectral norm of  $\bar{\mathbf{Q}}^{\top}\bar{\mathbf{E}}\bar{\mathbf{Q}}$ , following the proof of Lemma D.1 in [66].
First, we set  $\bar{\mathbf{E}} \coloneqq \mathcal{D}(\widehat{\mathbf{A}} - \bar{\mathbf{A}})$ , where  $\mathcal{D}$  is the dilation operator (see Section F.1.2) and  $\bar{\mathbf{Q}} = (\mathbf{Q} \mathbf{R})$ , and observe that
$$
\left\| \mathbf {Q} ^ {\top} \left(\widehat {\boldsymbol {\Lambda}} - \bar {\boldsymbol {\Lambda}}\right) \mathbf {R} \right\| _ {2} = \left\| \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \right\| _ {2} = \max  _ {\| v \| _ {2} \leq 1} \left| v ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} v \right|
$$
where the second equality follows from the Courant-Fischer min-max theorem. Now, let  $S_{\varepsilon}^{d-1}$  be an  $\varepsilon$ -net of the  $d-1$ -dimensional unit sphere  $S^{d-1} := \{ v : \| v \|_2 = 1 \}$ . By definition, for any  $v \in S^{d-1}$ , there exists some  $w(v) \in S_{\varepsilon}^{d-1}$  such that  $\| v - w(v) \|_2 < \varepsilon$  and
$$
\begin{array}{l} \left\| \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \right\| _ {2} = \max  _ {\| v \| _ {2} \leq 1} \left| v ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} v \right| \\ = \max  _ {\| v \| _ {2} \leq 1} \left| \left\{v ^ {\top} - w (v) + w (v) \right\} ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \left\{v - w (v) + w (v) \right\} \right| \\ \leq \left(\varepsilon^ {2} + 2 \varepsilon\right) \left\| \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} \right\| _ {2} + \max  _ {w \in S _ {\varepsilon} ^ {d - 1}} \left| w ^ {\top} \bar {\mathbf {Q}} ^ {\top} \bar {\mathbf {E}} \bar {\mathbf {Q}} w \right|. \\ \end{array}
$$
With  $\varepsilon = 1 / 3$ , we have