SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Using Lemma 8 with the iid variable  $y_{j} = (x_{j},\nu_{j\ell})$  and  $\mathbb{E}w_{\mathbf{S}}(\cdot ,x_j)\nu_{j\ell} = 0$  since  $\nu$  and  $X$  are independent, we obtain
$$
\forall \ell , \left\| \frac {1}{n} \sum_ {j} w _ {\mathbf {S}} (\cdot , x _ {j}) \nu_ {j \ell} \right\| _ {\infty} \xrightarrow [ n \to \infty ]{\mathcal {P}} 0
$$
B.2 Eigenvectors positional encodings: SignNet
In this whole section, the number of eigenvectors  $q$  is fixed, and we assume that  $\lambda_1^{\mathbf{S}}, \ldots, \lambda_{q+1}^{\mathbf{S}}$  are pairwise distinct. We first start by generic results that allow us to go from the graph matrix  $S$  to the Gram matrix.
Lemma 6 (Intermediate result for SignNet). Suppose that Assumption 2 holds, that  $\lambda_1^{\mathbf{S}},\dots,\lambda_{q+1}^{\mathbf{S}}$  are distinct, and that
$$
\max  _ {i = 1, \dots , q} \min  _ {s \in \{1, - 1 \}} \left\| s \sqrt {n} u _ {i} ^ {W} - \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} + \left| \lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}} \right| \xrightarrow [ n \rightarrow \infty ]{\mathcal {P}} 0. \tag {16}
$$
Then the result of Theorem 2 holds.
Proof. Let  $f \in \mathcal{F}_{\mathrm{Eig}}$ , written as (9). By Assumption 2,  $\| S - W \| \to 0$ . By Kato's inequality, we have that  $\sup_i \left| \lambda_i^S - \lambda_i^W \right| \to 0$ , and by hypothesis, the eigenvalues of  $W$  converge to those of  $\mathbf{S}$ . Given the hypotheses on the eigenvalues of  $\mathbf{S}$ , with probability going to 1 the  $q + 1$  first eigenvalues of  $S$  have single multiplicities. When it is the case, according to Davis-Kahan theorem (Theorem 8), for all  $i = 1, \ldots, q$  there is  $s_i \in \{-1, 1\}$  such that
$$
\max  _ {i} \left\| s _ {i} u _ {i} ^ {S} - u _ {i} ^ {W} \right\|\rightarrow 0 \tag {17}
$$
which, combined with our hypotheses, yields
$$
\max  _ {i = 1, \dots , q} \min  _ {s \in \{1, - 1 \}} \left\| s \sqrt {n} u _ {i} ^ {S} - \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\text {M S E}} \xrightarrow [ n \rightarrow \infty ]{\mathcal {P}} 0. \tag {18}
$$
For  $i = 1,\dots ,q$ , let  $f_{i}:\mathbb{R}\to \mathbb{R}^{p_{i}}$  be continuous functions and  $\varepsilon >0$ . By Lemma 3, there is  $f_{\gamma_i}^{\mathrm{MLP}}$  such that  $\left\| (f_{\gamma_i}^{\mathrm{MLP}} - f_i)\circ u_i^{\mathbf{S}}\right\|_{L^2}\leqslant \varepsilon /(2q)$ . Then, call  $L_{i}$  the (uniform) Lipschitz constant of  $f_{\gamma_i}^{\mathrm{MLP}}$  on  $\mathbb{R}$ . We have
$$
\begin{array}{l} \left\| \mathrm {P E} _ {\gamma} - \iota_ {X} \left[ (\mathbf {Q} f _ {i}) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \right\| _ {\mathrm {M S E}} = \left\| \left[ (\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}) (\sqrt {n} u _ {i} ^ {S}) \right] _ {i = 1} ^ {q} - \iota_ {X} \left[ (\mathbf {Q} f _ {i}) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \right\| _ {\mathrm {M S E}} \\ \leqslant \left\| \left[ \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \left(\sqrt {n} u _ {i} ^ {S}\right) \right] _ {i = 1} ^ {q} - \iota_ {X} \left[ \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \right\| _ {\text {M S E}} \\ + \left\| \iota_ {X} \left[ \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} - \iota_ {X} \left[ \left(\mathbf {Q} f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \right\| _ {\text {M S E}} \\ \leqslant \sum_ {i} \left\| \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \left(\sqrt {n} u _ {i} ^ {S}\right) - \iota_ {X} \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \circ u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} \\ + \left\| \iota_ {X} \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \circ u _ {i} ^ {\mathbf {S}} - \iota_ {X} \left(\mathbf {Q} f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right\| _ {\text {M S E}} \\ \leqslant 2 \sum_ {i} \min  _ {s _ {i} \in \{1, - 1 \}} \left\| f _ {\gamma_ {i}} ^ {\mathrm {M L P}} \left(s _ {i} \sqrt {n} u _ {i} ^ {S}\right) - \iota_ {X} f _ {\gamma_ {i}} ^ {\mathrm {M L P}} \circ u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} \\ + 2 \left\| \iota_ {X} \left(f _ {\gamma_ {i}} ^ {\mathrm {M L P}} - f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right\| _ {\text {M S E}} \\ \end{array}
$$
The first term goes to 0 in probability by what precedes, while for the second
$$
\sum_ {i} \left\| \iota_ {X} \left(f _ {\gamma_ {i}} ^ {\mathrm {M L P}} - f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} \xrightarrow [ n \to \infty ]{\mathcal {P}} \sum_ {i} \left\| \left(f _ {\gamma_ {i}} ^ {\mathrm {M L P}} - f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right\| _ {L ^ {2}} \leqslant \varepsilon / 2
$$
which proves that  $\mathcal{F}_{\mathrm{Eig}} \subset \mathcal{F}_{\mathrm{PE}}$ , and thus  $\overline{\mathcal{F}_{\mathrm{Eig}}} \subset \mathcal{F}_{\mathrm{PE}}$  since  $\mathcal{F}_{\mathrm{PE}}$  is closed by Lemma 5.
For the reverse inclusion, let  $f \in \mathcal{F}_{\mathrm{PE}}$ . By hypothesis, for all  $m \in \mathbb{N}$ , there is  $\gamma \in \Gamma$ , such that
$$
\mathbb {P} \left(\| \mathrm {P E} _ {\gamma} - \iota_ {X} f \| _ {\mathrm {M S E}} \geqslant 1 / m\right) \xrightarrow [ n \to \infty ]{} 0
$$
By what precedes, if we write  $\mathrm{PE}_{\gamma} = (\mathbf{Q}f_{\gamma_i}^{\mathrm{MLP}})(\sqrt{n} u_i^S)]_{i = 1}^q$  , we have
$$
\left\| \mathrm {P E} _ {\gamma} - \iota_ {X} \left[ \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \right\| _ {\mathrm {M S E}} \xrightarrow [ n \to \infty ]{\mathcal {P}} 0
$$
$$
\mathbb {P} \left(\left|\left|\left| \iota_ {X} (f - [ (\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}) \circ u _ {i} ^ {\mathbf {S}} ] _ {i = 1} ^ {q}) \right|\right| _ {\mathrm {M S E}} - \left|\left| f - [ (\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}) \circ u _ {i} ^ {\mathbf {S}} ] _ {i = 1} ^ {q}) \right|\right| _ {L ^ {2}} \right| \geqslant 1 / m\right)\rightarrow 0
$$
Hence, by a union bound and triangular inequality, we obtain the deterministic bound  $\left\| f - \left[(\mathbf{Q}f_{\gamma_i}^{\mathrm{MLP}})\circ u_i^{\mathbf{S}}\right]_{i = 1}^q\right\|_{L^2}\leqslant 3 / m$  Since  $\left[\left(\mathbf{Q}f_{\gamma_i}^{\mathrm{MLP}}\right)\circ u_i^{\mathbf{S}}\right]_{i = 1}^q\in \mathcal{F}_{\mathrm{Eig}}$  , we have  $f\in \overline{\mathcal{F}_{\mathrm{Eig}}}$
We must now prove the hypothesis of Lemma 6. This is done separately for p.s.d. kernel and SBM.
B.2.1 Positive semidefinite kernels
In this subsection, we assume that  $w_{\mathbf{S}}$  is p.s.d. The following result is adapted from [42].
Theorem 5 (Adapted from [42]). Suppose that Assumption 2 holds, that  $\lambda_1^{\mathbf{S}},\dots,\lambda_{q + 1}^{\mathbf{S}}$  are pairwise distinct, and that  $w_{\mathbf{S}}$  is p.s.d. (Ex. b). Then:
$$
\max  _ {i = 1, \dots , q} \min  _ {s \in \{1, - 1 \}} \left\| s \sqrt {n} u _ {i} ^ {W} - \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} + \left| \lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}} \right|\rightarrow 0 \tag {19}
$$
Proof. Denote by  $\mathcal{H}$  the RKHS associated with  $w_{\mathbf{S}}$ , and by  $T_{\mathcal{H}}: \mathcal{H} \to \mathcal{H}$  the kernel integral operator. Then, it is known [42] that the spectrum of  $\mathbf{S}$  and  $T_{\mathcal{H}}$  are the same (up to 0's), and the eigenfunctions (normalized in  $\mathcal{H}$ )  $v_{i}$  of  $T_{\mathcal{H}}$  corresponding to positive eigenvalues satisfy:
$$
v _ {i} ^ {\mathbf {S}} (x) = \left\{ \begin{array}{l l} \sqrt {\lambda_ {i} ^ {\mathbf {S}}} u _ {i} ^ {\mathbf {S}} (x) & \text {f o r} x \in \operatorname {s u p p} (P) \\ \frac {1}{\sqrt {\lambda_ {i} ^ {\mathbf {S}}}} \int w _ {\mathbf {S}} (x, y) u _ {i} ^ {\mathbf {S}} (y) d P (y) & \text {e l s e} \end{array} \right. \tag {20}
$$
Note that  $\lambda_1^{\mathbf{S}} > \ldots >\lambda_q^{\mathbf{S}} > \lambda_{q + 1}^{\mathbf{S}}\geqslant 0$  by hypothesis. Following [42], we know that
$$
\sup  _ {i} \left| \lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}} \right|\rightarrow 0 \tag {21}
$$
in probability, and that in particular, with probability going to one  $\lambda_i^W > 0$  for all  $i = 1, \ldots, q$ . Assuming this is satisfied for all  $i$ , we denote by  $v_i^W = \frac{1}{\sqrt{n\lambda_i^W}} \sum_j w_{\mathbf{S}}(\cdot, x_j) u_{i,j}^W \in \mathcal{H}$ . Then, using the fact that the eigenvalues of  $\mathbf{S}$  have single multiplicities, the proof of Theorem 12 in [42] tells us that for all  $m = 1, \ldots, q$ ,
$$
\sum_ {j = 1} ^ {m} \sum_ {i \geqslant m + 1} \left\langle v _ {i} ^ {\mathbf {S}}, v _ {j} ^ {W} \right\rangle_ {\mathcal {H}} ^ {2} + \sum_ {j \geqslant m + 1} \sum_ {i = 1} ^ {m} \left\langle v _ {i} ^ {\mathbf {S}}, v _ {j} ^ {W} \right\rangle_ {\mathcal {H}} ^ {2} \rightarrow 0 \tag {22}
$$
in probability. Since these are all nonnegative quantities, all partial sums go to 0. The first part (the term for  $j = m$ ) tells us that  $\sum_{i \geqslant m + 1} \langle v_i^{\mathbf{S}}, v_m^{W} \rangle_{\mathcal{H}}^2 \to 0$ , and the second part applied at  $m - 1$  (again the term with  $j = m$ ) gives us  $\sum_{i = 1}^{m - 1} \langle v_i^{\mathbf{S}}, v_m^{W} \rangle_{\mathcal{H}}^2 \to 0$ . Hence for all  $m = 1, \ldots, q$ ,
$$
\sum_ {i \neq m} \left\langle v _ {i} ^ {\mathbf {S}}, v _ {m} ^ {W} \right\rangle_ {\mathcal {H}} ^ {2} \rightarrow 0 \tag {23}
$$
Since  $(v_{i}^{\mathbf{S}})_{i}$  and  $(v_{i}^{W})_{i}$  are orthonormal basis of  $\mathcal{H}$ , we have  $\left\| v_m^W\right\|_{\mathcal{H}}^2 = 1 = \sum_i\langle v_i^{\mathbf{S}},v_m^W\rangle_{\mathcal{H}}^2$ , and thus  $\langle v_{m}^{\mathbf{S}},v_{m}^{W}\rangle_{\mathcal{H}}^{2}\to 1$ . By the reproducing property
$$
\left\langle v _ {m} ^ {\mathbf {S}}, v _ {m} ^ {W} \right\rangle_ {\mathcal {H}} = \frac {1}{\sqrt {\lambda_ {m} ^ {W}}} \frac {1}{\sqrt {n}} \sum_ {i} u _ {m, i} ^ {W} v _ {m} ^ {\mathbf {S}} (x _ {i}) = \sqrt {\frac {\lambda_ {m} ^ {\mathbf {S}}}{\lambda_ {m} ^ {W}}} \frac {1}{\sqrt {n}} \sum_ {i} u _ {m, i} ^ {W} u _ {m} ^ {\mathbf {S}} (x _ {i})
$$
By the convergence of  $\lambda_{m}^{W}$  we obtain that  $\left(\frac{1}{\sqrt{n}}\left\langle u_i^W,\iota_Xu_i^{\mathbf{S}}\right\rangle\right)^2\to 1$  in probability, and choosing the sign of  $u_{i}^{W}$  such that  $\langle u_i^W,\iota_Xu_i^{\mathbf{S}}\rangle \geqslant 0$ , we get  $\frac{1}{\sqrt{n}}\langle u_i^W,\iota_Xu_i^{\mathbf{S}}\rangle \to 1$ . Finally
$$
\left\| \sqrt {n} u _ {i} ^ {W} - \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} ^ {2} = 2 \left(1 - \frac {1}{\sqrt {n}} \left\langle u _ {i} ^ {W}, \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\rangle\right) + \left\| \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} ^ {2} - 1
$$
by the LLN,  $\left\| \iota_X u_i^{\mathbf{S}}\right\|_{\mathrm{MSE}}^2\to \left\| u_i^{\mathbf{S}}\right\|_{L^2}^2 = 1$  in probability, which concludes the proof.
By combining Theorem 5 with Lemma 6, we conclude the proof in the p.s.d. case.
B.2.2 SBM
Variants of the following result appear under (sometimes significantly) different formulations in the literature.
Theorem 6. Suppose that Assumption 2 holds, that  $\lambda_1^{\mathbf{S}},\ldots ,\lambda_{q + 1}^{\mathbf{S}}$  are pairwise distinct, and that  $\mathcal{X}$  is finite (Ex. a). Then:
$$
\max  _ {i = 1, \dots , q} \min  _ {s \in \{1, - 1 \}} \left\| s \sqrt {n} u _ {i} ^ {W} - \iota_ {X} u _ {i} ^ {\mathbf {S}} \right\| _ {\mathrm {M S E}} + \left| \lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}} \right| \xrightarrow [ n \to \infty ]{\mathcal {P}} 0. \tag {24}
$$
Proof. In the SBM case, functions are represented by vectors of size  $K$ . The operator  $\mathbf{S}$  acts as:  $\mathbf{S}f = C\mathrm{diag}(P_k)f$ , where  $C \stackrel{\mathrm{def.}}{=} [w_{\mathbf{S}}(k,\ell)]_{k\ell}$ . Therefore  $C\mathrm{diag}(P_k)u_i^{\mathbf{S}} = \lambda_i^{\mathbf{S}}u_i^{\mathbf{S}}$ , for  $i = 1,\dots ,K$ . Note that  $u_i^{\mathbf{S}}$  is orthonormal in  $L^2 (P)$ , that is,  $(u_i^{\mathbf{S}})^\top \mathrm{diag}(P_k)u_i^{\mathbf{S}} = 1$  and  $(u_i^{\mathbf{S}})^\top \mathrm{diag}(P_k)u_j^{\mathbf{S}} = 0$ . In particular,  $(\lambda_i^{\mathbf{S}},\mathrm{diag}(\sqrt{P_k})u_i^{\mathbf{S}})$  is an eigenvalue/eigenvector pair of the symmetric matrix  $C_P \stackrel{\mathrm{def.}}{=} \mathrm{diag}(\sqrt{P_k})C\mathrm{diag}(\sqrt{P_k})$ . Denoting by  $u_i^P \stackrel{\mathrm{def.}}{=} \mathrm{diag}(\sqrt{P_k})u_i^{\mathbf{S}}$  we have  $C_P = \sum_{i=1}^{K}\lambda_i^{\mathbf{S}}u_i^P(u_i^P)^\top$ .
Since the space  $\mathcal{X}$  is finite, each  $x_{i}$  is equal to a community label  $1\leqslant k_{i}\leqslant K$  . Define  $\Theta \in \{0,1 / \sqrt{n}\}^{n\times K}$  the community matrix, as:
$$
\left\{ \begin{array}{l l} \Theta_ {i, k _ {i}} = \frac {1}{\sqrt {n}} & \text {f o r a l l} 1 \leqslant i \leqslant n \\ \Theta_ {i, j} = 0 & \text {o t h e r w i s e} \end{array} \right. \tag {25}
$$
Then the Gram matrix is
$$
W = \Theta C \Theta^ {\top} \tag {26}
$$
Also note that  $\Theta^{\top}\Theta = \mathrm{diag}(\hat{P})$ , where  $\hat{P}_k = \frac{1}{n}\sum_i 1_{k_i = k} \to P_k$  almost surely by the LLN. Note that, when interpreting  $u_i^{\mathbf{S}}$  as a function on  $\mathcal{X} = \{1,\dots ,K\}$ , we have  $\sqrt{n}\Theta u_i^{\mathbf{S}} = \iota_X u_i^{\mathbf{S}}$ .
Defining  $\Theta_P = \Theta \mathrm{diag}(1 / \sqrt{P_k})$ , we have
$$
W = \Theta_ {P} C _ {P} \Theta_ {P} ^ {\top} = \sum_ {i} \lambda_ {i} ^ {\mathbf {S}} (\Theta_ {P} u _ {i} ^ {P}) (\Theta_ {P} u _ {i} ^ {P}) ^ {\top} = \sum_ {i} \lambda_ {i} ^ {\mathbf {S}} v _ {i} v _ {i} ^ {\top}
$$
where  $v_{i} = \Theta_{P}u_{i}^{P} = \Theta u_{i}^{\mathbf{S}}$  satisfies
$$
\left\| v _ {i} \right\| ^ {2} = \left(u _ {i} ^ {\mathbf {S}}\right) ^ {\top} \operatorname {d i a g} \left(\hat {P} _ {k}\right) u _ {i} ^ {\mathbf {S}} \rightarrow 1, \quad v _ {i} ^ {\top} v _ {j} = \left(u _ {i} ^ {\mathbf {S}}\right) ^ {\top} \operatorname {d i a g} \left(\hat {P} _ {k}\right) u _ {j} ^ {\mathbf {S}} \rightarrow 0 \tag {27}
$$
in probability. Hence the  $v_{i}$  are almost orthonormal but not exactly. Define their orthonormalization:
$$
\tilde {u} _ {1} = v _ {1}, \forall i = 2, \dots , K, \tilde {u} _ {i} = v _ {i} - \sum_ {j = 1} ^ {i - 1} \left(v _ {i} ^ {\top} u _ {j}\right) u _ {j} \text {a n d} u _ {i} = \frac {\tilde {u} _ {i}}{\| \tilde {u} _ {i} \|} \tag {28}
$$
such that  $u_{1}, \ldots, u_{K} \in \mathbb{R}^{n}$  are orthonormal and  $u_{i} \in \operatorname{Span}(v_{1}, \ldots, v_{i})$ . We define  $G = \sum_{i=1}^{K} \lambda_{i}^{\mathbf{S}} u_{i} u_{i}^{\top}$ , whose eigenvalues are the  $\lambda_{i}^{\mathbf{S}}$  and eigenvectors  $u_{i}$ . By the properties of  $v_{i}$  we have  $\sup_{1 \leq i \leq n} \| v_{i} - u_{i} \| \to 0$ , so  $\| W - G \| \to 0$ . Hence by Kato's inequality we have  $\sup_{1 \leq i \leq n} \left| \lambda_{i}^{W} - \lambda_{i}^{\mathbf{S}} \right| \to 0$ , and since the  $\lambda_{1}^{\mathbf{S}}, \ldots, \lambda_{q+1}^{\mathbf{S}}$  have unique multiplicity, again by Davis-Kahan theorem for all  $i = 1, \ldots, q$  we have  $\min_{s \in \{-1,1\}} \| s u_{i}^{W} - u_{i} \| \to 0$ , and by consequence  $\min_{s \in \{-1,1\}} \| s \sqrt{n} u_{i}^{W} - \sqrt{n} v_{i} \|_{\mathrm{MSE}} \to 0$ . We conclude by recalling that  $\sqrt{n} v_{i} = \sqrt{n} \Theta u_{i}^{\mathbf{S}} = \iota_{X} u_{i}^{\mathbf{S}}$ .
As before, we conclude by combining Theorem 6 with Lemma 6.
B.3 Distance-based encodings
Again, we start with an intermediate result, assuming some convergence in Frobenius norm that we will then show for our cases of interest.
Lemma 7 (Intermediate result for Distance-based encodings). Suppose that Assumption 2 holds, and that:
$$
\forall \varepsilon > 0, \exists \gamma_ {2}, \mathbb {P} \left(\left\| S _ {\gamma_ {2}} - W \right\| _ {\mathrm {F}} \geqslant \varepsilon\right)\rightarrow 0 \tag {29}
$$
Then the result of Theorem 3 holds.
Proof. Let  $f \in \mathcal{C}_{\mathrm{Lip}}([0,1]^q, \mathbb{R}^d)$  and  $\varepsilon > 0$ . By the universality theorem [40], let  $f_{\gamma_1}^{\mathrm{MLP}}$  such that  $\left\| f_{\gamma_1}^{\mathrm{MLP}} - f \right\|_{\infty} \leqslant \varepsilon$  on  $[0,1]^q$ . Since it is an MLP,  $f_{\gamma_1}^{\mathrm{MLP}}$  is uniformly Lipschitz on  $\mathbb{R}^q$ , call  $L$  its Lipschitz constant. Then, let  $f_{\gamma_2}^{\mathrm{MLP}}$  such that
$$
\mathbb {P} \left(\left\| S _ {\lambda_ {2}} - W \right\| _ {\mathrm {F}} \geqslant \varepsilon / L\right)\rightarrow 0 \tag {30}
$$
For convenience, define  $M_j^{S_{\gamma_2}} = [S_{\gamma_2}e_j,\dots,(S_{\gamma_2})^q e_j] \in \mathbb{R}^{n\times q}$ ,  $M_j^W = [We_j,\dots,W^q e_j]$ ,  $J(x,y) = [\mathbf{S}\delta_y(x),\dots,\mathbf{S}^q\delta_y(x)] \in [0,1]^q$ .
$$
\begin{array}{l} \left\| \mathrm {P E} - \iota_ {X} \int f (J (\cdot , x) d P (x) \right\| _ {\mathrm {M S E}} \\ \leqslant \left\| \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n \cdot M _ {j} ^ {S _ {\gamma_ {2}}}\right) - f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n \cdot M _ {j} ^ {W}\right) \right\| _ {\text {M S E}} \\ + \left\| \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (n \cdot M _ {j} ^ {W}) - \iota_ {X} \int f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x)) d P (x) \right\| _ {\text {M S E}} \\ + \left\| \iota_ {X} \int f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x)) d P (x) - \iota_ {X} \int f (J (\cdot , x)) d P (x) \right\| _ {\text {M S E}} \tag {31} \\ \end{array}
$$
The third term in (31) is bounded by  $\left\| f_{\gamma_1}^{\mathrm{MLP}} - f\right\|_{\infty}\leqslant \varepsilon$
The second term in (31) is
$$
\begin{array}{l} \left\| \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (n \cdot M _ {j} ^ {W}) - \iota_ {X} \int f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , y)) d P (x) \right\| _ {\text {M S E}} \\ \leqslant \left\| \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (n \cdot M _ {j} ^ {W}) - \iota_ {X} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x _ {j})) \right\| _ {\text {M S E}} \\ + \left\| \iota_ {X} \left(\frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x _ {j})) - \int f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x)) d P (x)\right) \right\| _ {\text {M S E}} \\ \leqslant L \sum_ {\ell = 1} ^ {q} \sup  _ {x, x ^ {\prime}} \left| \frac {1}{n ^ {\ell - 1}} \sum_ {i _ {1}, \dots , i _ {\ell - 1}} w _ {\mathbf {S}} \left(x, x _ {i _ {1}}\right) w _ {\mathbf {S}} \left(x _ {i _ {1}}, x _ {i _ {2}}\right) \dots w _ {\mathbf {S}} \left(x _ {i _ {\ell - 1}}, x ^ {\prime}\right) - \mathbf {S} ^ {\ell} \delta_ {x ^ {\prime}} (x) \right| \\ + \left\| \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x _ {j})) - \int f _ {\gamma_ {1}} ^ {\mathrm {M L P}} (J (\cdot , x)) d P (x) \right\| _ {\infty} \\ \end{array}
$$
where we have used the Lipschitz property of  $f_{\gamma_1}^{\mathrm{MLP}}$  and a supremum over  $x_i, x_j$  for the first term. The first term converges to 0 in probability by Lemma 9, while the second goes to 0, using the boundedness of  $f_{\gamma_1}^{\mathrm{MLP}}$  on  $[0,1]^q$ , by Lemma 8.
Finally, using Schwartz inequality, the first term in (31) is bounded by
$$
\begin{array}{l} \left\| n ^ {- 1} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n M _ {j} ^ {S _ {\gamma_ {2}}}\right) - f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n M _ {j} ^ {W}\right) \right\| _ {\text {M S E}} \\ \leqslant \frac {1}{\sqrt {n}} \sqrt {\sum_ {j} \left\| f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n M _ {j} ^ {S _ {\gamma_ {2}}}\right) - f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n M _ {j} ^ {W}\right) \right\| _ {\mathrm {M S E}} ^ {2}} \\ = \frac {1}{n} \sqrt {\sum_ {i j} \left\| f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n \left(M _ {j} ^ {S _ {\gamma_ {2}}}\right) _ {i , :}\right) - f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n \left(M _ {j} ^ {W}\right) _ {i , :}\right) \right\| ^ {2}} \\ \leqslant L \sqrt {\sum_ {i j} \left\| \left(M _ {j} ^ {S _ {\gamma_ {2}}}\right) _ {i , :} - \left(M _ {j} ^ {W}\right) _ {i , :} \right\| ^ {2}} \\ \leqslant L \sum_ {\ell} \left\| S _ {\gamma_ {2}} ^ {\ell} - W ^ {\ell} \right\| _ {\mathrm {F}} \leqslant L \sum_ {\ell} \left(\sum_ {p = 1} ^ {\ell - 1} \| S _ {\gamma_ {2}} \| _ {\mathrm {F}} ^ {p} \| W \| _ {\mathrm {F}} ^ {\ell - 1 - p}\right) \| S _ {\gamma_ {2}} - W \| _ {\mathrm {F}} \lesssim q ^ {2} \varepsilon \\ \end{array}
$$
with probability going to 1, using (30) and the fact that  $\| W\|_{\mathrm{F}}$  is bounded, and thus  $\| S_{\gamma_2}\|_{\mathrm{F}}$  as well with probability going to 1. Gathering everything,  $f\in \mathcal{F}_{\mathrm{PE}}$ , which concludes the proof.
We then prove this property for both p.s.d. kernel and SBM. The following Theorem is similar to Theorem 4, but under Assumption 2, which is true for ex. 1 and 2.
Theorem 7 (Theorem 4 reformulated). Suppose that Assumption 2 holds. For both p.s.d. kernel  $(Ex.b)$  or finite  $\mathcal{X}$ $(Ex.a)$ , for all  $\varepsilon >0$ , there is an MLP filter  $S_{\gamma} = h_{f^{\mathrm{MLP}}}^{(S)}(S)$  such that
$$
\forall \varepsilon > 0, \exists \gamma , \mathbb {P} \left(\left\| S _ {\gamma} - W \right\| _ {\mathrm {F}} \geqslant \varepsilon\right)\rightarrow 0 \tag {32}
$$
Proof. Note that  $\mathbf{S}$  is trace-class (both if  $w$  is p.s.d. or in the SBM case), such that  $\sum_{i}|\lambda_{i}^{\mathbf{S}}| < \infty$ . In particular,  $|\lambda_i^{\mathbf{S}}| = o(1 / \sqrt{i})$ . In the p.s.d. case, all  $\lambda_i^{\mathbf{S}}$  are nonnegative. We define  $\lambda_{\varepsilon} > 0$  and the support  $T = \{i\mid |\lambda_i^{\mathbf{S}}|\geqslant \lambda_\varepsilon \}$
i)  $\sum_{i\in T^c}(\lambda_i^{\mathbf{S}})^2\leqslant \varepsilon /2$  , which can be satisfied since S is trace class and therefore Hilbert-Schmidt
ii)  $\sqrt{2|T|}\sup_{i\in T^c}\left|\lambda_i^{\mathbf{S}}\right|\leqslant \varepsilon /4$  , which can be satisfied since  $|\lambda_i^{\mathbf{S}}| = o(1 / \sqrt{i})$
iii)  $\inf_{i\in T,j\in T^c}\left|\lambda_i^{\mathbf{S}} - \lambda_j^{\mathbf{S}}\right| > 0$  , which can be satisfied by choosing  $\lambda_{\varepsilon}$  in a gap in the spectrum of  $\mathbf{S}$ , since all eigenvalues but 0 have finite multiplicities.
We will first start by approximating  $W$  with an ideal filtered matrix  $\hat{S} = \sum_{i\in T}\lambda_i^S u_i^S (u_i^S)^\top$ . We define  $G = \sum_{i\in T}\lambda_i^W u_i^W (u_i^W)^\top$ . We decompose
$$
\left\| \hat {S} - W \right\| _ {\mathrm {F}} \leqslant \left\| \hat {S} - G \right\| _ {\mathrm {F}} + \| G - W \| _ {\mathrm {F}}
$$
For the first term, since this matrix is of rank at most  $2|T|$ , we have
$$
\left\| \hat {S} - G \right\| _ {\mathrm {F}} \leqslant \sqrt {2 | T |} \left\| \hat {S} - G \right\|
$$
We further decompose
$$
\left\| \hat {S} - G \right\| \leqslant \left\| \hat {S} - S \right\| + \| S - W \| + \| W - G \|
$$
The second term is bounded by Theorem 9, we have  $\| S - W \| \to 0$  in probability.
The third term is bounded by Kato inequality
$$
\sup  _ {i \in T ^ {c}} \lambda_ {i} ^ {W} \leqslant \sup  _ {i \in T ^ {c}} \lambda_ {i} ^ {\mathbf {S}} + \sup  _ {i} \left| \lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}} \right|
$$
the last term goes to 0 in probability.
The first term is bounded by  $\sup_{i\in T^c}\lambda_i^S$  , and by Kato's inequality
$$
\sup  _ {i \in T ^ {c}} \lambda_ {i} ^ {S} \leqslant \sup  _ {i \in T ^ {c}} \lambda_ {i} ^ {W} + \| S - W \|
$$
which is a combination of both previous cases.
At the end of the day, with probability going to 1,
$$
\left\| \hat {S} - G \right\| \leqslant \sqrt {2 | T |} 2 \sup  _ {i \in T ^ {c}} \lambda_ {i} ^ {\mathbf {S}} + o (1) \leqslant \varepsilon / 2 + o (1)
$$
$$
\| G - W \| _ {\mathrm {F}} ^ {2} = \sum_ {i \in T ^ {c}} \left(\lambda_ {i} ^ {W}\right) ^ {2} \leqslant 2 \left(\sum_ {i \in T ^ {c}} \left(\lambda_ {i} ^ {\mathbf {S}}\right) ^ {2} + \sum_ {i} \left(\lambda_ {i} ^ {W} - \lambda_ {i} ^ {\mathbf {S}}\right) ^ {2}\right)
$$