SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $C = C_1 + C_2 + C_3$ ,  $C_1 = \left(1 + \| g\|_{L^1 (\mathbb{R}^{d + 1})}\right)\sum_{m = 1}^{q}\| f\|_{W_2^\beta (\mathbb{R}^{d + 1})}\left\| \frac{\partial f_m}{\partial x_j}\right\|_{L^\infty (\mathbb{R}^{d + 1})}$ ,  $C_2 =$
$$
\sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \text {a n d} C _ {3} = \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}.
$$
$$
\begin{array}{l} T r (M _ {\varepsilon}) = c _ {\eta} 2 ^ {2 \beta} (1 + (\frac {v}{3}) ^ {2 \beta} e ^ {\frac {8 9}{\eta_ {0} v ^ {2}}}) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ = c _ {\eta} 2 ^ {2 \beta} \left(1 + 3 ^ {- 2 \beta} \left(\frac {\varepsilon}{9 6 \pi^ {2} 2 ^ {\beta - 2} C}\right) ^ {\frac {2 \beta}{\beta - 2}} e ^ {\frac {8 9}{\eta_ {0} v ^ {2}}}\right) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ = c _ {\eta} 2 ^ {2 \beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \left(1 + 3 ^ {- 2 \beta} \left(\frac {\varepsilon}{\widetilde {C}}\right) ^ {\frac {2 \beta}{\beta - 2}} e ^ {\frac {8 9}{\eta_ {0} \left(\frac {\varepsilon}{C}\right) ^ {\frac {2}{\beta - 2}}}}\right) \\ = \hat {C} | \eta | ^ {1 / 2} \left(1 + \varepsilon^ {\frac {2 \beta}{\beta - 2}} e ^ {\frac {\tilde {C} ^ {\prime}}{\eta_ {0} \varepsilon^ {\frac {2}{\beta - 2}}}}\right) \leq \hat {C} | \eta | ^ {1 / 2} \left(1 + \varepsilon^ {\frac {2 \beta}{\beta - 2}} \exp \left(\frac {\tilde {C} ^ {\prime}}{\eta_ {0}} \varepsilon^ {- \frac {2}{\beta - 2}}\right)\right) \\ \end{array}
$$
where  $\tilde{C} = 96\pi^2 2^{\beta -2}C$ $\tilde{C}^{\prime} = 89\tilde{C}^{\frac{2}{\beta - 2}}$  , and  $\hat{C} = \pi^{-d / 2}2^{2\beta}\sum_{i = 1}^{q}\| f_i\|_{W_2^\beta (\mathbb{R}^{d + 1})}^2\max \left(1,\frac{3^{-2\beta}}{\tilde{C}^{\frac{2\beta}{\beta - 2}}}\right)$
A.3 Compression Result
Let  $\mathcal{X} \subset \mathbb{R}^d$  be an open set with Lipschitz boundary, contained in the hypercube  $[-R,R]^d$  with  $R > 0$ . Given  $\tilde{x}_1,\ldots ,\tilde{x}_m \in \mathcal{X}$  be  $m$  points in  $[-R,R]^d$ . Define the base point matrix  $\tilde{X} \in \mathbb{R}^{m\times d}$  to be the matrix whose  $j$ -th row is the point  $\tilde{x}_j$ . The following result holds. We introduce the so-called fill distance [Wen04]
$$
h = \max  _ {x \in [ - R, R ] ^ {d}} \min  _ {z \in \tilde {X}} \| x - z \|. \tag {34}
$$
Now, we state some useful results below.
A.4 Useful Results
Lemma 4 ([RC21], Lemma C.1). Let  $T = (-R, R)^d$  and  $\eta \in \mathbb{R}_{++}^d$ . Let  $u \in \mathcal{H}_\eta$  satisfying  $u(\tilde{x}_1) = \dots = u(\tilde{x}_m) = 0$ . There exists three constants  $c, C, C'$  depending only on  $d$  (and in particular, independent from  $R, \eta, u, \tilde{x}_1, \ldots, \tilde{x}_m$ ), such that, when  $h \leq \sigma / C'$ , then,
$$
\left\| u \right\| _ {L ^ {\infty} (T)} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \left\| u \right\| _ {\mathcal {H} _ {\eta}},
$$
with  $q_{\eta} = \operatorname*{det}\left(\frac{1}{\eta_{+}}\operatorname{diag}(\eta)\right)^{-1 / 4}$  and  $\sigma = \min(R, \frac{1}{\sqrt{\eta_{+}}})$  and  $\eta_{+} = \max_{i=1,\ldots,d} \eta_{i}$ .
Lemma 5 (Norm of function's derivative with scattered zeros). Let  $T = (-R, R)^d$  and  $\eta \in \mathbb{R}_{++}^d$ . Let  $u \in \mathcal{H}_{\eta}$  satisfying  $u(\tilde{x}_1) = \dots = u(\tilde{x}_m) = 0$ . There exists three constants  $c, C, C'$  depending only on  $d$  (and in particular, independent from  $R, \eta, u, \tilde{x}_1, \ldots, \tilde{x}_m$ ), such that, when  $h \leq \sigma / C'$ , then,
$$
\| D ^ {j} u \| _ {L ^ {\infty} (T)} \leq G e ^ {- c \sigma / h} \log (c \sigma / h) \| u \| _ {\mathcal {H} _ {\eta}},
$$
with  $G$  be some positive constant and  $\sigma = \min \left(R, \frac{1}{\sqrt{\eta_{+}}}\right)$  and  $\eta_{+} = \max_{i=1,\ldots,d} \eta_{i}$ .
Proof. Here, we utilize Gagliardo-Nirenberg inequality [Nir11]. We have, for  $0 < k < \infty$ ,  $0 \leq j < \infty$  and  $\vartheta(j, m) = \frac{j}{k} + \frac{d}{2k} < 1$ , then
$$
\left\| D ^ {j} u \right\| _ {L ^ {\infty} (T)} \leq G \| u \| _ {L ^ {2} (T)} ^ {1 - \vartheta (j, k)} \| D ^ {k} u \| _ {L ^ {2} (T)} ^ {\vartheta (j, k)}, \tag {35}
$$
where  $G = \frac{1}{2^{d / 2}\pi^{d / 4 - 1 / 2}\sqrt{\Gamma(d / 2)(1 - \vartheta)^{1 - \vartheta}\vartheta^{\vartheta}k\sin\pi\vartheta}}$
Considering the case for  $j = 1$  and  $j = 2$ , we have,
$$
\left\| D ^ {1} u \right\| _ {L ^ {\infty} (T)} \leq G _ {1} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 2}{2 k}} \| D ^ {k} u \| _ {L ^ {2} (T)} ^ {\frac {d + 2}{2 k}},
$$
$$
\| D ^ {2} u \| _ {L ^ {\infty} (T)} \leq G _ {2} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 4}{2 k}} \| D ^ {k} u \| _ {L ^ {2} (T)} ^ {\frac {d + 4}{2 k}}
$$
where  $G_{1} = \frac{1}{2^{d / 2}\pi^{d / 4 - 1 / 2}\sqrt{\Gamma(d / 2)(1 - \frac{d + 2}{2k})^{1 - \frac{d + 2}{2k}}\left(\frac{d + 2}{2k}\right)^{\frac{d + 2}{2k}}k\sin\pi\left(\frac{d + 2}{2k}\right)}}$  and
$G_{2} = \frac{1}{2^{d / 2}\pi^{d / 4 - 1 / 2}\sqrt{\Gamma(d / 2)(1 - \frac{d + 4}{2k})^{1 - \frac{d + 4}{2k}}\left(\frac{d + 4}{2m}\right)^{\frac{d + 4}{2k}}k\sin\pi\left(\frac{d + 4}{2k}\right)}}.$  We here have assumed that  $k > \frac{d}{2} + 2$
From the defeinition, we have
$$
\| D ^ {1} u \| _ {L ^ {\infty} (T)} \leq G _ {1} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 2}{2 k}} \| D ^ {k} u \| _ {L ^ {2} (T)} ^ {\frac {d + 2}{2 k}},
$$
$$
\left\| D ^ {2} u \right\| _ {L ^ {\infty} (T)} \leq G _ {2} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 4}{2 k}} \left\| D ^ {k} u \right\| _ {L ^ {2} (T)} ^ {\frac {d + 4}{2 k}}.
$$
From the definition,
$$
\left\| D ^ {1} u \right\| _ {L ^ {\infty} (T)} \leq G _ {1} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 2}{2 k}} \| u \| _ {W _ {2} ^ {k} (T)} ^ {\frac {d + 2}{2 k}},
$$
$$
\| D ^ {2} u \| _ {L ^ {\infty} (T)} \leq G _ {2} \| u \| _ {L ^ {2} (T)} ^ {1 - \frac {d + 4}{2 k}} \| u \| _ {W _ {2} ^ {k} (T)} ^ {\frac {d + 4}{2 k}}.
$$
Now by Theorem 4.3 of [RZ10], we have for some constants  $B_{d}$  and  $B_{d}^{\prime}$  depending only on  $\mathbf{d}$ ,
$$
\| u \| _ {L ^ {2} (T)} \leq \frac {B _ {d} ^ {k} k ^ {k}}{k !} h ^ {k} \| u \| _ {W _ {2} ^ {k} (T)},
$$
when  $kh \leq \frac{R}{B_d'}$ . Hence, we have,
$$
\left\| D ^ {1} u \right\| _ {L ^ {\infty} (T)} \leq G _ {1} \frac {B _ {d} ^ {k} k ^ {k}}{k !} h ^ {k} \| u \| _ {W _ {2} ^ {k} (T)},
$$
$$
\left\| D ^ {2} u \right\| _ {L ^ {\infty} (T)} \leq G _ {2} \frac {B _ {d} ^ {k} k ^ {k}}{k !} h ^ {k} \| u \| _ {W _ {2} ^ {k} (T)}.
$$
From equation C.9 in [RC21, Proof of Lemma C.1], we have
$$
\| u \| _ {W _ {2} ^ {k} (T)} \leq \| u \| _ {\mathcal {H} _ {\eta}} c _ {\eta} ^ {1 / 2} (4 \eta_ {+}) ^ {k / 2} \sqrt {k !}.
$$
$$
\left\| D ^ {1} u \right\| _ {L ^ {\infty} (T)} \leq G _ {1} \frac {B _ {d} ^ {k} h ^ {k}}{k !} h ^ {k} \| u \| _ {\mathcal {H} _ {\eta}} c _ {\eta} ^ {1 / 2} \left(4 \eta_ {+}\right) ^ {k / 2} \sqrt {k !} = G _ {1} \frac {\left(2 \sqrt {\eta_ {+}} B _ {d} h k\right) ^ {k}}{\sqrt {k !}} \| u \| _ {\mathcal {H} _ {\eta}},
$$
$$
\| D ^ {2} u \| _ {L ^ {\infty} (T)} \leq G _ {2} \frac {B _ {d} ^ {k} k ^ {k}}{k !} h ^ {k} \| u \| _ {\mathcal {H} _ {\eta}} c _ {\eta} ^ {1 / 2} (4 \eta_ {+}) ^ {k / 2} \sqrt {k !} = G _ {2} \frac {(2 \sqrt {\eta_ {+}} B _ {d} h k) ^ {k}}{\sqrt {k !}} \| u \| _ {\mathcal {H} _ {\eta}}.
$$
Now, we make similar choice of the parameter  $h$  and  $k$  as in [RC21]. If we choose,  $C_3 = \frac{1}{\max(2B_d,B_d')},$ $h \leq \frac{C_3}{d + 4}\min (R,1 / \sqrt{\eta_+}), k = \lfloor s\rfloor$  and  $s = \frac{C_3}{h}\min (R,1 / \sqrt{\eta_+})$ , then we have  $hk \leq \frac{R}{B_d'}$  and  $2\sqrt{\eta_{+}} B_{d}hk \leq 1$ . Moreover,  $\frac{1}{\sqrt{k!}} \leq e^{-\frac{k}{2}\log \frac{k}{2}}$ . If we also assume that  $G = \max (G_1,G_2)$ , then for  $j = \{1,2\}$
$$
\| D ^ {j} u \| _ {L ^ {\infty} (T)} \leq G \frac {1}{\sqrt {k !}} \| u \| _ {\mathcal {H} _ {\eta}} \leq G e ^ {- s / 4} \log (s / 4) \| u \| _ {\mathcal {H} _ {\eta}}.
$$
The final result is obtained by writing  $s / 4 = c\sigma /h$  with  $\sigma = \min (R,1 / \sqrt{\eta_{+}})$  and  $c = C_3 / 4$ .
Lemma 6 (Lemma 3, page 28 [PRDVR20]). Let  $\mathcal{X} \subset \mathbb{R}^d$  with non-zero volume. Let  $\mathcal{H}$  be a reproducing kernel Hilbert space on  $\mathcal{X}$ , associated to a continuous uniformly bounded feature map  $\phi: \mathcal{X} \to \mathcal{H}$ . Let  $A: \mathcal{H} \to \mathcal{H}$  be a bounded linear operator. Then,
$$
\sup  _ {x \in \mathcal {X}} \| A \phi (x) \| _ {\mathcal {H}} \leq \sup  _ {\| f \| _ {\mathcal {H}} \leq 1} \| A ^ {*} f \| _ {C (\mathcal {X})}.
$$
In particular, if  $\mathcal{X} \subset \mathbb{R}^d$  is a non-empty open set, then  $\sup_{\|f\|_{\mathcal{H}} \leq 1} \|A\phi(x)\|_{\mathcal{H}} \leq \sup_{\|f\|_{\mathcal{H}} \leq 1} \|A^{*}f\|_{L^{\infty}(\mathcal{X})}$ .
Theorem 6 (Theorem C.3 [RC21]). Let  $R > 0, \eta \in \mathbb{R}_{++}^{d}, m \in \mathbb{N}$ . Let  $\mathcal{X} \subseteq T = (-R,R)^{d}$  be a non-empty open set and let  $\tilde{x}_1, \ldots, \tilde{x}_m$  be a set of distinct points. Let  $h > 0$  be the fill distance associated to the points w.r.t  $T$  (defined in eq. (34)). Let  $\tilde{P}: \mathcal{H}_{\eta} \to \mathcal{H}_{\eta}$  be the associated projection operator (see definition in eq. (38)). There exists three constants  $c, C, C'$ , such that, when  $h \leq \sigma / C'$ ,
$$
\sup _ {x \in \mathfrak {X}} \| (I - \tilde {P}) \phi_ {\eta} (x) \| _ {\mathcal {H} _ {\eta}} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}.
$$
Here  $q_{\eta} = \operatorname*{det}\left(\frac{1}{\eta_{+}}\operatorname {diag}(\eta)\right)^{-1 / 4}$  and  $\sigma = \min (R,\frac{1}{\sqrt{\eta_+}})$ ,  $\eta_{+} = \max_{i}\eta_{i}$ . The constants  $c, C', C''$  depend only on  $d$  and, in particular, are independent from  $R, \eta, \tilde{x}_1, \ldots, \tilde{x}_m$ .
Similar to the results in theorem 6, we will obtain results for derivatives.
Theorem 7. Let  $R > 0, \eta \in \mathbb{R}_{++}^{d}, m \in \mathbb{N}$ . Let  $\mathcal{X} \subseteq T = (-R, R)^{d}$  be a non-empty open set and let  $\tilde{x}_1, \ldots, \tilde{x}_m$  be a set of distinct points. Let  $h > 0$  be the fill distance associated to the points w.r.t  $T$  (defined in eq. (34)). Let  $\tilde{P}: \mathcal{H}_{\eta} \to \mathcal{H}_{\eta}$  be the associated projection operator (see definition in eq. (38)). There exists three constants  $c, C, C'$ , such that, when  $h \leq \sigma / C'$ ,
$$
\sup  _ {x \in \mathcal {X}} \| (I - \tilde {P}) D ^ {j} \phi_ {\eta} (x) \| _ {\mathcal {H} _ {\eta}} \leq C _ {1} q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}},
$$
for some positive constant  $C_1$ . Here  $q_{\eta} = \operatorname*{det}\left(\frac{1}{\eta_{+}}\operatorname{diag}(\eta)\right)^{-1 / 4}$  and  $\sigma = \min(R, \frac{1}{\sqrt{\eta_{+}}})$ ,  $\eta_{+} = \max_{i}\eta_{i}$ . The constants  $c, C', C''$  depend only on  $d$  and, in particular, are independent from  $R, \eta, \tilde{x}_1, \ldots, \tilde{x}_m$ .
Proof. If the function  $f$  lies in the RKHS, then using the property of the projection operator  $\tilde{P}$ , we have
$$
D ^ {j} (\tilde {P} f) (x _ {i}) = D ^ {j} \langle \tilde {P} f, \phi (x _ {i}) \rangle = D ^ {j} \langle f, \tilde {P} \phi (x _ {i}) \rangle = D ^ {j} \langle f, \phi (x _ {i}) \rangle = \langle f, D ^ {j} \phi (x _ {i}) = (D ^ {j} f) (x _ {i}).
$$
Hence,  $D^{j}f - \tilde{P}D^{j}f(\tilde{x}_{i}) = 0$  for all  $i \in \{1,\dots ,m\}$ . By lemma 4, we know that there exist three constants  $c,C,C'$  depending only on  $d$  such that when  $h \leq \sigma /C'$  we have that the following holds  $\| u\|_{L^{\infty}(T)} \leq Cq_{\eta}e^{-\frac{c\sigma}{h}\log \frac{c\sigma}{h}}$ , for any  $u \in \mathcal{H}_{\eta}$  such that  $u(\tilde{x}_1) = \dots = u(\tilde{x}_m) = 0$ . Since, for any  $u \in \mathcal{H}_{\eta}$ , we have that  $D^{j}f - \tilde{P}D^{j}f$  belongs to  $\mathcal{H}_{\eta}$  and satisfies such property, we can apply lemma 4 with  $u = (I - \tilde{P})D^{j}f$ , obtaining, under the same assumption on  $h$ ,
$$
\| (I - \tilde {P}) D ^ {j} f \| _ {L ^ {\infty} (T)} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| D ^ {j} f \| _ {\mathcal {H} _ {\eta}}, \quad \forall f \in \mathcal {H} _ {\eta},
$$
where we used the fact that  $\| (I - \tilde{P})D^{j}f\|_{\mathcal{H}_{\eta}}\leq \| I - \tilde{P}\| \| D^{j}f\|_{\mathcal{H}_{\eta}}$  and  $\| I - \tilde{P}\| \leq 1$ , since  $P$  is a projection operator and so also  $I - P$  satisfies this property. Derivative reproducing properties for kernel can be looked at in [Zho08]. The final result is obtained by applying lemma 6 with  $A = I - \tilde{P}$  from which we have
$$
\begin{array}{l} \sup  _ {x \in \mathcal {X}} \| (I - \tilde {P}) D ^ {j} \phi (x) \| _ {\mathcal {H}} \leq \sup  _ {\| D ^ {j} f \| \leq 1} \| (I - \tilde {P}) D ^ {j} f \| _ {L ^ {\infty} (T)} \leq \sup  _ {\| D ^ {j} f \| \leq 1} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| D ^ {j} f \| _ {\mathcal {H} _ {\eta}} \\ = C _ {1} q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}, \\ \end{array}
$$
for some positive  $C_1$ . The same result can also be obtained by the use of Lemma 5 with a condition that  $f$  is  $d/2 + 1$  differentiable function which is a slightly weaker condition on the order of smoothness.
A.5 Compression of a PSD model for Fokker-Planck
We have  $(\tilde{x}_1,\tilde{t}_1),\tilde{x}_2,\dots (\tilde{x}_m,\tilde{t}_m)\in \mathcal{X}\times \mathcal{T}$  which is an open bounded subset with Lipschitz boundary of the cube  $[-R,R]^d\times [0,T]$ .  $\tilde{X}$  be the base point matrix whose j-rows are the points  $\tilde{x}_j$ . Let us now consider the model  $p = f(\cdot ,M,\psi)$  and  $\tilde{p} = f(\cdot ;A_m,\tilde{X},\psi)$  where  $A_{m} = K_{(X,T)(X,T)}^{-1}\tilde{Z} M\tilde{Z}^{\star}K_{(X,T)(X,T)}^{-1}$ .
$$
\tilde {Z} u = \left(\psi \left(\tilde {x} _ {1}, \tilde {t} _ {1}\right) ^ {\top} u, \dots , \psi \left(\tilde {x} _ {m}, \tilde {t} _ {m}\right) ^ {\top} u\right). \tag {36}
$$
$$
\tilde {Z} ^ {\star} \alpha = \sum_ {i = 1} ^ {m} \psi \left(\tilde {x} _ {i}, \tilde {t} _ {i}\right) \alpha_ {i}. \tag {37}
$$
Hence, for any  $A\in \mathbb{R}^{m\times m}$
$$
\tilde {Z} ^ {\star} A \tilde {Z} = \sum_ {i, j = 1} ^ {m} A _ {i, j} \psi (\tilde {x} _ {i}, \tilde {t} _ {i}) \psi (\tilde {x} _ {i}, \tilde {t} _ {i}) ^ {\top}.
$$
Let us also define associated projection operator,
$$
\tilde {P} = \tilde {Z} ^ {\star} K _ {(X, T) (X, T)} ^ {- 1} \tilde {Z}. \tag {38}
$$
Some properties associated with it,
$$
\tilde {P} ^ {2} = \tilde {P}, \tilde {P} \tilde {Z} ^ {\star} = \tilde {Z} ^ {\star}, \tilde {Z} \tilde {P} = \tilde {Z}.
$$
Theorem 8. Let  $\eta \in \mathbb{R}_{++}^{d}$  and let  $\mathsf{M} \in \mathbb{S}_{+}(\mathcal{H}_{\eta})$ . Let  $\tilde{\mathcal{X}}$  be an open bounded subset with the Lipschitz boundary of the cube  $[-R,R]^d \times [0,R], R > 0$ . Let  $\tilde{x}_1, \ldots, \tilde{x}_m \in \tilde{\mathcal{X}}$  and  $\tilde{X}$  be the base point matrix whose  $j$ -rows are the points  $\tilde{x}_j$  with  $j = 1, \ldots, m$ . Consider the model  $p = f(\cdot; \mathsf{M}, \psi)$  and the compressed model  $\tilde{p} = f(\cdot; A_m, \tilde{X}, \eta)$  with
$$
A _ {m} = K _ {\tilde {X}, \tilde {X}, \eta} ^ {- 1} \tilde {Z} \mathsf {M} \tilde {Z} ^ {*} K _ {\tilde {X}, \tilde {X}, \eta} ^ {- 1},
$$
where  $\tilde{Z}:\mathcal{H}_{\eta}\to \mathbb{R}^{m}$  is defined in eq. (36) in terms of  $\tilde{X}_m$ . Let  $h$  be the fill distance (defined in eq. (34)) associated to the points  $\tilde{x}_1,\ldots ,\tilde{x}_m$ . The there exist three constants  $c,C,C^{\prime}$  depending only on  $d$  such that, when  $h\leq \sigma /C^{\prime}$ , with  $\sigma = \min(R,1/\sqrt{\eta_+})$ ,  $\eta_{+} = \max_{i=1,\dots,d}\eta_{i}, q_{\eta} = \operatorname*{det}\left(\frac{1}{\eta_{+}}\operatorname{diag}(\eta)\right)^{-1/4}$ , then we have
$$
\begin{array}{l} \left| \frac {\partial (p (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (p (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (p (x, t) - \tilde {p} (x, t)) \right| \\ \leq \left(2 + d R _ {\mu} + 2 d R _ {\mu} + 4 d ^ {2} R _ {d}\right) \| M \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} + \left(2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}\right) \sqrt {\| M \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ + 2 C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sqrt {\| M \| \tilde {f} _ {t} (x ; M , \psi)} + (2 R _ {\mu} + 4 d R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sqrt {\| M \| \tilde {f} _ {x _ {i}} (x ; M , \psi)} \\ + 2 R _ {d} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \sqrt {\| M \| \tilde {f} _ {x _ {j} x _ {i}} (x ; M , \psi)}, \\ \end{array}
$$
$$
w h e r e \tilde {f} _ {x _ {i}} (x; M, \psi) = \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} M \frac {\partial \psi (x , t)}{\partial x _ {i}}, \tilde {u} _ {x _ {i}} (x) = \left\| (I - \tilde {P}) \frac {\partial \psi (x , t)}{\partial x _ {i}} \right\| _ {\mathcal {H} _ {\eta}}, \tilde {f} _ {x _ {j} x _ {i}} (x; M, \psi) =
$$
$$
\frac {\partial^ {2} \psi (x , t)}{\partial x _ {j} x _ {i}} ^ {\top} M \frac {\partial^ {2} \psi (x , t)}{\partial x _ {j} x _ {i}} a n d \tilde {u} _ {x _ {j} x _ {i}} (x) = \left\| (I - \tilde {P}) \frac {\partial^ {2} \psi (x , t)}{\partial x _ {j} x _ {i}} \right\| _ {\mathcal {H} _ {\eta}}.
$$
Proof. We have,
$$
p (x, t) - \tilde {p} (x, t) = \psi (x, t) ^ {\top} (\tilde {P} M \tilde {P} - M) \psi (x, t). \tag {39}
$$
Let now consider every term separately.
$$
\begin{array}{l} \frac {\partial p (x , t)}{\partial t} - \frac {\partial \tilde {p} (x , t)}{\partial t} = 2 \frac {\partial \psi (x , t)}{\partial t} ^ {\top} M \psi (x, t) - 2 \frac {\partial \psi (x , t)}{\partial t} ^ {\top} \tilde {P} M \tilde {P} \psi (x, t) \\ = 2 \frac {\partial \psi (x , t)}{\partial t} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t). \tag {40} \\ \end{array}
$$
$$
\frac {\partial p (x , t)}{\partial x _ {i}} - \frac {\partial \tilde {p} (x , t)}{\partial x _ {i}} = 2 \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} M \psi (x, t) - 2 \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} \tilde {P} M \tilde {P} \psi (x, t)
$$
$$
= 2 \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t). \tag {41}
$$
$$
\begin{array}{l} \frac {\partial^ {2} p (x , t)}{\partial x _ {i} x _ {j}} - \frac {\partial^ {2} \tilde {p} (x , t)}{\partial x _ {i} x _ {j}} = 2 \psi_ {i} ^ {\prime} (x, t) ^ {\top} M \psi_ {j} ^ {\prime} (x, t) + 2 \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} M \psi (x, t) \\ - 2 \psi_ {i} ^ {\prime} (x, t) ^ {\top} \tilde {P} M \tilde {P} \psi_ {j} ^ {\prime} (x, t) - 2 \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} \tilde {P} M \tilde {P} \psi (x, t) \\ = 2 \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) + 2 \psi_ {i} ^ {\prime} (x, t) ^ {\top} (M - \tilde {P} M \tilde {P}) \psi_ {j} ^ {\prime} (x, t). \tag {42} \\ \end{array}
$$
We have,
$$
\tilde {P} M \tilde {P} - M = (I - \tilde {P}) M (I - \tilde {P}) - M (I - \tilde {P}) - (I - \tilde {P}) M.
$$
Since  $|a^{\top}ABAa| \leq \|Aa\|_{\mathcal{H}}^{2}\|B\|$  and  $|a^{\top}ABA| \leq \|Aa\|_{\mathcal{H}}\|B^{1/2}\|\|B^{1/2}a\|_{\mathcal{H}}$ , for any  $a$  in a Hilbert space  $\mathcal{H}$  and for  $A, B$  bounded symmetric linear operators with  $B \in \mathbb{S}_{+}(\mathcal{H})$ , by bounding the terms of the equation above, we have for any  $x \in \mathbb{R}^d$ ,
$$
\begin{array}{l} | \psi (x, t) ^ {\top} (\tilde {P} M \tilde {P} - M) \psi (x, t) | \leq 2 \| (I - \tilde {P}) \psi (x, t) \| _ {\mathcal {H} _ {\eta}} \| M \| ^ {1 / 2} \| M ^ {1 / 2} \psi (x, t) \| _ {\mathcal {H} _ {\eta}} \\ + \| (I - \tilde {P}) \psi (x, t) \| _ {\mathcal {H} _ {\eta}} ^ {2} \| M \| \\ = 2 c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) + c _ {M} \tilde {u} _ {1} (x) ^ {2}, \tag {43} \\ \end{array}
$$
where  $c_{M} = \| M\|$  and we denoted by  $\tilde{u}_1(x)$  the quantity  $\tilde{u}_1(x) = \| (I - \tilde{P})\psi (x,t)\|_{\mathcal{H}_{\eta}}$  and we noted that  $\| M^{1 / 2}\psi (x,t)\|_{\mathcal{H}_{\eta}}^2 = \psi (x,t)^\top M\psi (x,t) = f(x;M,\psi (x,t))$ . Now,
$$
\begin{array}{l} \left| \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| \leq \| (I - \tilde {P}) \psi (x, t) \| _ {\mathcal {H} _ {\eta}} \| M \| \left\| (I - \tilde {P}) \frac {\partial \psi (x , t)}{\partial x _ {i}} \right\| _ {\mathcal {H} _ {\eta}} \\ + \left\| (I - \tilde {P}) \frac {\partial \psi (x , t)}{\partial x _ {i}} \right\| _ {\mathcal {H} _ {\eta}} \| M \| ^ {1 / 2} \| M ^ {1 / 2} \psi (x, t) \| _ {\mathcal {H} _ {\eta}} + \left\| (I - \tilde {P}) \psi (x, t) \right\| _ {\mathcal {H} _ {\eta}} \| M \| ^ {1 / 2} \left\| M ^ {1 / 2} \frac {\partial \psi (x , t)}{\partial x _ {i}} \right\| _ {\mathcal {H} _ {\eta}} \\ = c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x), \tag {44} \\ \end{array}
$$
where  $\tilde{f}_{x_i}(x;M,\psi) = \frac{\partial\psi(x,t)}{\partial x_i}^\top M\frac{\partial\psi(x,t)}{\partial x_i}$  and  $\tilde{u}_{x_i}(x) = \left\| (I - \tilde{P})\frac{\partial\psi(x,t)}{\partial x_i}\right\|_{\mathcal{H}_\eta}$ .
Following similar steps, we get
$$
\begin{array}{l} \left| \frac {\partial \psi (x , t)}{\partial t} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| \leq c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {t} (x) + c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {t} (x) \\ + c _ {M} ^ {1 / 2} \tilde {f} _ {t} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x), \tag {45} \\ \end{array}
$$
where  $\tilde{f}_t(x;M,\psi) = \frac{\partial\psi(x,t)}{\partial t}^\top M\frac{\partial\psi(x,t)}{\partial t}$  and  $\tilde{u}_t(x) = \left\| (I - \tilde{P})\frac{\partial\psi(x,t)}{\partial t}\right\|_{\mathcal{H}_{\eta}}$
$$
\begin{array}{l} \left| \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| \leq c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {x _ {j} x _ {i}} (x) + c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {j} x _ {i}} (x) \\ + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {j} x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x), \tag {46} \\ \end{array}
$$
where  $\tilde{f}_{x_j x_i}(x;M,\psi) = \frac{\partial^2\psi(x,t)}{\partial x_j x_i}^\top M\frac{\partial^2\psi(x,t)}{\partial x_j x_i}$  and  $\tilde{u}_{x_j x_i}(x) = \left\| (I - \tilde{P})\frac{\partial^2\psi(x,t)}{\partial x_j x_i}\right\|_{\mathcal{H}_\eta}$ .
And,
$$
\begin{array}{l} \psi_ {i} ^ {\prime} (x, t) ^ {\top} (M - \tilde {P} M \tilde {P}) \psi_ {j} ^ {\prime} (x, t) \leq c _ {M} \tilde {u} _ {x _ {j}} (x) \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} f _ {x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {j}} (x) \\ + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {j}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {i}} (x). \tag {47} \\ \end{array}
$$
Hence, combining everything together, we have:
$$
\left| \frac {\partial p (x , t)}{\partial t} - \frac {\partial \tilde {p} (x , t)}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) p (x , t))}{\partial x _ {i}} - \frac {\partial (\mu_ {i} (x , t) \tilde {p} (x , t))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p (x, t) - D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \tilde {p} (x, t) \right|
$$