SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathcal {F} [ (- \Delta) ^ {s} f ] (\xi) = \| \xi \| ^ {2 s} \mathcal {F} [ f ] (\xi) \\ = \| \xi \| ^ {2 s} \left(\int \left(\sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} - x)} e ^ {j \omega_ {k} ^ {\top} (x _ {j} - x)} q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k}\right) e ^ {- j \xi^ {\top} x} d x\right) \\ = \| \xi \| ^ {2 s} \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} \left(\int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} - x)} e ^ {j \omega_ {k} ^ {\top} (x _ {j} - x)} e ^ {- j \xi^ {\top} x} d x\right) q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k} \\ = \| \xi \| ^ {2 s} \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} \left(\int e ^ {j \left(\omega_ {\ell} ^ {\top} x _ {i} + \omega_ {k} x _ {j}\right)}. e ^ {- j x ^ {\top} \left(\omega_ {\ell} + \omega_ {k} + \xi\right)} d x\right) q \left(\omega_ {\ell}\right) q \left(\omega_ {k}\right) d \omega_ {\ell} d \omega_ {k} \\ = \| \xi \| ^ {2 s} \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j \left(\omega_ {\ell} ^ {\top} x _ {i} + \omega_ {k} x _ {j}\right)} \left(\int_ {\mathbb {R} ^ {d}} e ^ {- j x ^ {\top} \left(\omega_ {\ell} + \omega_ {k} + \xi\right)} d x\right) q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k} \\ = \frac {1}{p ^ {2}} \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j (\omega_ {\ell} ^ {\top} x _ {i} + \omega_ {k} x _ {j})} \| \xi \| ^ {2 s} \delta (\omega_ {\ell} + \omega_ {k} + \xi) q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k}. \\ \end{array}
$$
$$
\begin{array}{l} (- \Delta) ^ {s} f (x) = \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j (\omega_ {\ell} ^ {\top} x _ {i} + \omega_ {k} x _ {j})} \left(\int_ {\mathbb {R} ^ {d}} \| \xi \| ^ {2 s} \delta (\omega_ {\ell} + \tilde {\omega} _ {k} + \xi) e ^ {- j x ^ {\top} \xi} d \xi\right) q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k} \\ = \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j (\omega_ {\ell} ^ {\top} x _ {i} + \omega_ {k} x _ {j})} \| \omega_ {\ell} + \tilde {\omega} _ {k} \| ^ {2 s} e ^ {j x ^ {\top} (\omega_ {\ell} + \tilde {\omega} _ {k})} q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k} \\ = \sum_ {i, j = 1} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} + x)} e ^ {j \omega_ {k} ^ {\top} (x _ {j} + x)} \| \omega_ {\ell} + \tilde {\omega} _ {k} \| ^ {2 s} q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k} \\ \end{array}
$$
$$
= \sum_ {i, j = 1} ^ {m} A _ {i j} \mathbb {E} [ \| \omega_ {\ell} + \omega_ {k} \| ^ {2 s} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} + x)} e ^ {j \omega_ {k} ^ {\top} (x _ {j} + x)} ].
$$
C Existence of an appropriate PSD matrix  $M_{\varepsilon}$  (Fractional Fokker-Planck Equation)
Let us redefine the mollifier function  $g$  defined as in [RC21].
$$
g (x) = \frac {2 ^ {- d / 2}}{V _ {d}} \| x \| ^ {- d} J _ {d / 2} (2 \pi \| x \|) J _ {d / 2} (4 \pi \| x \|), \tag {55}
$$
where  $J_{d / 2}$  is the Bessel function of the first kind of order  $d / 2$  and  $V_{d} = \int_{\| x\| \leq 1}dx = \frac{\pi^{d / 2}}{\Gamma(d / 2 + 1)}$ .  $g_{v}(x) = v^{-d}g(x / v)$ .  $\mathcal{F}[f](\cdot)$  denotes the fourier transform of  $f$ .
C.1 Useful Results (Fractional Fokker Planck Equation)
Lemma 7. Consider the definition of  $g_v(x) = v^{-d}g(x / v)$  where  $g$  is defined in equation (13). Let  $\beta > 2, q \in \mathbb{N}$ . Let  $f_1, \ldots, f_q \in W_2^\beta(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d)$  and the function  $p^\star = \sum_{i=1}^{q} f_i^2$ . Let  $\varepsilon \in (0,1]$  and let  $\eta \in \mathbb{R}_{++}^d$ . Let  $\phi_{X,T}$  be the feature map of the Gaussian kernel with bandwidth  $\eta$  and let  $\mathcal{H}_X \otimes \mathcal{H}_T$  be the associated RKHS. Then there exists  $\mathsf{M}_{\varepsilon} \in \mathbb{S}_+(H_X \otimes H_T)$  with  $\mathrm{rank}(\mathsf{M}_{\varepsilon}) \leq q$ , such that for the representation  $\hat{p}(x,t) = \phi_{X,T}^\top \mathsf{M}_{\varepsilon} \phi_{X,T}$ , following holds for  $v > 0$  and  $k \in \{1, \dots, d\}$ ,
$$
\| (- \Delta) ^ {s} (p ^ {\star} - \hat {p}) \| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq c _ {1} (2 v) ^ {(\beta - s)} \sum_ {i = 1} ^ {q} C _ {T _ {1}} ^ {(i)} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} + c _ {2} (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} C _ {T _ {2}} ^ {(i)} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2},
$$
for some positive constants  $c_{1}$  and  $c_{2}$ , where  $C_{T_1}^{(i)} = \left\| \mathcal{F}[f_i]\right\|_{L^1 (\mathbb{R}^{d + 1})} + \left\| \mathcal{F}[f_i]\right\|_{L^\infty (\mathbb{R}^{d + 1})}\| g\|_{L^1 (\mathbb{R}^{d + 1})}$  and  $C_{T_2}^{(i)} = \left\| \left\| \omega \right\|^{2s}\mathcal{F}(f_i)(\omega)\right\|_{L^1 (\mathbb{R}^{d + 1})}+$ $\left\| \left\| \omega \right\|^{2s}\mathcal{F}[f_i](\omega)\right\|_{L^1 (\mathbb{R}^{d + 1})}\left\| g\right\|_{L^1 (\mathbb{R}^{d + 1})}$  for  $i\in \{1,\dots ,q\}$ .
Proof. As in [RC21], let us denote,  $f_{i} \star g_{v}$  as  $f_{i,v}$ . We consider
$$
\begin{array}{l} \| (- \Delta) ^ {s} (p ^ {\star} - \hat {p}) \| _ {L ^ {2} (\mathbb {R} ^ {d})} = \| \mathcal {F} [ (- \Delta) ^ {s} (p ^ {\star} - \hat {p}) ] \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \\ = \| \| \omega \| ^ {2 s} \mathcal {F} [ p ^ {\star} - \hat {p} ] (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} = \left\| \| \omega \| ^ {2 s} \mathcal {F} \left[ \sum_ {i = 1} ^ {q} f _ {i} ^ {2} - f _ {i, v} ^ {2} \right] (\omega) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \\ = \left\| \| \omega \| ^ {2 s} \mathcal {F} \left[ \sum_ {i = 1} ^ {q} \left(f _ {i} - f _ {i, v}\right) \left(f _ {i} + f _ {i, v}\right) \right] (\omega) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} = \left\| \| \omega \| ^ {2 s} \sum_ {i = 1} ^ {q} \mathcal {F} \left[ \left(f _ {i} - f _ {i, v}\right) \left(f _ {i} + f _ {i, v}\right) \right] (\omega) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \\ \leq \sum_ {i = 1} ^ {q} \left| \left| \| \omega \| ^ {2 s} \mathcal {F} \left[ (f _ {i} - f _ {i, v}) (f _ {i} + f _ {i, v}) \right] (\omega) \right| \right| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} = \sum_ {i = 1} ^ {q} \left| \left| \| \omega \| ^ {2 s} \mathcal {F} \left[ (f _ {i} - f _ {i, v}) \right] (\omega) \star \mathcal {F} \left[ f _ {i} + f _ {i, v} \right] (\omega) \right| \right| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)}. \\ \end{array}
$$
Let us consider the function,
$$
\| \omega \| ^ {2 s} \underbrace {\mathcal {F} \left[ \left(f _ {i} - f _ {i , v}\right) \right] (\omega)} _ {:= m (\omega)} \star \underbrace {\mathcal {F} \left[ f _ {i} + f _ {i , v} \right] (\omega)} _ {:= n (\omega)} = \int \| \omega \| ^ {2 s} m (\omega - y) n (y) d \omega
$$
$$
\begin{array}{l} = \int \| \omega - y + y \| ^ {2 s} m (\omega - y) n (y) d \omega \leq c _ {1} \int \| \omega - y \| ^ {2 s} m (\omega - y) n (y) d y + c _ {2} \int \| y \| ^ {2 s} m (\omega - y) n (y) d y \\ = c _ {1} \left[ \| \omega \| ^ {2 s} m (\omega) \right] * n (\omega) + c _ {2} m (\omega) * \left[ \| \omega \| ^ {2 s} n (\omega) \right]. \\ \end{array}
$$
$$
\| (- \Delta) ^ {s} (p ^ {\star} - \hat {p}) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq \underbrace {c _ {1} \| [ \| \omega \| ^ {2 s} m (\omega) ] * n (\omega) \| _ {L ^ {2} (\mathbb {R} ^ {d})}} _ {:= \mathrm {T} _ {1}} + \underbrace {c _ {2} \| m (\omega) * [ \| \omega \| ^ {2 s} n (\omega) ] \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})}} _ {:= \mathrm {T} _ {2}}
$$
Now, we can apply Young's convolution inequality on  $\mathrm{T}_1$  and  $\mathrm{T}_2$ . Hence,
$$
\begin{array}{l} \mathrm {T} _ {1} \leq c _ {1} \| \| \omega \| ^ {2 s} m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \| n (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \\ = c _ {1} \| \left\| \omega \right\| ^ {2 s} m (\omega) \| _ {L ^ {2} (\mathbb {R} ^ {d})} \| \mathcal {F} [ f _ {i} + f _ {i, v} ] \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} \\ \leq c _ {1} \| \left\| \omega \right\| ^ {2 s} m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \mathcal {F} [ f _ {i, v} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \right] \\ = c _ {1} \| \left\| \omega \right\| ^ {2 s} m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} + \| \mathcal {F} [ f _ {i} ]. \mathcal {F} [ g _ {v} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \right] \\ \leq c _ {1} \| \| \omega \| ^ {2 s} m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \| \mathcal {F} [ g _ {v} ] \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \right]. \\ \end{array}
$$
In the last equation, we applied holder's inequality. From uniform continuity and the Riemann-Lebesgue lemma, we have
$$
\| \mathcal {F} [ g _ {v} ] \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \leq \| g _ {v} \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} = \int g _ {v} (v x) d x = \int t ^ {- d} | g (x / t) | d x = \int | g (x) | d x = \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}.
$$
$$
\mathrm {T} _ {1} \leq c _ {1} \| \| \omega \| ^ {2 s} m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \mathcal {F} [ f _ {i} ] \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \| g \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \right]
$$
$$
\begin{array}{l} \mathrm {T} _ {2} \leq c _ {2} \| m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \| \| \omega \| ^ {2 s} n (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \\ \leq c _ {2} \| m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i, v}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \right] \\ = c _ {2} \| m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \left[ \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \mathcal {F} (g _ {v}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \right]. \\ \end{array}
$$
In the above equation, we can apply holder's inequality to get,
$$
\begin{array}{l} \left. \mathrm {T} _ {2} \leq c _ {2} \| m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \| \mathcal {F} (g _ {v}) (\omega) \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \right] \right. \\ \leq c _ {2} \| m (\omega) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \left[ \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} + \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \| \mathcal {F} (g _ {v}) (\omega) \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \right]. \\ \end{array}
$$
$$
\mathrm {T} _ {2} \leq c _ {2} \| m (\omega) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \left[ \| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} + \| \| \omega \| ^ {2 s} \mathcal {F} [ f _ {i} ] (\omega) \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} \right]
$$
The term  $\| m(\omega)\|_{L^2 (\mathbb{R}^{d + 1})}$  and  $\| \| \omega \|^{2s}m(\omega)\|_{L^2 (\mathbb{R}^{d + 1})}$  can be bounded using similar technique as before. We will use the result from Lemma 1.
$$
\begin{array}{l} \left\| \left\| \omega \right\| ^ {2 s} m (\omega) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} ^ {2} \leq \int \left\| \omega \right\| ^ {4 s} \left| \mathcal {F} [ f _ {i} ] (\omega) \right| ^ {2} | | 1 - \mathcal {F} [ g ] (v \omega) | ^ {2} d \omega \\ \leq \int_ {v \| \omega \| \geq 1} \| \omega \| ^ {4 s} | \mathcal {F} [ f _ {i} ] (\omega) | ^ {2} | d \omega \\ = \int_ {v \| \omega \| \geq 1} \| \omega \| ^ {4 s} (1 + \| \omega \| ^ {2}) ^ {- \beta} (1 + \| \omega \| ^ {2}) ^ {\beta} | \mathcal {F} [ f _ {i} ] (\omega) | ^ {2} | d \omega \\ \leq 2 ^ {2 \beta} \sup  _ {v \| \omega \| \geq 1} \frac {\| \omega \| ^ {4 s}}{(1 + \| \omega \| ^ {2}) ^ {\beta}} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ \leq 2 ^ {2 \beta} \sup  _ {v \| \omega \| \geq 1} \frac {1}{(1 + \| \omega \| ^ {2}) ^ {\beta - 2 s}} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ \leq 2 ^ {2 \beta} \frac {v ^ {2 (\beta - 2 s)}}{(1 + v ^ {2}) ^ {\beta - 2 s}} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ \end{array}
$$
$$
= 2 ^ {2 s} \frac {(2 v) ^ {2 (\beta - 2 s)}}{(1 + v ^ {2}) ^ {\beta - 2 s}} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \leq 2 ^ {2 s} (2 v) ^ {2 (\beta - 2 s)} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2}.
$$
$$
\| (- \Delta) ^ {s} (p ^ {\star} - \hat {p}) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq c _ {1} (2 v) ^ {(\beta - 2 s)} \sum_ {i = 1} ^ {q} C _ {T _ {1}} ^ {(i)} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} + c _ {2} (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} C _ {T _ {2}} ^ {(i)} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2},
$$
where  $C_{T_1}^{(i)} = \left\| \mathcal{F}[f_i] \right\|_{L^1(\mathbb{R}^{d+1})} + \left\| \mathcal{F}[f_i] \right\|_{L^\infty(\mathbb{R}^{d+1})} \| g \|_{L^1(\mathbb{R}^{d+1})}$  and  $C_{T_2}^{(i)} =$
$$
\left\| \left\| \omega \right\| ^ {2 s} \mathcal {F} \left(f _ {i}\right) (\omega) \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} +
$$
$$
\left| \left| \left| \left| \omega \right| \right| ^ {2 s} \mathcal {F} [ f _ {i} ] (\omega) \right| \right| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} \| g \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)}.
$$
C.2 Proof of Theorem 4
From the previous lemma, we have,
for some positive  $c_{1}$  and  $c_{2}$  where  $C_{T_1}^{(i)} = \| \mathcal{F}[f_i]\|_{L^1 (\mathbb{R}^{d + 1})} + \| \mathcal{F}[f_i]\|_{L^\infty (\mathbb{R}^{d + 1})}\| g\|_{L^1 (\mathbb{R}^{d + 1})}$  and
$$
C _ {T _ {2}} ^ {(i)} = \left\| \| \omega \| ^ {2 s} \mathcal {F} (f _ {i}) (\omega) \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} +
$$
$\left\| \left\| \omega \right\|^{2s}\mathcal{F}[f_i](\omega)\right\|_{L^1 (\mathbb{R}^{d + 1})}\left\| g\right\|_{L^1 (\mathbb{R}^{d + 1})}$ . Now, if we choose, for some constant  $c$
$$
v = \left(\frac {\varepsilon}{c 2 ^ {\beta - 2 s} \sum_ {i = 1} ^ {q} (C _ {T _ {1}} ^ {(i)} + C _ {T _ {2}} ^ {(i)}) \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2}}\right) ^ {\frac {1}{\beta - 2 s}},
$$
$$
\left\| (- \Delta) ^ {s} \left(p ^ {\star} - \hat {p}\right) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq \varepsilon .
$$
We can bound the other two terms as using lemma 2. We have,
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d + 1}\right)} \| f _ {i} \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d + 1})}. \\ \end{array}
$$
$$
\begin{array}{l} \sum_ {i = 1} ^ {q} \left\| f _ {i} \right\| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \left\| g \right\| _ {L _ {1} (\mathbb {R} ^ {d + 1})}. H e n c e, \\ \left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq \varepsilon . \\ \end{array}
$$
Similarly, from lemma 2
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d + 1}\right)} \| f _ {i} \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d + 1})}. \\ \end{array}
$$
From [RC21, Theorem D.4 (Eq. D.27)], we have,
where  $C_1 = \sum_{i=1}^q \|f_i\|_{W_{2}^\beta(\mathbb{R}^{d+1})} \|f_i\|_{L^\infty(\mathbb{R}^{d+1})},$ $C_2 = \sum_{i=1}^q \|f_i\|_{W_{2}^\beta(\mathbb{R}^{d+1})} \left\| \frac{\partial f_i}{\partial t} \right\|_{L^\infty(\mathbb{R}^{d+1})} \|g\|_{L_1(\mathbb{R}^{d+1})}$  and  $C_3 = (1 + \|g\|_{L^1(\mathbb{R}^{d+1})}) \sum_{i=1}^q \|f_i\|_{W_{2}^\beta(\mathbb{R}^{d+1})} \|f_i\|_{L^\infty(\mathbb{R}^{d+1})}$ , then, we have
Hence, by combining everything together, we have
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} (\mu_ {i} (x, t) \hat {p} (x, t)) - (- \Delta) ^ {s} (\hat {p} (x, t)) \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq \varepsilon (2 + d (R _ {\mu} + R _ {\mu_ {p}})).
$$
$$
v = \left(\frac {\varepsilon}{9 6 \pi^ {2} 2 ^ {\beta - 2} C}\right) ^ {\frac {1}{\beta - 2 s}},
$$
where  $C$  depends on  $f_{i}$  s,  $\mathcal{F}[f_i]$  and it's derivative.
$$
\begin{array}{l} T r (M _ {\varepsilon}) = c _ {\eta} 2 ^ {2 \beta} (1 + (\frac {v}{3}) ^ {2 \beta} e ^ {\frac {8 9}{\eta_ {0} v ^ {2}}}) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ = c _ {\eta} 2 ^ {2 \beta} \left(1 + 3 ^ {- 2 \beta} \left(\frac {\varepsilon}{9 6 \pi^ {2} 2 ^ {\beta - 2} C}\right) ^ {\frac {2 \beta}{\beta - 2 s}} e ^ {\frac {8 9}{\eta_ {0} v ^ {2}}}\right) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \\ = c _ {\eta} 2 ^ {2 \beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2} \left(1 + 3 ^ {- 2 \beta} \left(\frac {\varepsilon}{\tilde {C}}\right) ^ {\frac {2 \beta}{\beta - 2 s}} e ^ {\frac {8 9}{\eta_ {0} \left(\frac {\varepsilon}{\tilde {C}}\right) ^ {\beta - 2 s}}} \right) \\ = \hat {C} | \eta | ^ {1 / 2} \left(1 + \varepsilon^ {\frac {2 \beta}{\beta - 2 s}} e ^ {\frac {\tilde {C} ^ {\prime}}{\eta_ {0} \varepsilon^ {\frac {2}{\beta - 2 s}}}}\right) \leq \hat {C} | \eta | ^ {1 / 2} \left(1 + \varepsilon^ {\frac {2 \beta}{\beta - 2 s}} \exp \left(\frac {\tilde {C} ^ {\prime}}{\eta_ {0} \varepsilon^ {- \frac {2}{\beta - 2 s}}}\right)\right) \\ \end{array}
$$
where  $\tilde{C} = 96\pi^{2}2^{\beta -2}C,\tilde{C} ' = 89\tilde{C}^{\frac{2}{\beta - 2}},$  and  $\hat{C}$ $= \pi^{-d / 2}2^{2\beta}\sum_{i = 1}^{q}\| f_i\|_{W_2^\beta (\mathbb{R}^{d + 1})}^2\max \left(1,\frac{3^{-2\beta}}{\tilde{C}^{\frac{2\beta}{\beta - 2s}}}\right).$
C.3 Approximation Properties of PSD Model for Fractional Laplacian (Proof of Theorem 4)
Proof. From the previous result in theorem 4, we know that there exists an  $\mathsf{M}_{\varepsilon} \in \mathbb{S}_{+}(\mathcal{H}_X \otimes \mathcal{H}_T)$  with  $\mathrm{rank}(\mathsf{M}_{\varepsilon}) \leq q$ , such that for the representation  $\hat{p}(x,t) = \phi_{X,T}^{\top}\mathsf{M}_{\varepsilon}\phi_{X,T}$ , following holds under assumption 2 on the coefficients of the fractional FPE,
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) \hat {p} (x, t)\right) + (- \Delta) ^ {s} \hat {p} (x, t) \right\| _ {L ^ {2} (\tilde {X})} = O (\varepsilon), \tag {56}
$$
$$
\frac {\partial p ^ {\star} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t)\right) + (- \Delta) ^ {s} p ^ {\star} (x, t) = 0. \tag {57}
$$
We know that, for given  $A_{m}$ ,  $\tilde{p}(x,t) = \psi(x,t)^{\top} \tilde{P} \mathsf{M}_{\varepsilon} \tilde{P} \psi(x,t)^{\top}$  where  $\tilde{P}$  is the projection operator defined earlier. We will utilize the Gagliardo-Nirenberg inequality to control the error in compression of fractional laplacian compressed model with respect to the original one. For any fixed  $t$ ,
$$
\left. (- \Delta) ^ {s} [ \hat {p} (x, t) - \tilde {p} (x, t) ] = (- \Delta) ^ {s} [ \psi (x, t) ^ {\top} (\tilde {P} \mathsf {M} _ {\varepsilon} \tilde {P} - \mathsf {M} _ {\varepsilon}) \psi (x, t) ] \right.
$$
By applying Gagliardo-Nirenberg inequality for fractional laplacian [MP18], we have
$$
\| (- \Delta) ^ {s} \beta_ {t} (x) \| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq \underbrace {\| \beta_ {t} (x) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} ^ {1 - \vartheta}} _ {:= T _ {1}} \cdot \underbrace {\| \Delta \beta_ {t} (x) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} ^ {\vartheta}} _ {:= T _ {2}},
$$
where  $\vartheta = s\leq 1$  and  $\Delta$  is normal laplacian operator. Let us consider the term  $T_{1}$  and  $T_{2}$  separately. In our case,  $\beta_t(x) = \psi (x,t)^\top (\tilde{P}\mathsf{M}_\varepsilon \tilde{P} -\mathsf{M}_\varepsilon)\psi (x,t)$ . In the proof of theorem 8, we have proved the bound on  $\Delta \beta_{t}(x)$  and  $\beta_t(x)$ . From equation (43), we have,
$$
\begin{array}{l} | \psi (x, t) ^ {\top} (\tilde {P} \mathsf {M} _ {\varepsilon} \tilde {P} - \mathsf {M} _ {\varepsilon}) \psi (x, t) | \leq 2 \| (I - \tilde {P}) \psi (x, t) \| _ {\mathcal {H} _ {\eta}} \| \mathsf {M} _ {\varepsilon} \| ^ {1 / 2} \| \mathsf {M} _ {\varepsilon} ^ {1 / 2} \psi (x, t) \| _ {\mathcal {H} _ {\eta}} \\ + \left\| (I - \tilde {P}) \psi (x, t) \right\| _ {\mathcal {H} _ {\eta}} ^ {2} \| \mathsf {M} _ {\varepsilon} \| \\ = 2 c _ {M} ^ {1 / 2} f (x; \mathsf {M} _ {\varepsilon}, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) + c _ {M} \tilde {u} _ {1} (x) ^ {2}, \\ \end{array}
$$
where  $c_{M} = \| \mathsf{M}_{\varepsilon}\|$  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  $\| \mathsf{M}_{\varepsilon}^{1 / 2}\psi (x,t)\|_{\mathcal{H}_\eta}^2 = \psi (x,t)^\top \mathsf{M}_{\varepsilon}\psi (x,t) = f(x;\mathsf{M}_{\varepsilon},\psi (x,t))$  . Similarly, we have shown that
$$
\frac {\partial^ {2} p (x , t)}{\partial x _ {i} x _ {j}} - \frac {\partial^ {2} \tilde {p} (x , t)}{\partial x _ {i} x _ {j}} = 2 \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} \left(\mathrm {M} _ {\varepsilon} - \tilde {P} \mathrm {M} _ {\varepsilon} \tilde {P}\right) \psi (x, t) + 2 \psi_ {i} ^ {\prime} (x, t) ^ {\top} \left(\mathrm {M} _ {\varepsilon} - \tilde {P} \mathrm {M} _ {\varepsilon} \tilde {P}\right) \psi_ {j} ^ {\prime} (x, t). \tag {58}
$$
From equations (46) and (47)
$$
\begin{array}{l} \left| \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} (\mathsf {M} _ {\varepsilon} - \tilde {P} \mathsf {M} _ {\varepsilon} \tilde {P}) \psi (x, t) \right| \leq c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {x _ {j} x _ {i}} (x) + c _ {M} ^ {1 / 2} f (x; \mathsf {M} _ {\varepsilon}, \psi) ^ {1 / 2} \tilde {u} _ {x _ {j} x _ {i}} (x) \\ + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {j} x _ {i}} (x; \mathsf {M} _ {\varepsilon}, \psi) ^ {1 / 2} \tilde {u} _ {1} (x), \tag {59} \\ \end{array}
$$