SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For any  $x \in \mathcal{X}$ , we denote  $\mu_p(x)$  the conditional mean embedding associated to the transition kernel defined as
$$
\mu_ {p} (x) := \mathbf {E} \big [ \phi (X _ {t + 1}) | X _ {t} = x \big ] = \int \phi (y) p (x, \mathrm {d} y)
$$
The following lemma provides a characterization of  $\Phi_{Y|X}^{*}$  in terms of the conditional mean embedding.
Lemma A.1: We have the following relations:
$$
\Phi_ {Y \mid X} f = \int_ {\mathcal {X}} f (x) \mu_ {p} (x) \mathrm {d} \pi (x), \quad f \in L _ {\pi} ^ {2} \tag {16}
$$
$$
\left(\Phi_ {Y | X} ^ {*} f\right) (x) = \langle f, \mu_ {p} (x) \rangle , \quad f \in \mathcal {H} \tag {17}
$$
$$
\Phi_ {Y | X} \Phi_ {Y | X} ^ {*} = \mathbf {E} _ {x \sim \pi} \mu_ {p} (x) \otimes \mu_ {p} (x) \tag {18}
$$
Proof of Lemma A.1: For the first property:
$$
\begin{array}{l} \left(\Phi_ {Y \mid X} ^ {*} f\right) (x) = \left(\mathcal {K} _ {\pi} \left(\Phi_ {X} ^ {*} f\right)\right) (x) (19) \\ = \int \left(\Phi_ {X} ^ {*} f\right) (y) p (x, \mathrm {d} y) (20) \\ = \int f (y) p (x, \mathrm {d} y) (21) \\ = \langle f, \int \phi (y) p (x, \mathrm {d} y) \rangle = \langle f, \mu_ {p} (x) \rangle (22) \\ \end{array}
$$
where we used that  $f$  and  $\Phi_X^* f$  coincide  $\pi$ -almost everywhere. The second property is a direct consequence of the definition of the adjoint. For (18), we simply use (17) and the definition of  $\Phi_{Y|X}$  to get
$$
\Phi_ {Y \mid X} \left(\Phi_ {Y \mid X} ^ {*} f\right) = \int \langle f, \mu_ {p} (z) \rangle \mu_ {p} (z) \mathrm {d} \pi (z) = \left(\int \mu_ {p} (z) \mu_ {p} (z) ^ {*} \mathrm {d} \pi (z)\right) f.
$$
A.3 Power spaces
We now define the  $\alpha$ -power space  $[\mathcal{H}]_{\pi}^{\alpha}$  in order to provide some intuition regarding Assumption 4.3. By Assumption 4.1,  $\operatorname{tr}(C) = \int \operatorname{tr}(\phi(x) \otimes \phi(x)) \, \mathrm{d}\pi(x) \leq K^2$  and thus  $C$  is trace-class (and compact). By [18], there exists a non-increasing summable sequence  $(\mu_i)_{i \in I}$  for an at most countable index set  $I$ , a family  $(e_i)_{i \in I} \in \mathcal{H}$  s.t.  $(\Phi_X^* e_i)_{i \in I}$  is an orthonormal basis of  $\overline{\operatorname{span} \Phi_X^*} \subseteq L_\pi^2$  and  $(\mu_i^{1/2} e_i)_{i \in I}$  is an orthonormal basis of  $(\ker \Phi_X^*)^\perp \subseteq \mathcal{H}$  such that
$$
C = \sum_ {i \in I} \mu_ {i} \langle \cdot , \mu_ {i} ^ {1 / 2} e _ {i} \rangle_ {\mathcal {H}} \mu_ {i} ^ {1 / 2} e _ {i}.
$$
For  $\alpha \geq 0$ , we now define the  $\alpha$ -power space as
$$
[ \mathcal {H} ] _ {\pi} ^ {\alpha} := \left\{\sum_ {i \in I} a _ {i} \mu_ {i} ^ {\alpha / 2} \Phi_ {X} ^ {*} e _ {i} \Bigg | (a _ {i}) _ {i \in I} \in \ell_ {2} (I) \right\} \subseteq L _ {\pi} ^ {2}
$$
equipped with norm
$$
\left\| \sum_ {i \in I} a _ {i} \mu_ {i} ^ {\alpha / 2} \Phi_ {X} ^ {*} e _ {i} \right\| _ {[ \mathcal {H} ] _ {\pi} ^ {\alpha}} := \| (a _ {i}) _ {i \in I} \| _ {\ell_ {2} (I)}.
$$
We can now make the following assumption regarding the embedding of the power spaces into  $L_{\pi}^{\infty}$ .
Assumption A.2 (Embedding): There exists  $\tau \in [\beta, 1]$  such that  $c_{\tau} \coloneqq \left\| [\mathcal{H}]_{\pi}^{\tau} \hookrightarrow L_{\pi}^{\infty} \right\|^{2} < \infty$ .
We stress that Assumption A.2 implies in particular Assumption 4.3, and is a common assumption in the literature, see for instance [18].
B Expression of the risk
We have the following risk decomposition.
Lemma B.1: The risk can alternatively be written
$$
\begin{array}{l} \mathcal {R} _ {\mathrm {H S}} (A) = \mathbf {E} _ {(x, y) \sim \rho} \| \phi (y) - A \phi (x) \| ^ {2} \\ = \mathcal {R} _ {\mathrm {H S}, 0} + \mathcal {E} _ {\mathrm {H S}} (A) \\ \end{array}
$$
$$
\begin{array}{l} w h e r e \quad \mathcal {R} _ {\mathrm {H S}, 0} := \| \Phi_ {X} \| _ {\mathrm {H S}} ^ {2} - \| \Phi_ {Y | X} \| _ {\mathrm {H S}} ^ {2} \\ = \int \left\| \mu_ {p} (x) - \phi (y) \right\| ^ {2} d \rho (x, y) \\ \end{array}
$$
$$
\begin{array}{l} a n d \quad \mathcal {E} _ {\mathrm {H S}} (A) := \| \Phi_ {Y | X} - A \Phi_ {X} \| _ {\mathrm {H S}} ^ {2} \\ = \int \left\| \mu_ {p} (x) - A \phi (x) \right\| ^ {2} \mathrm {d} \pi (x). \\ \end{array}
$$
where  $\inf_{A\in \mathrm{HS}(\mathcal{H})}\mathcal{E}_{\mathrm{HS}}(A) = 0$  and thus we interpret  $\mathcal{E}_{\mathrm{HS}}$  as the excess risk.
Proof of Lemma B.1: Let  $(h_i)_{i\in \mathbb{N}}$  be an orthonormal basis of  $\mathcal{H}$ . Then
$$
\begin{array}{l} \mathcal {E} _ {\mathrm {H S}} (A) := \left\| \Phi_ {Y | X} - A \Phi_ {X} \right\| _ {\mathrm {H S}} ^ {2} \\ = \sum_ {i \in \mathbb {N}} \| \Phi_ {Y | X} ^ {*} h _ {i} - \Phi_ {X} ^ {*} A ^ {*} h _ {i} \| _ {L _ {\pi} ^ {2}} ^ {2} \\ = \sum_ {i \in \mathbb {N}} \int \left(\left(\Phi_ {Y | X} ^ {*} h _ {i}\right) (x) - \langle A ^ {*} h _ {i}, \phi (x) \rangle_ {\mathcal {H}}\right) ^ {2} \mathrm {d} \pi (x) \\ \end{array}
$$
$$
\begin{array}{l} \text {(b y (1 7))} = \sum_ {i \in \mathbb {N}} \int \left(\langle h _ {i}, \mu_ {p} (x) \rangle_ {\mathcal {H}} - \langle h _ {i}, A \phi (x) \rangle_ {\mathcal {H}}\right) ^ {2} \mathrm {d} \pi (x) \\ = \int \left\| \mu_ {p} (x) - A \phi (x) \right\| ^ {2} \mathrm {d} \pi (x). \\ \end{array}
$$
It holds
$$
\begin{array}{l} \mathcal {R} _ {\mathrm {H S}, 0} = \int \| \mu_ {p} (x) - \phi (y) \| ^ {2} \mathrm {d} \rho (x, y) \\ = \int \left(\left\| \mu_ {p} (x) \right\| ^ {2} - 2 \langle \mu_ {p} (x), \phi (y) \rangle_ {\mathcal {H}} + \operatorname {t r} \phi (y) \phi (y) ^ {*}\right) \mathrm {d} \rho (x, y) \\ = \int \| \mu_ {p} (x) \| ^ {2} d \pi (x) - 2 \int \left\langle \mu_ {p} (x), \int \phi (y) p (x, d y) \right\rangle_ {\mathcal {H}} d \pi (x) + \int \int \operatorname {t r} \phi (y) \phi (y) ^ {*} p (x, d y) d \pi (x) \\ \stackrel {(i)} {=} - \int \| \mu_ {p} (x) \| ^ {2} d \pi (x) + \int \operatorname {t r} \left(\phi (y) \phi (y) ^ {*}\right) d \pi (y) \\ = - \operatorname {t r} \left(\int \mu_ {p} (x) \mu_ {p} (x) ^ {*} \mathrm {d} \pi (x)\right) + \operatorname {t r} (C) \\ = - \operatorname {t r} \left(\Phi_ {Y | X} \Phi_ {Y | X} ^ {*}\right) + \operatorname {t r} \left(\Phi_ {X} \Phi_ {X} ^ {*}\right) \\ \end{array}
$$
where we used the invariance property of  $\pi$  in  $(i)$  and Lemma A.1 for the last inequality. Then one can easily check that the sum of both corresponds to the full risk defined in (5):
$$
\begin{array}{l} \mathcal {R} _ {\mathrm {H S}, 0} + \mathcal {E} _ {\mathrm {H S}} (A) = \int \| \mu_ {p} (x) - \phi (y) \| ^ {2} \mathrm {d} \rho (x, y) + \int \| \mu_ {p} (x) - A \phi (x) \| ^ {2} \mathrm {d} \pi (x) \\ = \int \left(\| \mu_ {p} (x) \| ^ {2} - 2 \langle \mu_ {p} (x), \int \phi (y) p (x, \mathrm {d} y) \rangle + \int \| \phi (y) \| ^ {2} p (x, \mathrm {d} y)\right) \mathrm {d} \pi (x) \\ + \int \left(\| \mu_ {p} (x) \| ^ {2} - 2 \langle \mu_ {p} (x), A \phi (x) \rangle + \| A \phi (x) \| ^ {2}\right) d \pi (x) \\ = \int \left(\int \| \phi (y) \| ^ {2} p (x, \mathrm {d} y) - 2 \langle \int \phi (y) p (x, \mathrm {d} y), A \phi (x) \rangle + \| A \phi (x) \| ^ {2}\right) \mathrm {d} \pi (x) \\ = \int \left(\| \phi (y) \| ^ {2} - 2 \langle \phi (y), A \phi (x) \rangle + \| A \phi (x) \| ^ {2}\right) d \rho (x, y) \\ = \int \| \phi (y) - A \phi (x) \| ^ {2} \mathrm {d} \rho (x, y) = \mathcal {R} _ {\mathrm {H S}} (A). \\ \end{array}
$$
C Expression of the estimators
In this section we give proofs of propositions 3.1 to 3.3 on how to efficiently compute the Nyström estimators.
For all three - KRR, PCR and RRR - estimators, the starting point is their respective full estimator which can be derived by following the first-order optimality criterion for the following minimization problems
$$
\text {F u l l} \quad \hat {A} _ {\lambda} ^ {\mathrm {K R R}} = \underset {A \in \mathcal {H} \rightarrow \mathcal {H}} {\arg \min } \| \hat {\Phi} _ {Y | X} - A \hat {\Phi} _ {X} \| _ {\mathrm {H S}} ^ {2} + \lambda \| A \| _ {\mathrm {H S}} ^ {2} \tag {23}
$$
$$
\text {F u l l P C R :} \quad \hat {A} ^ {\mathrm {P C R}} = \underset {A \in \mathcal {H} \rightarrow \mathcal {H}} {\arg \min } \| \hat {\Phi} _ {Y | X} - A \Pi_ {r} \hat {\Phi} _ {X} \| _ {\mathrm {H S}} ^ {2} \tag {24}
$$
$$
\text {F u l l} \quad \hat {A} _ {\lambda} ^ {\text {R R R}} = \underset {A \in \mathcal {H} \rightarrow \mathcal {H}: \operatorname {r k} (A) \leq r} {\arg \min } \| \hat {\Phi} _ {Y | X} - A \hat {\Phi} _ {X} \| _ {\mathrm {H S}} ^ {2} + \lambda \| A \| _ {\mathrm {H S}} ^ {2} \tag {25}
$$
where  $\Pi_r$  is the orthogonal projection onto the top-r eigenvectors of  $\hat{C}$ .
To derive the Nyström estimators, we project the embedded data  $\hat{\Phi}_X$ ,  $\hat{\Phi}_{Y|X}$  onto the span of the embedded inducing points -  $P_{X}\hat{\Phi}_{X}$ ,  $P_{Y}\hat{\Phi}_{Y|X}$  - and then express the resulting estimators as  $\widetilde{\Phi}_YW\widetilde{\Phi}_X^*$  with  $W\in \mathbb{R}^{m\times m}$ . This form is particularly useful for later computing forecasts, eigenfunctions and Koopman modes with the estimator. In particular the following equalities for the projection (shown here for  $P_{X}$  but equivalently exist for  $P_Y$ )
$$
P _ {X} = P _ {X} P _ {X} = \widetilde {\Phi} _ {X} (\widetilde {\Phi} _ {X} ^ {*} \widetilde {\Phi} _ {X}) ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*} = \widetilde {\Phi} _ {X} ^ {* \dagger} \widetilde {\Phi} _ {X} ^ {*} = \widetilde {\Phi} _ {X} \widetilde {\Phi} _ {X} ^ {\dagger},
$$
and the characterization of  $P_{X}$  through the SVD of  $\widetilde{\Phi}_X = U\Sigma V^*$ , such that  $P_{X} = UU^{*}$ .
C.1 Nyström KRR
We begin with the Nyström KRR estimator, providing an alternative but equivalent description in lemma C.1.
Lemma C.1 (Expression of the KRR regularization): Let  $U$  be such that  $P_{X} = UU^{*}$ ,  $U^{*}U = I$ . Then it holds
$$
g _ {\mathrm {K R R}} (\hat {C}) := P _ {X} \left(P _ {X} \hat {C} P _ {X} + \lambda I\right) ^ {- 1} = U \left(U ^ {*} \hat {C} U + \lambda I\right) ^ {- 1} U ^ {*}. \tag {26}
$$
Proof of Lemma C.1: Using  $U^{*}U = I$ , it holds  $(U^{*}\hat{C} U + \lambda I)U^{*} = U^{*}(UU^{*}\hat{C}UU^{*} + \lambda I)$  and thus  $U^{*}(UU^{*}\hat{C}UU^{*} + \lambda I)^{-1} = (U^{*}\hat{C}U + \lambda I)^{-1}U^{*}$ . As a consequence,
$$
\begin{array}{l} g _ {\mathrm {K R R}} (\hat {C}) = P _ {X} \left(P _ {X} \hat {C} P _ {X} + \lambda I\right) ^ {- 1} \\ = U U ^ {*} \left(U U ^ {*} \hat {C} U U ^ {*} + \lambda I\right) ^ {- 1} \\ = U \left(U ^ {*} \hat {C} U + \lambda I\right) ^ {- 1} U ^ {*}. \\ \end{array}
$$
Then we can provide computatable formulas for Nyström KRR
Proposition C.2 (Nyström KRR): The Nyström KRR estimator, obtained by projection of eq. (23) is
$$
\begin{array}{l} \hat {A} _ {m, \lambda} ^ {K R R} = P _ {Y} \hat {C} _ {Y X} P _ {X} \left(P _ {X} \hat {C} P _ {X} + \lambda I\right) ^ {- 1} \\ = \widetilde {\Phi} _ {Y} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} \left(K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}\right) ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*}. \\ \end{array}
$$
Proof of Proposition C.2: Using the definition in eq. (26), and lemma C.1, we have
$$
\begin{array}{l} \hat {A} _ {m, \lambda} ^ {\mathrm {K R R}} = P _ {Y} \hat {C} _ {Y X} g _ {\mathrm {K R R}} (\hat {C}) \\ = P _ {Y} \hat {C} _ {Y X} U \left(U ^ {*} \hat {C} U + \lambda I\right) ^ {- 1} U ^ {*} \\ = P _ {Y} \hat {C} _ {Y X} U \Sigma V ^ {*} V \Sigma^ {- 1} (U ^ {*} \hat {C} U + \lambda I) ^ {- 1} \Sigma^ {- 1} V ^ {*} V \Sigma U ^ {*} \\ \end{array}
$$
Now using the fact that  $\Sigma, V, V^{*}$  and  $U^{*}\hat{C}U + \lambda I$  are full-rank, it holds [2, eq. (20)]
$$
\begin{array}{l} \hat {A} _ {m, \lambda} ^ {\mathrm {K R R}} = P _ {Y} \hat {C} _ {Y X} \widetilde {\Phi} _ {X} (V ^ {*}) ^ {\dagger} (\Sigma U ^ {*} \hat {C} U \Sigma + \lambda \Sigma^ {2}) ^ {\dagger} V ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*} \\ = P _ {Y} \hat {C} _ {Y X} \tilde {\Phi} _ {X} \left(V \Sigma U ^ {*} \hat {C} U \Sigma V ^ {*} + \lambda V \Sigma^ {2} V ^ {*}\right) ^ {\dagger} \tilde {\Phi} _ {X} ^ {*}. \\ \end{array}
$$
Finally, by definition of  $P_{Y}$ ,  $\hat{C}_{YX}$  and  $\hat{C}$ ,
$$
\begin{array}{l} \hat {A} _ {m, \lambda} ^ {\mathrm {K R R}} = \widetilde {\Phi} _ {Y} \left(\widetilde {\Phi} _ {Y} ^ {*} \widetilde {\Phi} _ {Y}\right) ^ {\dagger} \widetilde {\Phi} _ {Y} ^ {*} \hat {\Phi} _ {Y | X} \hat {\Phi} _ {X} ^ {*} \widetilde {\Phi} _ {X} \left(\widetilde {\Phi} _ {X} ^ {*} \hat {\Phi} _ {X} \hat {\Phi} _ {X} ^ {*} \widetilde {\Phi} _ {X} + n \lambda \widetilde {\Phi} _ {X} ^ {*} \widetilde {\Phi} _ {X}\right) ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*} \\ = \widetilde {\Phi} _ {Y} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} \left(K _ {X, \tilde {X}} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}\right) ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*}. \\ \end{array}
$$
Remark C.1 (Alternative derivation of the Nyström KRR estimator): Note that the Nyström KRR estimator can equivalently be derived as the solution to a variational problem similar to eq. (23), where the operator  $A$  is restricted to operate between spaces  $\mathcal{H}_{\widetilde{X}} \coloneqq \operatorname{span} \widetilde{\Phi}_X$  and  $\mathcal{H}_{\widetilde{Y}} \coloneqq \operatorname{span} \widetilde{\Phi}_Y$ .
C.2 Nyström PCR
Define the following filter on the spectrum of  $P_X\hat{C} P_X$ :  $g_{\mathrm{PCR}}(\hat{C}) = \llbracket P_X\hat{C} P_X\rrbracket_r^\dagger$ , which truncates it to the first  $r$  components before taking the pseudo-inverse. The Nyström PCR estimator, obtained by projection of eq. (24) is
$$
\hat {A} _ {m} ^ {\mathrm {P C R}} = P _ {Y} \hat {C} _ {Y X} g _ {\mathrm {P C R}} (\hat {C}). \tag {27}
$$
The next proposition provides an efficiently implementable version of the PCR estimator.
Proposition C.3 (Nyström PCR): The sketched PCR estimator eq. (27) satisfies
$$
\hat {A} _ {m} ^ {P C R} = \widetilde {\Phi} _ {Y} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} [   [ K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} K _ {X, \tilde {X}} ]   ] _ {r} \widetilde {\Phi} _ {X} ^ {*} \tag {28}
$$
Proof of Proposition C.3: We begin by computing the decomposition of  $P_X \hat{C} P_X$  which is necessary to obtain  $g_{\mathrm{PCR}}(\hat{C})$ . The following expressions are equivalent [43, Proposition 3] for determining its eigenvectors  $\tilde{h}$  and eigenvalues  $\lambda$ :
$$
U U ^ {*} \hat {C} U U ^ {*} \tilde {h} = \lambda \tilde {h}
$$
$$
U ^ {*} \hat {C} U h = \lambda h, \quad \tilde {h} = U h.
$$
Let the truncated eigenvalues be  $\Lambda_r = \mathrm{diag}[\lambda_1, \dots, \lambda_r]$  and the eigenvectors be  $H_r = [h_1, \dots, h_r]$ . Then  $\tilde{H}_r = U H_r$  must be normalized such that  $\tilde{H}_r^* \tilde{H}_r = H_r^* U^* U H_r = I$ . The rank- $\mathbf{r}$  truncation  $\llbracket P_X \hat{C} P_X \rrbracket_r$  is a projection onto  $\tilde{H}_r \tilde{H}_r^*$ :
$$
\llbracket P _ {X} \hat {C} P _ {X} \rrbracket_ {r} ^ {\dagger} = (U U ^ {*} \hat {C} U U ^ {*} (U H _ {r}) (U H _ {r}) ^ {*}) ^ {\dagger} = (U H \Lambda H ^ {*} H _ {r} H _ {r} ^ {*} U ^ {*}) ^ {\dagger} = U H _ {r} \Lambda_ {r} ^ {- 1} H _ {r} ^ {*} U ^ {*}
$$
where we used that  $U^{*}\hat{C}U = H\Lambda H^{*}$ .
Now substitute  $U = \widetilde{\Phi}_X V \Sigma^{-1}$  to simplify the eigendecomposition of  $U^* \hat{C} U$ :
$$
\Sigma^ {- 1} V ^ {*} K _ {\tilde {X}, X} K _ {X, \tilde {X}} V \Sigma^ {- 1} h = \lambda h
$$
$$
V \Sigma^ {- 2} V ^ {*} K _ {\tilde {X}, X} K _ {X, \tilde {X}} d = \lambda d, \quad h = \Sigma^ {- 1} V ^ {*} K _ {\tilde {X}, X} K _ {X, \tilde {X}} d. \tag {29}
$$
where  $V\Sigma^{-2}V^{*} = K_{\tilde{X},\tilde{X}}^{\dagger}$ . Denote by  $D_r = [d_1,\dots ,d_r]$  the truncated eigenvectors such that  $H_{r} = \Sigma^{-1}V^{*}K_{\tilde{X},X}K_{X,\tilde{X}}D_{r}$ , normalized such that  $H^{*}H = D^{*}K_{\tilde{X},X}K_{X,\tilde{X}}K_{\tilde{X},\tilde{X}}^{\dagger}K_{\tilde{X},X}K_{X,\tilde{X}}D = I$
$$
\begin{array}{l} U H _ {r} \Lambda_ {r} ^ {- 1} H _ {r} ^ {*} U ^ {*} = \widetilde {\Phi} _ {X} K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} K _ {X, \tilde {X}} D _ {r} \Lambda_ {r} ^ {- 1} D _ {r} ^ {*} K _ {\tilde {X}, X} K _ {X, \tilde {X}} K _ {\tilde {X}, \tilde {X}} ^ {\dagger} \widetilde {\Phi} _ {X} ^ {*} \\ = \widetilde {\Phi} _ {X} D _ {r} \Lambda_ {r} D _ {r} ^ {*} \widetilde {\Phi} _ {X} ^ {*} \\ = \widetilde {\Phi} _ {X} [   [ K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} K _ {X, \tilde {X}} ]   ] _ {r} \widetilde {\Phi} _ {X} ^ {*}. \\ \end{array}
$$
Finally, we can plug the pieces together to get
$$
P _ {Y} \hat {C} _ {Y X} [ [ P _ {X} \hat {C} P _ {X} ] ] _ {r} ^ {\dagger} = \widetilde {\Phi} _ {Y} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} [ [ K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} K _ {X, \tilde {X}} ] ] _ {r} \widetilde {\Phi} _ {X} ^ {*}.
$$
Remark C.2 (Variational problem for Nyström PCR): Note that, unlike the NysKRR estimator, the variational problem for NysPCR where the operator is restricted to  $A: \mathcal{H}_{\widetilde{X}} \to \mathcal{H}_{\widetilde{Y}}$  is not equivalent to the one obtained in proposition C.3 by projecting the covariance operator. In fact, the former does not take the full covariance into account when computing the low-rank projection, but just the Nyström points.
C.3 Nyström RRR
The Nyström RRR estimator does not correspond to a specific spectral filter. We can nonetheless compute it starting from the expression of the exact empirical estimator [29], projecting the covariance operators, and rearranging the expression to result in a finite-dimensional procedure.
Proposition C.4 (Nyström RRR): The sketched RRR estimator can be written as
$$
\hat {A} _ {m, \lambda} ^ {R R R} = \llbracket P _ {Y} \hat {C} _ {Y X} P _ {X} \left(P _ {X} \hat {C} P _ {X} + \lambda I\right) ^ {- 1 / 2} \rrbracket_ {r} \left(P _ {X} \hat {C} P _ {X} + \lambda I\right) ^ {- 1 / 2}. \tag {30}
$$
$$
(K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}) ^ {\dagger} K _ {\tilde {X}, X} K _ {Y, \tilde {Y}} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} w _ {i} = \sigma_ {i} ^ {2} w _ {i}
$$
for the first  $r$  eigenvectors  $W_{r} = [w_{1},\ldots ,w_{r}]$ , normalized such that  $W_{r}^{*}K_{\tilde{X},X}K_{Y,\tilde{Y}}K_{\tilde{Y},Y}^{\dagger}K_{\tilde{Y},Y}K_{X,\tilde{X}}W_{r} = I$ . Then let  $D_{r} := K_{Y,\tilde{Y}}^{\dagger}K_{\tilde{Y},Y}K_{X,\tilde{X}}W_{r}$  and  $E_{r} := (K_{\tilde{X},X}K_{X,\tilde{X}} + n\lambda K_{\tilde{X},\tilde{X}})^{\dagger}K_{\tilde{X},X}K_{Y,\tilde{Y}}U_{r}$ , such that the following holds
$$
\hat {A} _ {m, \lambda} ^ {R R R} = \widetilde {\Phi} _ {Y} D _ {r} E _ {r} ^ {*} \widetilde {\Phi} _ {X} ^ {*}. \tag {31}
$$
Proof of Proposition C.4: Let  $B \coloneqq n^{1/2} P_Y \hat{C}_{YX} P_X (P_X \hat{C} P_X + \lambda I)^{-1/2}$ . The computationally intensive part for this estimator is in evaluating the rank-r truncation  $[[B]]_r$ . Its singular values and left singular vectors can be obtained by solving the symmetric eigenvalue problem  $BB^* q_i = \sigma_i^2 q_i$ . We rewrite  $BB^*$
$$
\begin{array}{l} B B ^ {*} = P _ {Y} \hat {\Phi} _ {Y | X} \hat {\Phi} _ {X} ^ {*} P _ {X} (P _ {X} \hat {\Phi} _ {X} \hat {\Phi} _ {X} ^ {*} P _ {X} + n \lambda I) ^ {- 1} P _ {X} \hat {\Phi} _ {X} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \\ = P _ {Y} \hat {\Phi} _ {Y | X} \hat {\Phi} _ {X} ^ {*} P _ {X} \hat {\Phi} _ {X} (\hat {\Phi} _ {X} ^ {*} P _ {X} \hat {\Phi} _ {X} + n \lambda I) ^ {- 1} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \\ = P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} (K _ {X, \tilde {X}} K _ {\tilde {X}, \tilde {X}} ^ {\dagger} K _ {\tilde {X}, X} + n \lambda I) ^ {- 1} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \\ = P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} K _ {\tilde {X}, \tilde {X}} ^ {\dagger} (K _ {\tilde {X}, X} K _ {X, \tilde {X}} K _ {\tilde {X}, \tilde {X}} + n \lambda I) ^ {- 1} K _ {\tilde {X}, X} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \\ = P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} (K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}) ^ {\dagger} K _ {\tilde {X}, X} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \\ \end{array}
$$
where the second and fourth equalities are applications of the push-through identity, the third by definition of projections and kernel matrices, and the last by collecting  $K_{\tilde{X},\tilde{X}}$ . By construction, the non-trivial eigenfunctions of  $BB^{*}$  are in the range of  $P_{Y}\hat{\Phi}_{Y|X}K_{X,\tilde{X}}$ , therefore we can set  $q_{i} = P_{Y}\hat{\Phi}_{Y|X}K_{X,\tilde{X}}w_{i}$  for some  $w_{i}\in \mathbb{R}^{m}$ , and solve the following eigenvalue problem instead
$$
\begin{array}{l} P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} (K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}) ^ {\dagger} K _ {\tilde {X}, X} \hat {\Phi} _ {Y | X} ^ {*} P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} w _ {i} = \sigma_ {i} ^ {2} P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} w _ {i} \\ (K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}) ^ {\dagger} K _ {\tilde {X}, X} K _ {Y, \tilde {Y}} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} w _ {i} = \sigma_ {i} ^ {2} w _ {i} \\ \end{array}
$$
where we have simplified the left term of both sides of the equation.
The eigenfunctions of  $BB^{*}$  are therefore  $q_{i} = P_{Y}\hat{\Phi}_{Y|X}K_{X,\bar{X}}w_{i}$ , which must be normalized as
$$
\| q _ {i} \| ^ {2} = w _ {i} ^ {\top} K _ {\tilde {X}, X} K _ {Y, \tilde {Y}} K _ {\tilde {Y}, \tilde {Y}} ^ {\dagger} K _ {\tilde {Y}, Y} K _ {X, \tilde {X}} w _ {i} = 1.
$$
Thanks to this normalization, the projector onto the  $r$  leading left singular vectors of  $B$  is  $Q_{r}Q_{r}^{*}$  where  $Q_{r} = [q_{1},\ldots ,q_{r}]$ . Then the NysRRR estimator can be written as
$$
Q _ {r} Q _ {r} ^ {*} B (P _ {X} \hat {C} P _ {X} + \lambda I) ^ {- 1 / 2}
$$
$$
\begin{array}{l} B (P _ {X} \hat {C} P _ {X} + \lambda I) ^ {- 1 / 2} = P _ {Y} \hat {C} _ {Y X} P _ {X} (P _ {X} \hat {C} P _ {X} + \lambda I) ^ {- 1} \\ = P _ {Y} \hat {\Phi} _ {Y | X} K _ {X, \tilde {X}} (K _ {\tilde {X}, X} K _ {X, \tilde {X}} + n \lambda K _ {\tilde {X}, \tilde {X}}) ^ {- 1} \widetilde {\Phi} _ {X} ^ {*}. \\ \end{array}
$$
with the same techniques we used for rewriting  $BB^{*}$ . Finally, let  $D_{r}$  and  $E_{r}$  as in the statement. We can apply the projection to obtain
$$
Q _ {r} Q _ {r} ^ {*} B \left(P _ {X} \hat {\Phi} _ {X} \hat {\Phi} _ {X} ^ {*} P _ {X} + n \lambda I\right) ^ {- 1 / 2} = \widetilde {\Phi} _ {Y} D _ {r} E _ {r} ^ {*} \widetilde {\Phi} _ {X} ^ {*}.
$$
D Forecasting & Koopman Modes
The three estimators considered in Section C are all of the form
$$
\hat {A} _ {\lambda} = \widetilde {\Phi} _ {Y} W \widetilde {\Phi} _ {X} ^ {*}, \qquad W \in \mathbb {R} ^ {m \times m}.
$$
We will use this generic form to provide expressions for the following operations:
1. producing forecasts of the dynamical system at a future time,
2. computing the approximate eigenvalues and eigenfunctions of the Koopman operator,
3. computing the Koopman modes.
D.1 Forecasting
Given a new data-point  $x \in \mathcal{X}$  and an observable function  $g \in \mathcal{H}$  (note that this can simply be the identity function), we can approximate the one-step-ahead expectation  $\mathbf{E}\big[g(X_{t + 1})|X_t = x\big] =$
$(\mathcal{K}_{\pi}g)(\pmb {x})$  by using the obtained estimators  $\hat{A}^*$  . Note that by the reproducing property  $\widetilde{\Phi}_Y^* g = [g(\pmb {y}_i),\dots ,g(\pmb {y}_m)]^\top =: g_m$  , then
$$
(\hat {A} ^ {*} g) (x) = (\widetilde {\Phi} _ {X} W ^ {\top} \widetilde {\Phi} _ {Y} ^ {*} g) ({\pmb x}) = (\widetilde {\Phi} _ {X} W ^ {\top} g _ {m}) ({\pmb x}) = \sum_ {i = 1} ^ {m} (W ^ {\top} g _ {m}) _ {i} k (\tilde {{\pmb x}} _ {i}, {\pmb x}).
$$
D.2 Eigenfunctions and eigenvalues