SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\sup  _ {f: | f | \leq U, \operatorname {L i p} (f) \leq \ell} \left| \mathbb {E} \left[ \frac {1}{N + 1} \sum_ {n = 0} ^ {N} f \left(X _ {n}\right) - f \left(S _ {n}\right) \right] \right| \leq \mathrm {D} _ {\mathrm {B L}} (Y, W) + \mathrm {D} _ {\mathrm {B L}} (Z, W).
$$
By [5, Theorem 4.1], both terms on the right-hand side converge to 0 as  $N\to \infty$
Lemma A.1. If  $\forall i\in [n],a_i,b_i > 0$  , then
$$
\left| \log \frac {a _ {1} + \cdots + a _ {n}}{b _ {1} + \cdots + b _ {n}} \right| \leq \max  _ {1 \leq i \leq n} \left\{\left| \log \frac {a _ {i}}{b _ {i}} \right| \right\}
$$
Proof. Define  $A \coloneqq \sum_{i=1}^{n} a_i$ ,
$$
\log \frac {a _ {1} + \cdots + a _ {n}}{b _ {1} + \cdots + b _ {n}} = - \log \frac {b _ {1} + \cdots + b _ {n}}{a _ {1} + \cdots + a _ {n}} = - \log \sum_ {i = 1} ^ {n} \frac {a _ {i}}{A} \frac {b _ {i}}{a _ {i}}
$$
Jensen's inequality yields that
$$
\log \frac {a _ {1} + \cdots + a _ {n}}{b _ {1} + \cdots + b _ {n}} \leq \sum_ {i = 1} ^ {n} \frac {a _ {i}}{A} \log \frac {a _ {i}}{b _ {i}} \leq \max  _ {1 \leq i \leq n} \left\{\log \frac {a _ {i}}{b _ {i}} \right\}.
$$
Applying the same technique to the other direction yields the result.
Proof of Theorem 4.6. First, shadowing and the triangle inequality yields
$$
\begin{array}{l} | \log \hat {q} (x) - \log q (x) | \leq | \log q (x) - \log q (s) | + | \log \hat {q} (x) - \log q (s) | \\ \leq \epsilon L _ {q, \epsilon} (x) + | \log \hat {q} (x) - \log q (s) |. \\ \end{array}
$$
By Lemma A.1, we have
$$
\begin{array}{l} \left| \log \hat {q} (x) - \log q (s) \right| \leq \max  _ {0 \leq n \leq N} \left| \log \frac {q _ {0} (\hat {x} _ {- n}) / \prod_ {j = 1} ^ {n} J (\hat {x} _ {- j})}{q _ {0} (s _ {- n}) / \prod_ {j = 1} ^ {n} J (s _ {- j})} \right| \\ = \max  _ {0 \leq n \leq N} | \log \hat {q} _ {n} (x) - \log q _ {n} (s) |, \\ \end{array}
$$
where  $q_{n}$  is the density of the length-  $n$  normalizing flow with constant forward flow map  $F$ , and  $\hat{q}_n$  is its approximation via the pseudo-orbit. Using the same technique as in the proof of Theorem 4.3,
$$
\begin{array}{l} \max  _ {0 \leq n \leq N} | \log \hat {q} _ {n} (x) - \log q _ {n} (s) | \leq \max  _ {0 \leq n \leq N} \epsilon \cdot \left(L _ {q _ {0}, \epsilon} (\hat {x} _ {- n}) + \sum_ {j = 1} ^ {n} L _ {J, \epsilon} (\hat {x} _ {- j})\right) \\ \leq \epsilon \cdot \left(\max  _ {0 \leq n \leq N} L _ {q _ {0}, \epsilon} (\hat {x} _ {- n}) + \sum_ {n = 1} ^ {N} L _ {J, \epsilon} (\hat {x} _ {- n})\right). \\ \end{array}
$$
Combining this with the earlier bound yields the first stated result. To obtain the second result in the infinite shadowing setting, we repeat the process of bounding  $\max_{0\leq n\leq N}|\log \hat{q}_n(x) - \log q_n(s)|$ , but rather than expressing each supremum around the pseudo-orbit  $\hat{x}_{-n}$ , we express it around the shadowing orbit  $s_{-n}$  to find that
$$
\max  _ {0 \leq n \leq N} | \log \hat {q} _ {n} (x) - \log q _ {n} (s) | \leq \epsilon \cdot \left(\max  _ {0 \leq n \leq N} L _ {q _ {0}, \epsilon} (s _ {- n}) + \sum_ {n = 1} ^ {N} L _ {J, \epsilon} (s _ {- n})\right).
$$
We then bound  $\max_{0\leq n\leq N}$  with a sum and merge this with the first bound to find that
$$
\limsup_{N\to \infty}\frac{1}{N} |\log \hat{q} (x) - \log q(x)|\leq \epsilon \cdot \limsup_{N\to \infty}\left(\frac{1}{N} L_{q,\epsilon}(x) + \frac{1}{N}\sum_{n = 0}^{N}L_{q_{0},\epsilon}(s_{-n}) + \frac{1}{N}\sum_{n = 1}^{N}L_{J,\epsilon}(s_{-n})\right).
$$
Since  $L_{q,\epsilon}(x) = o(N)$  as  $N\to \infty$  , the first term decays to 0. The latter two terms are ergodic averages under the backward dynamics initialized at  $s$  ; if the pointwise ergodic theorem [42], [43, p. 212] holds at  $s$  , then the result follows. However, the pointwise ergodic theorem holds only  $\pi$  -almost surely for each of  $L_{q_0,\epsilon}$  and  $L_{J,\epsilon}$  . Denote  $\mathcal{Z}$  to be the set of  $\pi$  -measure zero for which the theorem does not apply. Suppose there is a set  $\mathcal{A}\subseteq \mathcal{X}$  such that  $q_{0}(\mathcal{A}) > 0$  , and  $x\in \mathcal{A}$  implies that  $s\in \mathcal{Z}$  . But then  $\xi_0(\mathcal{Z}) > 0$  , which contradicts  $\xi_0\ll \pi$  Therefore the pointwise ergodic theorem applies to  $s$  for  $q_{0}$  -almost every  $x\in \mathcal{X}$
Proof of Theorem 4.7. By definition,  $\operatorname{ELBO}(q||p) = \mathbb{E}[E(X)]$ , and by hypothesis the distribution  $q_0$  of the initial flow state is equal to the distribution  $\xi_0$  of the joint shadowing orbit initial state, so  $\mathbb{E}[E(X)] = \mathbb{E}[E(S)]$ . Hence  $\left|\operatorname{ELBO}(q||p) - \mathbb{E}\left[\hat{E}(X)\right]\right| \leq \mathbb{E}\left[\left|E(S) - \hat{E}(X)\right|\right]$ . So applying the triangle inequality, we find that
$$
\left| E (s) - \hat {E} (x) \right| \leq \left| \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \log \frac {p (s _ {n})}{p (\hat {x} _ {n})} \right| + \left| \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \log \frac {\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac {q _ {0} (\hat {x} _ {n - j})}{\prod_ {i = 1} ^ {j} J (\hat {x} _ {n - i})}}{\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac {q _ {0} (s _ {n - j})}{\prod_ {i = 1} ^ {j} J (s _ {n - i})}} \right|
$$
For the first sum, each term is bounded using the local Lipschitz constant,
$$
\begin{array}{l} \left| \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \log \frac {p \left(s _ {n}\right)}{p \left(\hat {x} _ {n}\right)} \right| \leq \frac {1}{N + 1} \sum_ {n = 0} ^ {N} | \log p \left(s _ {n}\right) - \log p \left(\hat {x} _ {n}\right) | \\ \leq \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \epsilon L _ {p, \epsilon} (\hat {x} _ {n}). \\ \end{array}
$$
For the second, we apply Lemma A.1 and then use the local Lipschitz constants,
$$
\begin{array}{l} \left| \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \log \frac {\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac {q _ {0} (\hat {x} _ {n - j})}{\prod_ {i = 1} ^ {j} J (\hat {x} _ {n - i})}}{\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac {q _ {0} (s _ {n - j})}{\prod_ {i = 1} ^ {j} J (s _ {n - i})}} \right| \leq \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \left| \log \frac {\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac {q _ {0} (\hat {x} _ {n - j})}{\prod_ {i = 1} ^ {j} J (\hat {x} _ {n - i})}}{\frac {1}{N + 1} \sum_ {j = 0} ^ {N} \frac { q _ {0} (s _ {n - j})}{\prod_ {i = 1} ^ {j} J (s _ {n - i})}} \right| \\ \leq \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \max _ {0 \leq j \leq N} \left| \log \frac {\frac {q _ {0} (\hat {x} _ {n - j})}{\prod_ {i = 1} ^ {j} J (\hat {x} _ {n - i})}}{\frac {q _ {0} (s _ {n - j})}{\prod_ {i = 1} ^ {j} J (s _ {n - i})}} \right| \\ \leq \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \max _ {0 \leq j \leq N} | \log q _ {0} (\hat {x} _ {n - j}) - \log q _ {0} (s _ {n - j}) | + \sum_ {i = 1} ^ {j} | \log J (\hat {x} _ {n - i}) - \log J (s _ {n - i}) | \\ \leq \epsilon \left(\frac {1}{N + 1} \sum_ {n = 0} ^ {N} \max  _ {0 \leq j \leq N} L _ {q _ {0}, \epsilon} (\hat {x} _ {n - j}) + \sum_ {i = 1} ^ {j} L _ {J, \epsilon} (\hat {x} _ {n - i})\right) \\ \leq \epsilon \left(\max  _ {- N \leq n \leq N} L _ {q _ {0}, \epsilon} (\hat {x} _ {n}) + \frac {1}{N + 1} \sum_ {n = 0} ^ {N} \sum_ {i = 1} ^ {N} L _ {J, \epsilon} (\hat {x} _ {n - i})\right) \\ \leq \epsilon \left(\max  _ {- N \leq n \leq N} L _ {q _ {0}, \epsilon} (\hat {x} _ {n}) + \sum_ {n = - N} ^ {N} L _ {J, \epsilon} (\hat {x} _ {n})\right). \\ \end{array}
$$
Combining this with the previous bound and taking the expectation yields the first result,
$$
\left| \operatorname {E L B O} (q | | p) - \mathbb {E} \left[ \hat {E} (X) \right] \right| \leq \epsilon \cdot \mathbb {E} \left[ \frac {1}{N + 1} \sum_ {n = 0} ^ {N} L _ {p, \epsilon} (\hat {X} _ {n}) + \max  _ {- N \leq n \leq N} L _ {q _ {0}, \epsilon} (\hat {X} _ {n}) + \sum_ {n = - N} ^ {N} L _ {J, \epsilon} (\hat {X} _ {n}) \right].
$$
Once again to arrive at the second result, we redo the analysis but center the supremum Lipschitz constants around the shadowing orbit points, and replace the maximum with a sum, yielding the bound
$$
\begin{array}{l} \left| \operatorname {E L B O} (q | | p) - \mathbb {E} \left[ \hat {E} (X) \right] \right| \leq \epsilon \cdot \mathbb {E} \left[ \frac {1}{N + 1} \sum_ {n = 0} ^ {N} L _ {p, \epsilon} \left(S _ {n}\right) + \max  _ {- N \leq n \leq N} L _ {q _ {0}, \epsilon} \left(S _ {n}\right) + \sum_ {n = - N} ^ {N} L _ {J, \epsilon} \left(S _ {n}\right) \right] \\ \leq \epsilon \cdot \mathbb {E} \left[ \frac {1}{N + 1} \sum_ {n = 0} ^ {N} L _ {p, \epsilon} (S _ {n}) + \sum_ {n = - N} ^ {N} L _ {q _ {0}, \epsilon} (S _ {n}) + \sum_ {n = - N} ^ {N} L _ {J, \epsilon} (S _ {n}) \right]. \\ \end{array}
$$
Now since  $\xi_0 = q_0$  by assumption,  $S_{n}\stackrel {d}{=}X_{n}$  for each  $n\in \mathbb{N}$ , so
$$
\left| \operatorname {E L B O} (q \| p) - \mathbb {E} \left[ \hat {E} (X) \right] \right| \leq \epsilon \cdot \mathbb {E} \left[ \frac {1}{N + 1} \sum_ {n = 0} ^ {N} L _ {p, \epsilon} (X _ {n}) + \sum_ {n = - N} ^ {N} L _ {q _ {0}, \epsilon} (X _ {n}) + \sum_ {n = - N} ^ {N} L _ {J, \epsilon} (X _ {n}) \right].
$$
We divide by  $N$  and take the limit supremum:
$$
\begin{array}{l} \limsup_{N\to \infty}\frac{1}{N}\left|\operatorname {ELBO}\left(q\| p\right) - \mathbb{E}\left[\hat{E} (X)\right]\right| \\ \leq \epsilon \lim  _ {N \rightarrow \infty} \sup  _ {\epsilon} \cdot \mathbb {E} \left[ \frac {1}{N (N + 1)} \sum_ {n = 0} ^ {N} L _ {p, \epsilon} (X _ {n}) + \frac {1}{N} \sum_ {n = - N} ^ {N} L _ {q _ {0}, \epsilon} (X _ {n}) + \frac {1}{N} \sum_ {n = - N} ^ {N} L _ {J, \epsilon} (X _ {n}) \right]. \\ \end{array}
$$
Consider just the term
$$
\mathbb {E} \left[ \frac {1}{N} \sum_ {n = 0} ^ {N} L _ {J, \epsilon} \left(X _ {n}\right) \right] = \int \frac {1}{N} \sum_ {n = 0} ^ {N} L _ {J, \epsilon} \left(F ^ {n} x\right) q _ {0} (\mathrm {d} x) = \int \frac {1}{N} \sum_ {n = 0} ^ {N} L _ {J, \epsilon} \left(F ^ {n} x\right) \frac {\mathrm {d} q _ {0}}{\mathrm {d} \pi} (x) \pi (\mathrm {d} x).
$$
[43, Theorem 8.10 (vi)] asserts that as long as  $L_{J,\epsilon} \in L^{m_1}(\pi)$  and  $\frac{\mathrm{d}q_0}{\mathrm{d}\pi} \in L^{m_2}(\pi)$  for  $1/m_1 + 1/m_2 = 1$ ,  $1 \leq m_1 < \infty$ , then
$$
\lim  _ {N \rightarrow \infty} \int \frac {1}{N} \sum_ {n = 0} ^ {N} L _ {J, \epsilon} (F ^ {n} x) \frac {\mathrm {d} q _ {0}}{\mathrm {d} \pi} (x) \pi (\mathrm {d} x) = \int L _ {J, \epsilon} (x) \pi (\mathrm {d} x) \int \frac {\mathrm {d} q _ {0}}{\mathrm {d} \pi} \pi (\mathrm {d} x) = \mathbb {E} [ L _ {J, \epsilon} (X) ], X \sim \pi .
$$
Applying this result to each term above yields the stated result.
A.4 Norm of matrix right inverse
Notice that the operator  $A$  in Theorem 5.1 has a following matrix representation:
$$
A = \left[ \begin{array}{c c c c c} - D _ {1} & I & & & \\ & - D _ {2} & I & & \\ & & \ddots & \ddots & \\ & & & - D _ {N} & I \end{array} \right] \in \mathbb {R} ^ {d N \times d (N + 1)}, \text {w h e r e} D _ {k} = \nabla F _ {k} (\hat {x} _ {k - 1}), \forall k \in [ N ]. \tag {8}
$$
Proposition A.2 ([33, Corollary 4.2]). Let  $A$  be as defined in Eq. (8). Suppose  $(F_{n})_{n = 1}^{N}$  are all differentiable. Then
$$
A ^ {\dagger} := A ^ {T} \left(A A ^ {T}\right) ^ {- 1} \in \arg \min  _ {X} \| X \|, \quad \text {s u b j e c t} A X = I, \tag {9}
$$
and  $\| A^{\dagger}\| = \sigma_{\min}(A)^{-1}$ , where  $\sigma_{\min}(A)$  denotes the smallest singular value of  $A$ .
Proof of Proposition A.2. Note that  $A$  has full row rank due to the identity blocks. In this case,  $AA^T$  is invertible, thus  $A^T (AA^T)^{-1}$  is a valid right inverse of  $A$ . Eq. (9) is then obtained by a direct application of [33, Corollary 4.2].
To obtain the norm of  $A^\dagger$ , since  $A$  has full row rank,
$$
A = U \Sigma V ^ {T}, \quad \text {w h e r e} U \in \mathbb {R} ^ {d N \times d N}, V \in \mathbb {R} ^ {d (N + 1) \times d (N + 1)} \text {a r e o r t h o n o r m a l m a t r i c e s ,}
$$
$$
\Sigma \in \mathbb {R} ^ {d N \times d (N + 1)} \text {i s a r e c t a n g u l a r d i a g o n a l m a t r i x w i t h f u l l r o w r a n k}.
$$
$$
\begin{array}{l} \| A ^ {\dagger} \| = \left\| V \Sigma^ {T} U ^ {T} \left(U \Sigma V ^ {T} V \Sigma^ {T} U ^ {T}\right) ^ {- 1} \right\| \\ = \left\| V \Sigma^ {T} U ^ {T} U \left(\Sigma \Sigma^ {T}\right) ^ {- 1} U ^ {T} \right\| \\ = \left\| \Sigma \left(\Sigma \Sigma^ {T}\right) ^ {- 1} \right\| \\ = \sigma_ {\min } (A) ^ {- 1} \\ \end{array}
$$
Notice that since  $AA^T$  is an invertible Hermitian matrix,  $\sigma_{\mathrm{min}}(A)$  is strictly positive.
B Discussion about the shadowing window  $\epsilon$
B.1 Numerical error when computing  $\epsilon$
As mentioned in Section 5, due to the floating-point error  $\delta$  involved in evaluating  $\nabla F_{k}$ , the evaluation of  $A$  is perturbed by the numerical error. This consequently leads to an inaccuracy of the estimation of  $\lambda = \| A_r^{-1}\|$ . Here  $A_r^{-1}$  denotes the right inverse of  $A$ . In this section, we explain how to calibrate the calculation of  $\lambda$  to take into account of this error.
This computational challenge, rooted in numerical inaccuracies, was previously acknowledged in Coomes et al. [17, Section 3.4], with a provided remedy. Suppose we compute terms in  $\nabla F_k(\hat{x}_{k-1})$  with a floating-point error  $\delta$ . This introduces  $O(\sqrt{N}\delta)$  Frobenius norm error into the practically calculated  $A$ . Here we denote the digital computed  $A$  by  $\tilde{A}$ ,  $\tilde{\lambda} \coloneqq \|\tilde{A}_r^{-1}\|$ , and  $\tilde{\delta} \coloneqq \|A - \tilde{A}\|$ . According to Coomes et al. [17, Eq. 45], we can employ the following upper bound on  $\lambda$ :
$$
\lambda \leq (1 - \tilde {\delta} \tilde {\lambda}) ^ {- 1} \tilde {\lambda} = \left(1 - O (\sqrt {N \delta}) \tilde {\lambda}\right) ^ {- 1} \tilde {\lambda}.
$$
Substituting values of  $\delta$ ,  $N$ , and computed  $\tilde{\lambda}$  from our experiments (refer to Fig. 7), this amounts to an inconsequential relative error of around  $(1 - 10^{-9})^{-1} \approx 1$  on  $\lambda$ .
It is also worth discussing the shadowing window computation methods proposed in Coomes et al. [17, Section 3.4]. As opposed to our generic eigenvalue computation introduced in Section 5, the computation procedure presented in Coomes et al. [17, Section 3.4] can potentially offer better scalability. However, the methodology in Coomes et al. [17] operates under the heuristic that a dynamical system with shadowing is intrinsically hyperbolic, or nearly so. Specifically, the "hyperbolic splitting" discussed in Coomes et al. [44, P.413, Appendix B] (i.e., the choice of  $\ell$ ), and the "hyperbolic threshold" choice in Coomes et al. [44, P.415, Appendix B] (i.e., the choice of  $p$ ), both implicitly assume that the dynamics either exhibit hyperbolicity or tend closely to it. Such presuppositions might be overly restrictive for generic variational flows, given that the underlying dynamics are general time-inhomogeneous systems. When the underlying dynamics are not (nearly) hyperbolic, applying strategies from Coomes et al. [17] could potentially lead to a poor estimation of the shadowing window.
B.2 Dependence of  $\epsilon$  on  $N$
One may notice that  $\epsilon$  depends on  $N$  implicitly through  $\lambda_{\mathrm{min}}(AA^T)^{-1/2}$ . Recall that  $A$  can be represented as a matrix
$$
A = \left[ \begin{array}{c c c c c} - D _ {1} & I & & & \\ & - D _ {2} & I & & \\ & & \ddots & \ddots & \\ & & & - D _ {N} & I \end{array} \right] \in \mathbb {R} ^ {d N \times d (N + 1)}, \text {w h e r e} D _ {k} = \nabla F _ {k} (\hat {x} _ {k - 1}), \forall k \in [ N ].
$$
And the shadowing window is given by  $\epsilon = 2\delta \lambda_{\mathrm{min}}(AA^T)^{-1 / 2}$ . If  $\lambda_{\mathrm{min}}(AA^T)^{-1 / 2}$  scales badly with  $N$ , our  $O(N\epsilon)$  error bounds become vacuous. Therefore, in this section, we collect some insights about the scaling of  $\epsilon$  over increasing  $N$ . It is important to note that understanding the explicit quantitative relationship between  $\epsilon$  and  $N$  for general nonautonomous dynamical systems is quite challenging. Although current literature of dynamical systems focuses on studying conditions so that  $\epsilon$  does not scale with  $N$ , these conditions are typically difficult to verify in practice or are only applicable in restrictive settings such as time-homogeneous linear dynamical systems. For a more systematic study, see [17, 20, 28]. The examples provided below merely serve as synthetic illustrations for intuition and should not be taken as definitive statements for all cases.
Example B.1 (Scaling map in  $\mathbb{R}$ ). Consider the time-homogeneous dynamics formed by  $T: \mathbb{R} \mapsto \mathbb{R}$  such that  $\forall x \in \mathbb{R}, T(x) = Cx$  for some  $C > 0$ . Then the corresponding shadowing window of a  $\delta$ -pseudo-orbit is
$$
\epsilon = 2 \delta \left[ (C - 1) ^ {2} + 2 C \left(1 - \cos \frac {\pi}{N + 1}\right) \right] ^ {- \frac {1}{2}}.
$$
Therefore, for all  $C \neq 1$ , as  $N \to \infty$ ,  $\epsilon = O(\delta)$ . For  $C = 1$ ,  $\epsilon = O(N\delta)$ . The formula for  $\lambda_{\min}(AA^T)$  above arises because in this case,  $AA^T$  has the following symmetric tridiagonal form,
$$
A A ^ {T} = \left[ \begin{array}{c c c c c} C ^ {2} + 1 & - C & & & \\ - C & C ^ {2} + 1 & - C & & \\ & \ddots & \ddots & \ddots & \\ & & - C & C ^ {2} + 1 & - C \\ & & & - C & C ^ {2} + 1 \end{array} \right],
$$
whose eigenvalues have the following closed form [45, Page 19-20]:
$$
\lambda_ {k} \left(A A ^ {T}\right) = C ^ {2} + 1 - 2 C \cos \frac {k \pi}{N + 1}, \quad k = 1, \dots , N.
$$
Example B.2 (Uniform hyperbolic linear map[46, Theorem 12.3 and p. 171]). Consider the time-homogeneous linear dynamical system in  $\mathbb{R}^d$ ,  $T: x \mapsto Ax$ , where  $A \in \mathbb{R}^{d \times d}$  is similar to a block-diagonal matrix  $\mathrm{diag}(A_1, A_2)$  such that  $\|A_1\| \leq \lambda$  and  $\|A_2^{-1}\| \leq \lambda$  for some  $\lambda \in (0,1)$ . Then for all  $N \in \mathbb{N}$ , any  $\delta$ -pseudo-orbit is  $\epsilon$ -shadowed, with  $\epsilon = \frac{1 + \lambda}{1 - \lambda} \delta$ .
As illustrated in Examples B.1 and B.2, it is possible for the  $\epsilon$ -shadowing property to hold in a way that  $\epsilon$  does not depend on the flow length  $N$  and remains in a similar magnitude as  $\delta$ . This phenomenon is referred to as the infinite shadowing property [17, 19, 32] as mentioned in the beginning of Section 4.3. The infinite shadowing property is shown to be generic in the space of all dynamical systems [32], meaning that dynamical systems exhibiting this property form a dense subset of the collection of all dynamical systems. However, while genericity offers valuable insights into the typical behavior of dynamical systems, it does not imply that all systems possess the property. As previously explained, only a small class of systems has been shown to have the infinite shadowing property, and verifying this property for a specific given system remains a challenging task.
C Normalizing flow via Hamiltonian dynamics
While past work on normalizing flows [2, 3, 24-26] typically builds a flexible flow family by composing numerous parametric flow maps that are agnostic to the target distribution, recent target-informed variational flows focus on designing architectures that take into account the structures of the target distribution. A common approach for constructing such target-informed flows is to use Langevin and Hamiltonian dynamics as part of the flow transformations [6, 7, 47-50]. In this section, we begin with a concise overview of two Hamiltonian-based normalizing flows: Hamiltonian variational auto-encoder (HVAE) [7] and Hamiltonian flow (HamFlow) [6]. Subsequently, we present corollaries derived from Theorems 4.3 and 4.4, specifically applying these general results to HVAE and HamFlow. Finally, we offer empirical evaluations of our theory for HamFlow on the same synthetic examples as those examined for MixFlow.
Hamiltonian dynamics (Eq. (10)) describes the evolution of a particle's position  $\theta_{t}$  and momentum  $\rho_{t}$  within a physical system. This movement is driven by a differentiable negative potential energy  $\log p(\theta_t)$  and kinetic energy  $\frac{1}{2}\rho_t^T\rho_t$ :
$$
\frac {\mathrm {d} \rho_ {t}}{\mathrm {d} t} = \nabla \log \pi (\theta_ {t}) \quad \frac {\mathrm {d} \theta_ {t}}{\mathrm {d} t} = \rho_ {t}. \tag {10}
$$
For  $t \in \mathbb{R}$ , we define the mappings  $H_{t}: \mathbb{R}^{2d} \to \mathbb{R}^{2d}$ , transforming  $(\theta_{s}, \rho_{s}) \mapsto (\theta_{t+s}, \rho_{t+s})$  according to the dynamics of Eq. (10). Intuitively, traversing the trajectory of  $H_{t}$ ,  $t \in \mathbb{R}$  allows particles to efficiently explore the distribution of interest's support, thereby motivating the use of  $H_{t}$  as a flow map. In practice, the exact Hamiltonian flow  $H_{t}$  is replaced by its numerical counterpart—leapfrog integration—which involves  $K$  iterations of the subsequent steps:
$$
\hat {\rho} \leftarrow \rho + \frac {\epsilon}{2} \nabla \log \pi (\theta) \qquad \qquad \theta \leftarrow \theta + \epsilon \hat {\rho} \qquad \qquad \rho \leftarrow \hat {\rho} + \frac {\epsilon}{2} \nabla \log \pi (\theta).
$$
Here  $\epsilon$  denotes the step size of the leapfrog integrator. We use  $T_{K,\epsilon}:\mathbb{R}^{2d}\to \mathbb{R}^{2d}$  to define the map by sequencing the above three steps  $K$  times. There are two key properties of  $T_{K,\epsilon}$  that make it suitable for normalizing flow constructions.  $T_{K,\epsilon}$  is invertible (i.e.,  $T_{K,\epsilon}^{-1} = T_{K, - \epsilon}$ ) and has a simple Jacobian determinant of the form:  $|\operatorname *{det}\nabla T_{K,\epsilon}| = 1$ . This enables a straightforward density transformation formula: for any density  $q_0(\theta ,\rho)$  on  $\mathbb{R}^{2d}$ , pushforward of  $q$  under  $T_{K,\epsilon}$  has the density  $q_{0}(T_{K, - \epsilon}(\theta ,\rho))$ .
Given the above properties, Caterini et al. [7] and Chen et al. [6] developed flexible normalizing flows—HVAE and HamFlow, respectively—utilizing flow transformations through alternating compositions of multiple  $T_{K,\epsilon}$  and linear transformations. Notably, the momentum variable  $\rho$  introduced in the Hamiltonian dynamics Eq. (10) necessitates that both HVAE and HamFlow focus on approximating the augmented target distribution
$\bar{\pi} (\theta ,\rho)\propto \pi (\theta)\cdot \exp (-\frac{1}{2}\rho^T\rho)$  on  $\mathbb{R}^{2d}$ . This represents the joint distribution of the target  $\pi$  and an independent momentum distribution  $\mathcal{N}(0,I_d)$ .
Specifically, flow layers of HVAE alternate between the leapfrog transformation  $T_{K,\epsilon}$  and a momentum scaling layer:
$$
(\theta , \rho) \mapsto (\theta , \gamma \rho), \quad \gamma \in \mathbb {R} ^ {+}.
$$
And those of HamFlow alternate between  $T_{K,\epsilon}$  and a momentum standardization layer:
$$
(\theta , \rho) \mapsto (\theta , \Lambda (\rho - \mu)), \quad \Lambda \in \mathbb {R} ^ {d \times d} \text {i s l o w e r t r i a n g u l a r w i t h p o s i t i v e d i a g o n a l s}, \mu \in \mathbb {R} ^ {d}.
$$
C.1 Error analysis for Hamiltonian-based flows
We have provided error analysis for generic normalizing flows in Section 4.2. In this section, we specialize the error bounds for density evaluation (Theorem 4.3) and ELBO estimation (Theorem 4.4) for the two Hamiltonian-based flows.  $\pi$  denotes the target distribution of interest on  $\mathbb{R}^d$ , and  $\bar{p}$  denotes the augmented target distribution on  $\mathbb{R}^{2d}$ .  $p$  and  $\bar{p}$  respectively denote the unnormalized density of  $\pi$  and  $\bar{\pi}$ .
Corollary C.1. Let  $q$  be either HVAE or HamFlow of  $N$  layers with  $q_0 = \mathcal{N}(0, I_{2d})$ . Suppose the backward dynamics has the  $(\epsilon, \delta)$ -shadowing property. Then
$$
\forall x \in \mathbb {R} ^ {2 d}, | \log \hat {q} (x) - \log q (x) | \leq \epsilon \cdot \left(L _ {q, \epsilon} (x) + \| \hat {x} _ {- N} \|\right) + \epsilon^ {2}.
$$
Proof of Corollary C.1. Since the Jacobian determinant of each HVAE/HamFlow layer is constant,
$$
L _ {J _ {n}, \epsilon} (x) = 0, \quad \forall n \in [ N ] \text {a n d} x \in \mathbb {R} ^ {2 d}. \tag {11}
$$
And since  $q_{0}$  is a standard normal distribution,