SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathfrak {u} (m) := \left\{A \in \mathbb {C} ^ {m \times m}: A ^ {*} + A = 0 \right\},
$$
$$
\mathfrak {s p} (2 m, \mathbb {C}) := \left\{A \in \mathbb {C} ^ {2 m \times 2 m}: A ^ {*} J _ {m} + J _ {m} A = 0 \right\}.
$$
The unitary group is compact and is a group of isometries of matrix multiplication with respect to the Hilbert-Schmidt norm. Such properties are crucial for establishing theorems and properties related to the path characteristic function (PCF), as discussed in subsequent sections.
The compact symplectic group  $\mathrm{Sp}(m)$  is the simply-connected maximal compact real Lie subgroup of  $\mathrm{Sp}(2m,\mathbb{C})$ . It is the real form of  $\mathrm{Sp}(2n,\mathbb{C})$ , and satisfies
$$
\operatorname {S p} (m) = \operatorname {S p} (2 m, \mathbb {C}) \cap U (2 m).
$$
Note that  $U(m)$  and  $\mathrm{Sp}(m)$  are both real Lie groups, albeit they have complex entries in general.
A.3 Unitary feature of a path
Recall Definition 2.1 for the unitary feature, reproduced below:
Definition A.4. Let  $M: \mathbb{R}^d \to \mathfrak{u}(m)$  be a linear map and let  $X \in \mathrm{BV}\left([0,T];\mathbb{R}^d\right)$  be a BV-path. The unitary feature [a.k.a. the path development on the unitary group  $U(m)$ ] of  $\mathbf{x}$  under  $M$  is the solution to the equation
$$
\mathrm {d} \mathbf {y} _ {t} = \mathbf {y} _ {t} \cdot M (\mathrm {d} \mathbf {x} _ {t}) \quad \text {f o r a l l} t \in [ 0, T ] \text {w i t h} \mathbf {Y} _ {0} = I _ {m}.
$$
We write  $\mathcal{U}_M(\mathbf{x})\coloneqq \mathbf{y}_T$
Definition 2.1 is motivated by [7, §4]. Consider  $M \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(\mathcal{H}_{\mathrm{fd}})\right)$  with  $\mathcal{H}_{\mathrm{fd}}$  ranging over all finite-dimensional complex Hilbert spaces. Extend  $M$  by naturality to the tensor algebra over  $\mathbb{R}^d$ ; that is, define  $\widetilde{M}: T\left(\left(\mathbb{R}^d\right)\right) \equiv \bigoplus_{k=0}^{\infty} \left(\mathbb{R}^d\right)^{\otimes k} \to \mathfrak{u}(\mathcal{H}_{\mathrm{fd}})$  by linearity and the following rule:
$$
\widetilde {M} \left(v _ {1} \otimes \dots \otimes v _ {k}\right) := M \left(v _ {1}\right) \dots M \left(v _ {k}\right) \quad \text {f o r a n y} k \in \mathbb {N} \text {a n d} v _ {1}, \dots , v _ {k} \in \mathbb {R} ^ {d}.
$$
Then denote by  $\mathcal{A}\left(\mathbb{R}^d\right)$  the totality of such  $\widetilde{M}$ . Any element in  $\mathcal{A}\left(\mathbb{R}^d\right)$  is a unitary representation of the Lie group  $\mathcal{G}\left(\left(\mathbb{R}^d\right)\right) := \{\text{group-like elements in } T\left(\left(\mathbb{R}^d\right)\right)\}$ . See [7, p.4059].
The following two lemmas are contained in [26].
Lemma A.5. [Multiplicativity] Let  $X \in \mathrm{BV}\left([0,s],\mathbb{R}^d\right)$  and  $Y \in \mathrm{BV}\left([s,t],\mathbb{R}^d\right)$ . Denote by  $X * Y$  their concatenation:  $(X * Y)(v) = X(v)$  for  $v \in [0,s]$  and  $Y(v) - Y(s) + X(s)$  for  $v \in [s,t]$ . Then  $\mathcal{U}_M(X * Y) = \mathcal{U}_M(X) \cdot \mathcal{U}_M(Y)$  for all  $M \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right)$ .
We shall compute by Lemma A.5 and Example 2.2 the unitary feature of piecewise linear paths.
Lemma A.6 (Invariance under time-reparametrisation). Let  $X \in \mathrm{BV}([0, T], \mathbb{R}^d)$  and let  $\lambda : t \mapsto \lambda_t$  be a non-decreasing  $\mathcal{C}^1$ -diffeomorphism from  $[0, T]$  onto  $[0, S]$ . Define  $X_t^\lambda \coloneqq X_{\lambda_t}$  for  $t \in [0, T]$ . Then, for all  $M \in \mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m))$  and for every  $s, t \in [0, T]$ , it holds that  $\mathcal{U}_M(X_{\lambda_s, \lambda_t}) = \mathcal{U}_M(X_{s,t}^\lambda)$ .
A key property of the unitary feature is that it completely determines the law of random paths:
Theorem A.7 (Uniqueness of unitary feature). For any two paths  $\mathbf{X}_1 \neq \mathbf{X}_2$  in  $\mathcal{X}$ , there exists an  $M \in \mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m))$  with some  $m \in \mathbb{N}$  such that  $\mathcal{U}_M(\mathbf{X}_1) \neq \mathcal{U}_M(\mathbf{X}_2)$ .
Proof. For  $\mathbf{X}_1 \neq \mathbf{X}_2$  in  $\mathcal{X}$ , by uniqueness of signature over BV-paths (cf. [15]) one has  $\operatorname{Sig}(\mathbf{X}_1) \neq \operatorname{Sig}(\mathbf{X}_2)$  in  $\mathcal{G}\left(\left(\mathbb{R}^d\right)\right)$ . Here we use the fact that the signatures of BV-paths are group-like elements in the tensor algebra. Then, as  $\mathcal{A}\left(\mathbb{R}^d\right)$  separates points over  $\mathcal{G}\left(\left(\mathbb{R}^d\right)\right)$  (cf. [7, Theorem 4.8]), there is  $M \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right)$  such that  $\widetilde{M}\left[\operatorname{Sig}(\mathbf{X}_1)\right] \neq \widetilde{M}\left[\operatorname{Sig}(\mathbf{X}_2)\right]$ ; hence  $M(\mathbf{X}_1) \neq M(\mathbf{X}_2)$ . Therefore, by considering the  $U(m)$ -valued equation  $\mathrm{dY}_t = \mathbf{Y}_t \cdot M(\mathrm{dX}_t)$  with  $\mathbf{Y}_0 = I_m$ , we conclude that  $\mathcal{U}_M(\mathbf{X}_1) \neq \mathcal{U}_M(\mathbf{X}_2)$ .
Theorem A.8 (Universality of unitary feature). Let  $\mathcal{K} \subset \mathrm{BV}\left([0,T];\mathbb{R}^d\right)$  be a compact subset. For any continuous function  $f: \mathcal{K} \to \mathbb{C}$  and any  $\epsilon > 0$ , there exists an  $m_{\star} \in \mathbb{N}$  and finitely many  $M_1, \dots, M_N \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m_{\star})\right)$  as well as  $L_1, \dots, L_N \in \mathcal{L}\left(U(m_{\star}); \mathbb{C}\right)$ , such that
$$
\sup  _ {\mathbf {X} \in \mathcal {K}} \left| f (\mathbf {X}) - \sum_ {i = 1} ^ {N} L _ {i} \circ \mathcal {U} _ {M _ {i}} (\mathbf {X}) \right| <   \epsilon . \tag {12}
$$
Proof. It follows from [26, Theorem A.4] and the universality of signature in [8] that Eq. (12) holds with  $M_j \in \mathcal{L}\left(\mathbb{R}^d, \bigoplus_{m \in \mathbb{N}} \mathfrak{u}(m)\right)$  and  $L_j \in \mathcal{L}\left(\bigoplus_{m \in \mathbb{N}} U(m); \mathbb{C}\right)$  and  $\epsilon/2$  in place of  $\epsilon$ . By a simple approximation via restricting the ranges of  $M_j$  and domains of  $L_j$ , we may obtain (without relabelling)  $M_j \in \mathcal{L}\left(\mathbb{R}^d, \bigoplus_{m=0}^{m_\star} \mathfrak{u}(m)\right)$  and  $L_j \in \mathcal{L}\left(\bigoplus_{m=0}^{m_\star} U(m); \mathbb{C}\right)$  that verify Eq. (12). We conclude by the flag structure of  $U(1) \subset U(2) \subset U(3) \subset \ldots$  and  $\mathfrak{u}(1) \subset \mathfrak{u}(2) \subset \mathfrak{u}(3) \subset \ldots$ .
B Path Characteristic loss
B.1 Path Characteristic function
Theorem B.1. Let  $\mathbf{X}$  be a  $\mathcal{X}$ -valued random variable with associated probability measure  $\mathbb{P}_{\mathbf{X}}$ . The path characteristic function  $\Phi_{\mathbf{X}}$  uniquely characterises  $\mathbb{P}_{\mathbf{X}}$ .
Proof. Assume that  $\mathbb{P}_{\mathbf{X}_1} \neq \mathbb{P}_{\mathbf{X}_2}$ . Then  $\operatorname{Sig}(\mathbf{X}_1) \neq \operatorname{Sig}(\mathbf{X}_2)$  by the uniqueness of signature over BV-paths (cf. [15]). It is proved in [26, Lemma 2.6] that  $\mathcal{U}_M(\mathbf{X}_i) = \widetilde{M}(\operatorname{Sig}(\mathbf{X}_i))$  for any  $M \in \mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m)); i \in \{1, 2\}$ . Hence  $\Phi_{\mathbf{X}_i} = \int_{\chi} \widetilde{M}(\operatorname{Sig}(\mathbf{x})) \mathrm{d}\mathbb{P}_{\mathbf{X}_i}(\mathbf{x})$ . But as in the proof of Theorem A.7,  $\mathcal{A}(\mathbb{R}^d)$  separates points over  $\mathcal{G}((\mathbb{R}^d))$  (cf. [7, Theorem 4.8]) and the signature of BV-paths lies in  $\mathcal{G}((\mathbb{R}^d))$ . Therefore, there is an  $M \in \mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m))$  such that  $\Phi_{\mathbf{X}} \neq \Phi_{\mathbf{Y}}$ .
B.2 Distance metric via path characteristic function
Lemma B.2. PCFD $_{\mathcal{M}}$  in Eq. (5) defines a pseudometric on the path space  $\mathcal{X}$  for any  $m \in \mathbb{N}$  and  $\mathcal{M} \in \mathcal{P}\left(\mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right)\right)$ . In addition, suppose that  $\{\mathcal{M}_j\}_{j \in \mathbb{N}}$  is a countable dense subset in  $\mathcal{P}\left(\mathcal{L}\left(\mathbb{R}^d, \bigoplus_{m \in \mathbb{N}} \mathfrak{u}(m)\right)\right)$ . Then the following defines a metric on  $\mathcal{X}$ :
$$
\widetilde {\operatorname {P C F D}} (\mathbf {X}, \mathbf {Y}) := \sum_ {j = 1} ^ {\infty} \frac {\min  \left\{1 , \operatorname {P C F D} _ {\mathcal {M} _ {j}} (\mathbf {X} , \mathbf {Y}) \right\}}{2 ^ {j}}. \tag {13}
$$
In Lemma B.2 above,  $\mathcal{L}\left(\mathbb{R}^d,\bigoplus_{m\in \mathbb{N}}\mathfrak{u}(m)\right)\cong \left(\mathbb{R}^d\right)^*\widehat{\otimes}_{\pi}\left(\bigoplus_{m\in \mathbb{N}}\mathfrak{u}(m)\right)$  where  $\widehat{\otimes}_{\pi}$  is the completion of the projective tensor product and  $\bigoplus_{m\in \mathbb{N}}\mathfrak{u}(m)$  is a Banach space under the norm  $\| T\| := \sum_{m\in \mathbb{N}}\left\| T^{(m)}\right\|_{\mathrm{HS}} < \infty$ . Here  $T^{(m)}$  denotes the  $m^{\mathrm{th}}$ -projection of  $T$  on  $\mathfrak{u}(m)$ . Therefore, such a sequence  $\{\mathcal{M}_j\}_{j\in \mathbb{N}}$  always exists since  $\mathcal{P}\left(\mathcal{L}\left(\mathbb{R}^d,\bigoplus_{m\in \mathbb{N}}\mathfrak{u}(m)\right)\right)$ , being the space of Borel probability measures over a Polish space, is itself a Polish space. See [33].
Proof. Non-negativity, symmetry, and that  $\mathrm{PCFD}_{\mathcal{M}}(\mathbf{X},\mathbf{X}) = 0$  are clear. That  $\mathrm{PCFD}_{\mathcal{M}}(\mathbf{X},\mathbf{Y})\leq$ $\mathrm{PCFD}_{\mathcal{M}}(\mathbf{X},\mathbf{Z}) + \mathrm{PCFD}_{\mathcal{M}}(\mathbf{Z},\mathbf{Y})$  follows from the triangle inequality of the Hilbert-Schmidt norm and the linearity of expectation. This shows that  $\mathrm{PCFD}_{\mathcal{M}}$  is a pseudometric for each  $\mathcal{M}$
In addition,  $\mathrm{PCFD}_{\mathcal{M}}(\mathbf{X},\mathbf{Y}) = 0$  implies that
$$
\begin{array}{l} \int_ {\mathcal {L} \left(\mathbb {R} ^ {d}, \mathrm {u} (m)\right)} d _ {\mathrm {H S}} ^ {2} \left(\Phi_ {\mathbf {X}} (M), \Phi_ {\mathbf {Y}} (M)\right) \mathrm {d} \mathbb {P} _ {\mathcal {M}} (M) \\ = \int_ {\mathcal {L} (\mathbb {R} ^ {d}, u (m))} \left\| \mathbb {E} \left[ \mathcal {U} _ {M} (\mathbf {X}) \right] - \mathbb {E} \left[ \mathcal {U} _ {M} (\mathbf {Y}) \right] \right\| _ {\mathrm {H S}} ^ {2} \mathrm {d} \mathbb {P} _ {\mathcal {M}} (M) = 0. \\ \end{array}
$$
So, if  $\mathbb{P}_{\mathcal{M}}$  is supported on the whole of  $\mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right)$ , then  $\Phi_{\mathbf{X}}(M) = \Phi_{\mathbf{Y}}(M)$  for any  $M\sim \mathbb{P}_{\mathcal{M}}$ . Now, by density of  $\{\mathcal{M}_j\}_{j\in \mathbb{N}}$  in  $\mathcal{P}\left(\mathcal{L}\left(\mathbb{R}^d,\bigoplus_{m\in \mathbb{N}}\mathfrak{u}(m)\right)\right)$ , there exists a subsequence  $\{\mathcal{M}_{j(m)}\}$  such that  $\mathcal{M}_{j(m)}$  has full support on  $\mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right)$  for each  $m\in \mathbb{N}$ . Thus,  $\widetilde{\mathrm{PCFD}} (\mathbf{X},\mathbf{Y}) = 0$  implies that  $\Phi_{\mathbf{X}} = \Phi_{\mathbf{Y}}$  on a dense subset of  $\mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right)$  for every  $m\in \mathbb{N}$ . We conclude by the characteristic Theorem 3.2 and a continuity argument.
Lemma B.3 (Lemma 3.5). Let  $\mathcal{M}$  be an  $\mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m))$ -valued random variable. Then for any BV  $([0, T]; \mathbb{R}^d)$ -valued random variables  $\mathbf{X}$  and  $\mathbf{Y}$ , it holds that
$$
\operatorname {P C F D} _ {\mathcal {M}} (\mathbf {X}, \mathbf {Y}) \leq 2 m ^ {2}.
$$
Proof. As  $(\mathbb{C}^{m\times m},\| \bullet \|_{\mathrm{HS}})$  is a Hilbert space, from the Pythagorean theorem one deduces that  $d_{\mathrm{HS}}^2 (\Phi_{\mathbf{X}}(M),\Phi_{\mathbf{Y}}(M))\leq \| \Phi_{\mathbf{X}}(M)\|_{\mathrm{HS}}^2 +\| \Phi_{\mathbf{Y}}(M)\|_{\mathrm{HS}}^2$  . Both  $\Phi_{\mathbf{X}}(M),\Phi_{\mathbf{Y}}(M)$  are expectations of  $U(m)$  -valued random variables, and  $\| U\|_{\mathrm{HS}}:= \sqrt{\operatorname{tr}(UU^{*})} = \sqrt{\operatorname{tr}(I_m)} = \sqrt{m}$  for  $U\in U(m)$  Thus  $d_{\mathrm{HS}}(\Phi_{\mathbf{X}}(M),\Phi_{\mathbf{Y}}(M))\leq \sqrt{2} m$  . We take expectation over  $\mathbb{P}_{\mathcal{M}}$  to conclude.
The result below is formulated in terms of the Hilbert-Schmidt norm of matrices in  $\mathbb{C}^{m\times m}$ . Any other norm on  $\mathbb{C}^{m\times m}$  is equivalent to that, modulo a constant depending on  $m$  only. In fact, the strict inequality  $\| T\|_{\mathrm{op}}\leq \| T\|_{\mathrm{HS}}$  for  $T\in \mathbb{C}^{m\times m}$  holds. See, e.g., [48, Lemma 3.1.10, p.55].
Lemma B.4. Let  $L, \tilde{L} : [a,b] \to \mathbb{R}^d$  be two linear paths, and let  $M \in \mathfrak{u}(m)^d \coloneqq \mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right)$  as before. Denote by  $\| \bullet \|_{\mathrm{e}}$  the usual Euclidean norm on  $\mathbb{R}^d$  and  $\| |M||$  the operator norm of  $M: (\mathbb{R}^d, \| \bullet \|_{\mathrm{e}}) \to (\mathfrak{u}(m), \| \bullet \|_{\mathrm{HS}})$ . Then we have
$$
\left\| e ^ {M (L (t))} - e ^ {M (\tilde {L} (t))} \right\| _ {\mathrm {H S}} \leq \| | M \| | \left\| L (t) - \tilde {L} (t) \right\| _ {\mathrm {e}} \quad \text {f o r e a c h} t \in [ a, b ].
$$
Proof. Let  $\Gamma(t, s) \coloneqq M((1 - s)L(t) + s\tilde{L}(t))$  with  $t \in [a, b]$  and  $s \in [0, 1]$ . This is the linear interpolation between  $\Gamma(t, 0) = M \circ L(t)$  and  $\Gamma(t, 1) = M \circ \tilde{L}(t)$ . Then we have
$$
\begin{array}{l} \left\| e ^ {M L (t)} - e ^ {M \tilde {L} (t)} \right\| _ {\mathrm {H S}} = \left\| \int_ {0} ^ {1} \frac {\partial e ^ {\Gamma}}{\partial s} (t, s) \mathrm {d} s \right\| _ {\mathrm {H S}} \\ = \left\| \int_ {0} ^ {1} \int_ {0} ^ {1} e ^ {(1 - r) \Gamma (t, 1 - s)} \frac {\partial \Gamma}{\partial s} (t, s) e ^ {r \Gamma (t, s)} d r d s \right\| _ {\mathrm {H S}}, \\ \end{array}
$$
thanks to an identity for differentiation of matrix exponential and the inequality  $\| T_1T_2\|_{\mathrm{HS}}\leq$ $\| T_1\|_{\mathrm{op}}\| T_2\|_{\mathrm{HS}}$ . Here  $e^{(1 - r)\Gamma (t,1 - s)}$  and  $e^{r\Gamma (t,s)}$  take values in  $U(m)$ , hence of operator norm 1 for any parameters  $t,s,r$ . So we infer that
$$
\begin{array}{l} \left\| e ^ {M L (t)} - e ^ {M \tilde {L} (t)} \right\| _ {\mathrm {H S}} \leq \int_ {0} ^ {1} \int_ {0} ^ {1} \left\| \frac {\partial \Gamma}{\partial s} (t, s) \right\| _ {\mathrm {H S}} \mathrm {d} r \mathrm {d} s \\ = \left\| M (\tilde {L} (t) - L (t)) \right\| _ {\mathrm {H S}} \leq \| | M \| \| \left\| \tilde {L} (t) - L (t) \right\| _ {\mathrm {e}}, \\ \end{array}
$$
where the first inequality holds for Bochner integrals. See [46, Corollary 1, p.133].
Lemma B.5 (Subadditivity of unitary feature). Let  $X, Y \in \mathrm{BV}\left([0, T]; \mathbb{R}^d\right)$  be BV-paths, and  $\mathcal{U}_M$  be the unitary feature associated with  $M \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right) = \mathfrak{u}(m)^d$ . For any  $0 < t < T$  we have
$$
\| \mathcal {U} _ {M} (X) - \mathcal {U} _ {M} (Y) \| _ {\mathrm {H S}} \leq \| \mathcal {U} _ {M} (X _ {0, t}) - \mathcal {U} _ {M} (Y _ {0, t}) \| _ {\mathrm {H S}} + \| \mathcal {U} _ {M} (X _ {t, T}) - \mathcal {U} _ {M} (Y _ {t, T}) \| _ {\mathrm {H S}}.
$$
Proof. We apply the multiplicative property of unitary feature in Lemma A.5, the triangle inequality, and the unitary invariance of the Hilbert-Schmidt norm to estimate that
$$
\begin{array}{l} \left\| \mathcal {U} _ {M} (X) - \mathcal {U} _ {M} (Y) \right\| _ {\mathrm {H S}} \\ = \| \mathcal {U} _ {M} \left(X _ {0, t}\right) \cdot \mathcal {U} _ {M} \left(X _ {t, T}\right) - \mathcal {U} _ {M} \left(Y _ {0, t}\right) \cdot \mathcal {U} _ {M} \left(Y _ {t, T}\right) \| _ {\mathrm {H S}} \\ \leq \left\| \left(\mathcal {U} _ {M} \left(X _ {0, t}\right) - \mathcal {U} _ {M} \left(Y _ {0, t}\right)\right) \cdot \mathcal {U} _ {M} \left(X _ {t, T}\right) \right\| _ {\mathrm {H S}} + \left\| \mathcal {U} _ {M} \left(Y _ {0, t}\right) \left(\mathcal {U} _ {M} \left(X _ {t, T}\right) - \mathcal {U} _ {M} \left(Y _ {t, T}\right)\right) \right\| _ {\mathrm {H S}} \\ = \| \mathcal {U} _ {M} (X _ {0, t}) - \mathcal {U} _ {M} (Y _ {0, t}) \| _ {\mathrm {H S}} + \| \mathcal {U} _ {M} (X _ {t, T}) - \mathcal {U} _ {M} (Y _ {t, T}) \| _ {\mathrm {H S}}. \\ \end{array}
$$
Proposition B.6. For  $\mathbf{X},\mathbf{Y}\in \mathcal{X}$ , the unitary feature  $\mathcal{U}_M(\mathbf{X})$  with  $M\in \mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right) = \mathfrak{u}(m)^d$  satisfies
$$
\left\| \mathcal {U} _ {M} (\mathbf {X}) - \mathcal {U} _ {M} (\mathbf {Y}) \right\| _ {\mathrm {H S}} \leq \| | M | \| \text {T o t . V a r .} [ \mathbf {X} - \mathbf {Y} ],
$$
where  $\mathrm{Tot.} \mathrm{Var.}[\mathbf{X} - \mathbf{Y}]$  denotes the total variation over  $[0, T]$  of the path  $\mathbf{X} - \mathbf{Y}$ .
Proof. Given BV-paths  $\mathbf{X}$ ,  $\mathbf{Y}$  with the same initial point, there are piecewise linear approximations  $\{\mathbf{X}^n\}$ ,  $\{\mathbf{Y}^n\}$  with common partition  $0 = t_0 < t_1 < \dots < t_n = T$  converging respectively to  $\mathbf{X}$  and  $\mathbf{Y}$  in the  $p$ -variation metric;  $p \in (1,2)$ . Applying Lemma B.5 recursively, we obtain that
$$
\left\| \mathcal {U} _ {M} \left(\mathbf {X} ^ {n}\right) - \mathcal {U} _ {M} \left(\mathbf {Y} ^ {n}\right) \right\| _ {\mathrm {H S}} \leq \sum_ {i = 0} ^ {n - 1} \left\| \mathcal {U} _ {M} \left(\mathbf {X} _ {t _ {i}, t _ {i + 1}} ^ {n}\right) - \mathcal {U} _ {M} \left(\mathbf {Y} _ {t _ {i}, t _ {i + 1}} ^ {n}\right) \right\| _ {\mathrm {H S}}
$$
By definition of unitary feature and Lemma B.4, one deduces that
$$
\left\| \mathcal {U} _ {M} \left(\mathbf {X} ^ {n}\right) - \mathcal {U} _ {M} \left(\mathbf {Y} ^ {n}\right) \right\| _ {\mathrm {H S}} \leq \sum_ {i = 0} ^ {n - 1} \| | M | \| \left\| \mathbf {X} _ {t _ {i}, t _ {i + 1}} ^ {n} - \mathbf {Y} _ {t _ {i}, t _ {i + 1}} ^ {n} \right\| _ {\mathrm {e}}.
$$
We may now conclude by taking supremum over all partitions and sending  $n \to \infty$  with the limit. This is a consequence of continuity of the Itô map with respect to the driving path in  $p$ -variation topology, since the vector field in (4) is Lipschitz ([27, Theorem 1.20]).
Theorem B.7 (Dependence on continuous parameter). Let  $\mathcal{X}$  and  $\mathcal{Z}$  be subsets of BV  $[0,T];\mathbb{R}^d$ $(\Theta ,\rho)$  be a metric space,  $\mathbb{Q}$  be a Borel probability measure on  $\mathcal{Z}$ , and  $\mathcal{M}$  be a Borel probability measure on  $\mathfrak{u}(m)^d$ . Assume that  $g:\Theta \times \mathcal{Z}\to \mathcal{X}$ ,  $(\theta ,\mathbf{Z})\mapsto g_{\theta}(\mathbf{Z})$  is Lipschitz in  $\theta$  such that  $\mathrm{Tot.Var.}[g_{\theta}(\mathbf{Z}) - g_{\theta '}(\mathbf{Z})]\leq \omega (\mathbf{Z})\rho (\theta ,\theta ')$ . In addition, suppose that  $\mathbb{E}_{M\sim \mathbb{P}_M}\left[||M||^2\right] < \infty$  and  $\mathbb{E}_{\mathbf{Z}\sim \mathbb{Q}}[\omega (\mathbf{Z})] < \infty$ . Then PCFD $_{\mathcal{M}}$ $(g_{\theta}(\mathbf{Z}),\mathbf{X})$  is Lipschitz in  $\theta$ .
Moreover, it holds that
$$
| \mathrm {P C F D} _ {\mathcal {M}} (g _ {\theta} (\mathbf {Z}), \mathbf {X}) - \mathrm {P C F D} _ {\mathcal {M}} (g _ {\theta^ {\prime}} (\mathbf {Z}), \mathbf {X}) | \leq \sqrt {\mathbb {E} _ {M \sim \mathbb {P} _ {\mathcal {M}}} [ | | M | | ^ {2} ]} \mathbb {E} _ {\mathbf {Z} \sim \mathbb {Q}} [ \omega (\mathbf {Z}) ] \rho (\theta , \theta^ {\prime})
$$
for any  $\theta, \theta' \in \Theta$ ,  $\mathbf{Z} \in \mathcal{Z}$ ,  $\mathbf{X} \in \mathcal{X}$ , and  $\mathcal{M} \in \mathcal{P}\left(\mathfrak{u}(m)^d\right)$ .
Proof. As  $\mathrm{PCFD}_{\mathcal{M}}$  is a pseudometric (Lemma B.2), we have
$$
\left| \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta} (\mathbf {Z}), \mathbf {X}\right) - \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta^ {\prime}} (\mathbf {Z}), \mathbf {X}\right) \right| \leq \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta} (\mathbf {Z}), g _ {\theta^ {\prime}} (\mathbf {Z})\right).
$$
We may control the right-hand side as follows, using subsequentially the definitions of PCFD and PCF, Proposition B.6, and the assumptions in this theorem:
$$
\begin{array}{l} \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta} (\mathbf {Z}), g _ {\theta^ {\prime}} (\mathbf {Z})\right) \\ = \left\{\int_ {\mathfrak {u} (m) ^ {d}} \left\| \Phi_ {g _ {\theta} (\mathbf {Z})} (M) - \Phi_ {g _ {\theta^ {\prime}} (\mathbf {Z})} (M) \right\| _ {\mathrm {H S}} ^ {2} \mathrm {d} \mathbb {P} _ {\mathcal {M}} \right\} ^ {\frac {1}{2}} \\ = \left\{\int_ {\mathfrak {u} (m) ^ {d}} \left\| \int_ {\mathcal {Z}} \left[ \mathcal {U} _ {M} \left(g _ {\theta} (\mathbf {Z})\right) - \mathcal {U} _ {M} \left(g _ {\theta^ {\prime}} (\mathbf {Z})\right) \right] \mathrm {d} \mathbb {Q} (\mathbf {Z}) \right\| _ {\mathrm {H S}} ^ {2} \mathrm {d} \mathbb {P} _ {\mathcal {M}} \right\} ^ {\frac {1}{2}} \\ \leq \left\{\int_ {\mathrm {u} (m) ^ {d}} \| | M | \| ^ {2} \left\{\int_ {\mathcal {Z}} \operatorname {T o t. V a r.} \left[ g _ {\theta} (\mathbf {Z}) - g _ {\theta^ {\prime}} (\mathbf {Z}) \right] \mathrm {d} \mathbb {Q} (\mathbf {Z}) \right\} ^ {2} \mathrm {d} \mathbb {P} _ {\mathcal {M}} \right\} ^ {\frac {1}{2}} \\ \leq \sqrt {\mathbb {E} _ {M \sim \mathbb {P} _ {\mathcal {M}}} \left[ | | M | | ^ {2} \right]} \left\{\int_ {\mathcal {Z}} \omega (\mathbf {Z}) \rho \left(\theta , \theta^ {\prime}\right) \mathrm {d} \mathbb {Q} (\mathbf {Z}) \right\}. \\ \end{array}
$$
The unitary feature is universal in the spirit of the Stone-Weierstrass theorem; i.e., continuous functions on paths can be uniformly approximated by linear functionals on unitary features.
As PCFD metrises the weak topology on the space of path-valued random variables, it emerges as a more sensible distance metric for training time series generations than metrics without this property; e.g., the Jensen-Shannon divergence.
Theorem B.8 (Metrisation of weak-star topology). Let  $\mathcal{K} \subset \mathcal{X}$  be a compact subset. Suppose that  $\{\mathcal{M}_j\}_{j \in \mathbb{N}}$  is a countable dense subset in  $\mathcal{P}$  ( $\mathcal{L}\left(\mathbb{R}^d, \bigoplus_{m \in \mathbb{N}} \mathfrak{u}(m)\right)$ ). Then  $\widetilde{\mathrm{PCFD}}$  defined by Eqn. 13 metrises the weak-star topology on  $\mathcal{P}(\mathcal{K})$ . That is,  $\widetilde{\mathrm{PCFD}}(\mathbf{X}_n, \mathbf{X}) \to 0 \iff \mathbf{X}_n \xrightarrow{d} \mathbf{X}$  as  $n \to \infty$ , where  $\xrightarrow{d}$  denotes convergence in distribution of random variables.
The metrisability of  $\mathcal{P}(\mathcal{K})$  follows from general theorems in functional analysis:  $\mathcal{K}$  is a compact metric space, hence  $C^0 (\mathcal{K})$  is separable ([13, Lemma 3.23]). Then, viewing  $\mathcal{P}(\mathcal{K})$  as the unit circle in  $[C^0 (\mathcal{K})]^*$  via Riesz representation, we infer from [13, Proposition 3.24] that  $\mathcal{P}(\mathcal{K})$  is metrisable in the weak-star topology, which is equivalent to the distributional convergence of random variables.
Proof. The backward direction is straightforward. By the Riesz representation theorem of Radon measures, the distributional convergence is equivalent to that  $\int_{\mathcal{K}}f\mathrm{d}\mathbb{P}_{\mathbf{X}_n}\to \int_{\mathcal{K}}f\mathrm{d}\mathbb{P}_{\mathbf{X}}$  for all continuous  $f\in C(\mathcal{K})$ . Thus  $\left\| \int_{\mathcal{K}}\mathcal{U}_M\mathrm{d}\mathbb{P}_{\mathbf{X}_n} - \int_{\mathcal{K}}\mathcal{U}_M\mathrm{d}\mathbb{P}_{\mathbf{X}}\right\|_{\mathrm{HS}}\to 0$ , namely that  $\Phi_{\mathbf{X}_n}[M]\to \Phi_{\mathbf{X}}[M]$  for each  $M\in \mathcal{L}\left(\mathbb{R}^d;\mathfrak{u}(m)\right)$ . The unitary feature  $\mathcal{U}_M$  is bounded as it is  $U(m)$ -valued for some  $m$ , so we deduce from the dominated convergence theorem that  $\widetilde{\operatorname{PCFD}} (\mathbf{X}_n,\mathbf{X})\to 0$ .
Conversely, suppose that  $\widetilde{\mathrm{PCFD}} (\mathbf{X}_n,\mathbf{X})\to 0$  . Then