SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Hang Lou
Department of Mathematics University College London hang.lou.19@ucl.ac.uk
Siran Li
Department of Mathematics  
Shanghai Jiao Tong University  
s14025@nyu.edu
Hao Ni
Department of Mathematics  
University College London  
h.ni@ucl.ac.uk
Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. Towards this goal, a key step is the development of an effective discriminator to distinguish between time series distributions. We propose the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance. On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality.
Generative Adversarial Networks (GANs) have been a powerful tool for generating complex data distributions, e.g., image data. The original GAN suffers from optimisation instability and mode collapse, partially remedied later by an alternative training scheme using integral probability metric (IPM) in lieu of Jensen-Shannon divergence. The IPMs, e.g., metrics based on Wasserstein distances or Maximum Mean Discrepancy (MMD), consistently yield good measures between generated and real data distributions, thus resulting in more powerful GANs on empirical data ([14, 2, 24]).
More recently, [1] proposed an IPM based on the characteristic function (CF) of measures on  $\mathbb{R}^d$ , which has the characteristic property, boundedness, and differentiability. Such properties enable the GAN constructed using this IPM as discriminator ("CF-GAN") to stabilise training and improve generative performance. However, ineffective in capturing the temporal dependency of sequential data, such CF-metric fails to address high-frequency cases due to the curse of dimensionality. To tackle this issue, we take the continuous time perspective of time series and lift discrete time series to the path space ([28, 29, 23]). This allows us to treat time series of variable length, unequal sampling, and high frequency in a unified approach. We propose a path characteristic function (PCF) distance to characterise distributions on the path space, and propose the corresponding PCF distance as a novel IPM to quantify the distance between measures on the path space.
Built on top of the unitary feature of paths ([26]), our proposed PCF has theoretical foundations deeply rooted in the rough path theory ([7]), which exploits the non-commutativity and the group structure of the unitary feature to encode information on order of paths. The CF may be regarded as the special case of PCF with linear random path and  $1 \times 1$  unitary matrix. We show that the  $PCF$  distance (PCFD) possesses favourable analytic properties, including boundedness and differentiability in model parameters, and we establish the linkages between PCFD and MMD. These results vastly generalise classical theorems on measures on  $\mathbb{R}^d$  ([1]), with much more technically involved proofs due to the infinite-dimensionality of path space.
On the numerical side, we design an efficient algorithm which, by optimising the trainable parameters of PCFD, maximises the discriminative power and improves the stability and efficiency of GAN training. Inspired by [25, 41], we integrate the proposed PCF into the IPM-GAN framework, utilising an auto-encoder architecture specifically tailored to sequential data. This model design enables our algorithm to generate and reconstruct realistic time series simultaneously, which has advantages in diverse applications, including privacy preservation ([35]) and semantic representation extraction for downstream tasks ([10]). To assess the efficacy of our PCF-GAN, we conduct extensive numerical experiments on several standard time series benchmarking datasets for both generation and reconstruction tasks.
We summarize key contributions of this work below:
- proposing a new metric for the distributions on the path space via PCF;
- providing theoretical proofs for analytic properties of the proposed loss metric which benefit GAN training;
- introducing a novel PCF-GAN to generate & reconstruct time series simultaneously; and
- reporting substantial empirical results validating the out-performance of our approach, compared with several state-of-the-art GANs with different loss functions on various time series generation and reconstruction tasks.
Related work. Given the wide practical use of, and challenges for, realistic time series synthesis ([3, 4]), various approaches are proposed to improve the quality of GANs for synthetic time series generation. Several works, e.g., [43, 45, 36], are devoted to improving the discriminator of GANs to be better suited to distributions induced by time series. Among them, COT-GAN in [43] shares a similar philosophy with PCF-GAN by introducing a novel discriminator based on causal optimal transport (which can be seen as an improved variant of the Sinkhorn divergence tailored to sequential data), while TimeGAN ([45]) shares a similar auto-encoder structure, which improves the generator's quality and enables time series reconstruction. Unlike PCF-GAN, the reconstruction and generation modules of TimeGAN are separated, whereas it has additional stepwise supervised loss and discriminative loss. In a different vein, CEGEN[36], GT-GAN [17], COSCI-GAN [39], and EWGAN[37] focus primarily on the design of network framework and generator architecture, which achieve state-of-the-art results on several benchmarking datasets.
The characteristic function of a measure on  $\mathbb{R}^d$ , namely that the Fourier transform, plays a central role in probability theory and analysis. The path characteristic function (PCF) is a natural extension of the characteristic function to the path space.
2.1 Characteristic function distance (CFD) between random variables in  $\mathbb{R}^d$
Let  $X$  be an  $\mathbb{R}^d$ -valued random variable with the law  $\mu = \mathbb{P} \circ X^{-1}$ . The characteristic function of  $X$ , denoted as  $\Phi_X: \mathbb{R}^d \to \mathbb{C}$ , maps each  $\lambda \in \mathbb{R}^d$  to the expectation of its complex unitary transform:  $\Phi_X: \lambda \longmapsto \mathbb{E}_{X \sim \mu}[e^{i\langle \lambda, X \rangle}]$ . Here  $U_\lambda: \mathbb{R}^d \to \mathbb{C}, x \mapsto e^{i\langle \lambda, x \rangle}$  is the solution to the linear controlled differential equation:
$$
\mathrm {d} U _ {\lambda} (x) = i U _ {\lambda} (x) \langle \lambda , \mathrm {d} x \rangle , \quad U _ {\lambda} (\mathbf {0}) = 1, \tag {1}
$$
where  $\mathbf{0}$  is the zero vector in  $\mathbb{R}^d$  and  $\langle \cdot ,\cdot \rangle$  is the Euclidean inner product on  $\mathbb{R}^d$
References [11, 16] studied the squared characteristic function distance (CFD) between two  $\mathbb{R}^d$ -valued random variables  $X$  and  $Y$  with respect to another probability distribution  $\Lambda$  on  $\mathbb{R}^d$ :
$$
\operatorname {C F D} _ {\boldsymbol {\Lambda}} ^ {2} (X, Y) = \mathbb {E} _ {Z \sim \boldsymbol {\Lambda}} \left[ \left| \Phi_ {X} (Z) - \Phi_ {Y} (Z) \right| ^ {2} \right]. \tag {2}
$$
It is proved in [25, 1] that if the support of  $\Lambda$  is  $\mathbb{R}^d$ , then  $\mathrm{CFD}_{\Lambda}$  is a distance metric, so that  $\mathrm{CFD}_{\Lambda}^{2}(X,Y) = 0$  if and only if  $X$  and  $Y$  have the same distribution. This justifies the usage of  $\mathrm{CFD}_{\Lambda}^{2}$  as a discriminator for GAN training to learn finite-dimensional random variables from data.
2.2 Unitary feature of a path
Let BV  $\left([0,T];\mathbb{R}^d\right)$  be the space of  $\mathbb{R}^d$  -valued paths of bounded variation over  $[0,T]$ . Consider
$$
\mathcal {X} := \left\{\bar {\mathbf {x}}: [ 0, T ] \rightarrow \mathbb {R} ^ {d + 1}: \bar {\mathbf {x}} (t) = (t, \mathbf {x} (t)) \text {f o r} t \in [ 0, T ]; \mathbf {x} \in \mathrm {B V} ([ 0, T ]; \mathbb {R} ^ {d}); \mathbf {x} (0) = 0 \right\}. \tag {3}
$$
For a discrete time series  $x = (t_i, x_i)_{i=0}^N$ , where  $0 = t_0 < t_1 < \dots < t_N = T$  and  $x_i \in \mathbb{R}^d$  ( $i \in \{0, \dots, N\}$ ), we can embed it into some  $\mathbf{x} \in \mathcal{X}$  whose evaluation at  $(t_i)_{i=1}^N$  coincides with  $x$ . This is well suited for sequence-valued data in the high-frequency limit with finer time-discretisation and is often robust in practice ([27, 26]). Such embeddings are not unique. In this work, we adopt the linear interpolation for embedding, following [23, 18, 32].
Let  $\mathbb{C}^{m\times m}\coloneqq \{m\times m$  complex matrices},  $I_{m}$  be the identity matrix, and  $\ast$  be conjugate transpose. Write  $U(m)$  and  $\mathfrak{u}(m)$  for the Lie group of  $m\times m$  unitary matrices and its Lie algebra, resp.:
$$
U (m) = \left\{A \in \mathbb {C} ^ {m \times m}: A ^ {*} A = I _ {m} \right\}, \qquad \mathfrak {u} (m) := \left\{A \in \mathbb {C} ^ {m \times m}: A ^ {*} + A = 0 \right\}.
$$
Definition 2.1. Let  $\mathbf{x} \in \mathrm{BV}\left([0,T];\mathbb{R}^d\right)$  be a continuous path and  $M: \mathbb{R}^d \to \mathfrak{u}(m)$  be a linear map. The unitary feature of  $\mathbf{x}$  under  $M$  is the solution  $\mathbf{y}: [0,T] \to U(m)$  to the following equation:
$$
\mathrm {d} \mathbf {y} _ {t} = \mathbf {y} _ {t} \cdot M (\mathrm {d} \mathbf {x} _ {t}), \quad \mathbf {y} _ {0} = I _ {m}. \tag {4}
$$
We write  $\mathcal{U}_M(\mathbf{x})\coloneqq \mathbf{y}_T$  , i.e., the endpoint of the solution path.
By a slight abuse of notations,  $\mathcal{U}_M(\mathbf{x})$  is also called the unitary feature of  $\mathbf{x}$  under  $M$ . Unitary feature is a special case of the Cartan/path development, for which one may consider paths taking values in any Lie group  $G$ . We take only  $G = U(m)$  here;  $m \neq d$  in general ([6, 30]).
Example 2.2. For  $M \in \mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right)$  and  $\mathbf{x} \in \mathrm{BV}\left([0,T];\mathbb{R}^d\right)$  linear,  $\mathcal{U}_M(X) = e^{M(\mathbf{x}_T - \mathbf{x}_0)}$ . In particular, when  $m = 1$ ,  $\mathfrak{u}(1)$  is reduced to  $i\mathbb{R}$  and  $M(y) = i\langle \lambda_M, y \rangle$  for some  $\lambda_M \in \mathbb{R}^d$ .
Motivated by the universality and characteristic property of unitary features ([7], see Appendix A.3), we constructed a unitary layer which transforms any  $d$ -dimensional time series  $x = (x_0, \dots, x_N)$  to the unitary feature of its piecewise linear interpolation  $\mathbf{X}$ . It is a special case of the path development layer [26], when Lie algebra is chosen as  $\mathfrak{u}(m)$ . In fact, the explicit formula holds:  $\mathcal{U}_M(\mathbf{X}) = \prod_{i=1}^{N+1} \exp(M(\Delta x_i))$ , where  $\Delta x_i := x_i - x_{i-1}$  and  $\exp$  is the matrix exponential.
**Convention 2.3.** The space  $\mathcal{L}\left(\mathbb{R}^d, \mathfrak{u}(m)\right)$  in which  $M$  of Eq. (4) resides is isomorphic to  $\mathfrak{u}(m)^d$ , where  $\mathfrak{u}(m)$  is Lie algebra isomorphic to  $\mathbb{R}^{\frac{m(m-1)}{2}}$ . For each  $\theta \in \mathfrak{u}(m)^d$  given by anti-Hermitian matrices  $\{\theta^{(i)}\}_{i=1}^d$ , a linear map  $M$  is uniquely induced:  $M(x) = \sum_{i=1}^d \theta^{(i)}\langle x, e_i \rangle$ ,  $\forall x \in \mathbb{R}^d$ .
3 Path characteristic function loss
3.1 Path characteristic function (PCF)
The unitary feature of a path  $\mathbf{x} \in \mathcal{X}$  plays a role similar to that played by  $e^{i\langle x, \lambda \rangle}$  to an  $\mathbb{R}^d$ -valued random variable. Thus, for a random path  $\mathbf{X}$ , the expected unitary feature can be viewed as the characteristic function for measures on the path space ([7]).
Definition 3.1. Let  $\mathbf{X}$  be an  $\mathcal{X}$ -valued random variable and  $\mathbb{P}_{\mathbf{X}}$  be its measure. The path characteristic function (PCF) of  $\mathbf{X}$  of order  $m \in \mathbb{N}$  is the map  $\Phi_{\mathbf{X}}^{(m)}: \mathcal{L}(\mathbb{R}^d, \mathfrak{u}(m)) \to \mathbb{C}^{m \times m}$  given by
$$
\Phi_ {\mathbf {X}} (M) := \mathbb {E} [ \mathcal {U} _ {M} (\mathbf {X}) ] = \int_ {\mathcal {X}} \mathcal {U} _ {M} (\mathbf {x}) \mathrm {d} \mathbb {P} _ {\mathbf {X}} (\mathbf {x}).
$$
The path characteristic function (PCF)  $\Phi_{\mathbf{X}}: \bigoplus_{m=0}^{\infty} \mathcal{L}\left(\mathbb{R}^{d}, \mathfrak{u}(m)\right) \to \bigoplus_{m=0}^{\infty} \mathbb{C}^{m \times m}$  is defined by the natural grading:  $\Phi_{\mathbf{X}}|_{\mathcal{L}(\mathbb{R}^{d}, \mathfrak{u}(m))} = \Phi_{\mathbf{X}}^{(m)}$  for each  $m \in \mathbb{N}$ .
In the above,  $\mathcal{U}_M(\mathbf{x})\in U(m)$  is the unitary feature of the path  $\mathbf{x}$  under  $M$ . See Definition 2.1.
Similarly to the characteristic function of  $\mathbb{R}^d$ -valued random variables, the PCF always exists. Moreover, we have the following important result, whose proof is presented in Appendix A.
Theorem 3.2 (Characteristicity). Let  $\mathbf{X}$  and  $\mathbf{Y}$  be  $\mathcal{X}$ -valued random variables. They have the same distribution (denoted as  $\mathbf{X} \stackrel{d}{=} \mathbf{Y}$ ) if and only if  $\Phi_{\mathbf{X}} = \Phi_{\mathbf{Y}}$ .
3.2 A new distance measure via PCF
We now introduce a novel and natural distance metric, which measures the discrepancy between distributions on the path space via comparing their PCFs. Throughout,  $d_{\mathrm{HS}}$  denotes the metric associated with the Hilbert-Schmidt norm  $\| \bullet \|_{\mathrm{HS}}$  on  $\mathbb{C}^{m\times m}$ :
$$
d _ {\mathrm {H S}} (A, B) := \sqrt {\| A - B \| _ {\mathrm {H S}} ^ {2}} = \sqrt {\operatorname {t r} \left[ (A - B) (A - B) ^ {*} \right]}.
$$
Definition 3.3. Let  $\mathbf{X},\mathbf{Y}:[0,T]\to \mathbb{R}^d$  be stochastic processes and  $\mathbb{P}_{\mathcal{M}}$  be a probability distribution on  $\mathfrak{u}(m)^d\coloneqq \mathcal{L}\left(\mathbb{R}^d,\mathfrak{u}(m)\right)$  (recall Convention 2.3). Define the squared PCF-based distance (PCFD) between  $\mathbf{X}$  and  $\mathbf{Y}$  with respect to  $\mathbb{P}_{\mathcal{M}}$  as
$$
\operatorname {P C F D} _ {\mathcal {M}} ^ {2} (\mathbf {X}, \mathbf {Y}) = \mathbb {E} _ {M \sim \mathbb {P} _ {\mathcal {M}}} \left[ d _ {\mathrm {H S}} ^ {2} \left(\Phi_ {\mathbf {X}} (M), \Phi_ {\mathbf {Y}} (M)\right) \right]. \tag {5}
$$
We shall not distinguish between  $\mathcal{M}$  and  $\mathbb{P}_{\mathcal{M}}$  for simplicity.
PCFD exhibits several mathematical properties, which provide the theoretical justification for its efficacy as the discriminator on the space of measures on the path space, leading to empirical performance boost. First, PCFD has the characteristic property.
Lemma 3.4 (Separation of points). Let  $\mathbf{X},\mathbf{Y}\in \mathcal{P}(\mathcal{X})$  and  $\mathbf{X}\neq \mathbf{Y}$ . Then there exists  $m\in \mathbb{N}$ , such that if  $\mathcal{M}$  is a  $\mathfrak{u}(m)^d$ -valued random variable with full support, then  $\operatorname{PCFD}_{\mathcal{M}}(\mathbf{X},\mathbf{Y})\neq 0$ .
Furthermore,  $\mathrm{PCFD}_{\mathcal{M}}$  has a simple uniform upper bound for any fixed  $m \in \mathbb{N}$ :
Lemma 3.5. Let  $\mathcal{M}$  be a  $\mathfrak{u}(m)^d$ -valued random variable. Then, for any BV  $([0,T];\mathbb{R}^d)$ -valued random variables  $\mathbf{X}$  and  $\mathbf{Y}$ , it holds that  $\mathrm{PCFD}_{\mathcal{M}}^2 (\mathbf{X},\mathbf{Y})\leq 2m^2$ .
Under mild conditions, PCFD is a.e. differentiable with respect to a continuous parameter, thus ensuring the feasibility of gradient descent in training.
Theorem 3.6 (Lipschitz 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  $\mathbb{E}_{M \sim \mathbb{P}_{\mathcal{M}}}[||M||^2] < \infty$  and  $\mathbb{E}_{\mathbf{Z} \sim \mathbb{Q}}[\omega(\mathbf{Z})] < \infty$ . Then  $\mathrm{PCFD}_{\mathcal{M}}(g_{\theta}(\mathbf{Z}), \mathbf{X})$  is Lipschitz in  $\theta$ . Moreover, it holds that
$$
| \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta} (\mathbf {Z}), \mathbf {X}\right) - \operatorname {P C F D} _ {\mathcal {M}} \left(g _ {\theta^ {\prime}} (\mathbf {Z}), \mathbf {X}\right) | \leq \sqrt {\mathbb {E} _ {M \sim \mathbb {P} _ {\mathcal {M}}} \left[ \| M \| ^ {2} \right]} \mathbb {E} _ {\mathbf {Z} \sim \mathbb {Q}} \left[ \omega (\mathbf {Z}) \right] \rho (\theta , \theta^ {\prime})
$$
for any  $\theta, \theta' \in \Theta$ ,  $\mathbf{Z} \in \mathcal{Z}$ ,  $\mathbf{X} \in \mathcal{X}$ , and  $\mathcal{M} \in \mathcal{P}$  ( $\mathfrak{u}(m)^d$ ).
Remark 3.7. The parameter space  $(\Theta, \rho)$  is usually taken to be  $\mathbb{R}^{\bar{d}}$  for some  $\bar{d} \in \mathbb{N}$ . In this case, by Rademacher's theorem  $\mathrm{PCFD}_{\mathcal{M}}(g_{\theta}(\mathbf{Z}), \mathbf{X})$  is a.e. differentiable in  $\theta$ .
Similarly to metrics on measures over  $\mathbb{R}^d$  (cf. [2, 24]), we construct a metric based on PCFD, denoted as  $\widetilde{\mathrm{PCFD}}$ , on the space  $\mathcal{P}(\mathcal{X})$  of Borel probability measures over the path space, and we prove that it metrises the weak-star topology on  $\mathcal{P}(\mathcal{X})$ . Throughout,  $\stackrel{\mathrm{d}}{\rightarrow}$  denotes the convergence in law.
Theorem 3.8 (Informal, convergence in law). Let  $\{\mathbf{X}_n\}_{n\in \mathbb{N}}$  and  $\mathbf{X}$  be  $\mathcal{X}$ -valued random variables with measures supported in a compact subset of  $\mathcal{X}$ . Then  $\widetilde{\mathrm{PCFD}} (\mathbf{X}_n,\mathbf{X})\to 0\iff \mathbf{X}_n\overset {d}{\to}\mathbf{X}$
The formal statement and proof can be found in Lemma B.2 and Theorem B.8 in the Appendix.
Similar to [40] for  $\mathbb{R}^d$ , we prove that PCFD can be interpreted as an MMD with a specific kernel  $\kappa$  (see Appendix B.3). Example B.12 illustrates that the PCFD has the superior test power for hypothesis testing on stochastic processes compared with CF distance on the flattened time series.
3.3 Computing PCFD under empirical measures
Now, we shall illustrate how to compute the PCFD on the path space.
Let  $\bar{\mathbf{X}}\coloneqq \{\mathbf{x}^i\}_{i = 1}^n$  and  $\bar{\mathbf{Y}}\coloneqq \{\mathbf{y}^i\}_{i = 1}^{n'}$  be i.i.d. drawn respectively from  $\mathcal{X}$ -valued random variables  $\mathbf{X}$  and  $\mathbf{Y}$ . First, for any linear map  $M\in \mathfrak{u}(m)^d$ , the empirical estimator of  $\Phi_{\mathbf{X}}(M)$  is the average of unitary features of all observations  $\bar{\mathbf{X}} = \{\mathbf{x}_i\}_{i = 1}^n$ , i.e.,  $\Phi_{\bar{\mathbf{X}}} (M) = \frac{1}{n}\sum_{i = 1}^{n}\mathcal{U}_M(\mathbf{x}_i)$ . We then parameterise the  $\mathfrak{u}(m)^d$ -valued random variable  $\mathcal{M}$  via the empirical measure  $\mathcal{M}_{\theta_M}$ , i.e.,  $\mathcal{M}_{\theta_M} = \sum_{i = 1}^{k}\delta_{M_i}$ , where  $\theta_M\coloneqq \{M_i\}_{i = 1}^k\in \mathfrak{u}(m)^{d\times k}$  are the trainable model parameters. Finally, define the corresponding empirical path characteristic function distance (EPCFD) as
$$
\operatorname {E P C F D} _ {\theta_ {M}} \left(\bar {\mathbf {X}}, \bar {\mathbf {Y}}\right) = \sqrt {\frac {1}{k} \sum_ {i = 1} ^ {k} \left\| \boldsymbol {\Phi} _ {\bar {\mathbf {X}}} \left(M _ {i}\right) - \boldsymbol {\Phi} _ {\bar {\mathbf {Y}}} \left(M _ {i}\right) \right\| _ {\mathrm {H S}} ^ {2}}. \tag {6}
$$
Our approach to approximating  $\mathcal{M}$  via the empirical distribution differs from that in [25], where  $\mathcal{M}$  is parameterised by mixture of Gaussian distributions. In §4.1 and §5, it is shown that, by optimising the empirical distribution, a moderately sized  $k$  is sufficient for achieving superior performance, in contrast to a larger sample size required by [25].
Figure 1: Flowchart of calculating the PCF  $\Phi_{\mathbf{X}}(M_{\theta})$
4 PCF-GAN for time series generation
4.1 Training of the EPCFD
In this subsection, we apply the EPCFD to GAN training for time series generation as the discriminator. We train the generator to minimise the EPCFD between true and synthetic data distribution, whereas the empirical distribution of  $\mathcal{M}$  characterised by  $\theta_{M} \in \mathfrak{u}(m)^{d \times k}$  is optimised by maximising EPCFD.
By an abuse of notation, let  $\mathcal{X} \coloneqq \mathbb{R}^{d \times n_T}$  ( $\mathcal{Z} \coloneqq \mathbb{R}^{e \times n_T}$ , resp.) denote the data (noise, resp.) space, composed of  $\mathbb{R}^d$  ( $\mathbb{R}^e$ , resp.) time series of length  $n_T$ . As discussed in §2.2,  $\mathcal{X}$  and  $\mathcal{Z}$  can be viewed as path spaces via linear interpolation. Like the standard GANs, our model is comprised of a generator  $G_{\theta_g}: \mathcal{Z} \to \mathbb{R}^{d \times n_T}$  and the discriminator EPCFD $_{\theta_M}$ :  $\mathbb{P}(\mathcal{X}) \times \mathbb{P}(\mathcal{X}) \to \mathbb{R}^+$ , where  $\theta_M \in \mathfrak{u}(m)^{k \times d}$  is the model parameter of the discriminator, which fully characterises the empirical measure of  $\mathcal{M}$ . The pre-specified noise random variable  $\mathbf{Z} = (Z_{t_i})_{i=0}^{n_T-1}$  is the discretised Brownian motion on  $[0,1]$  with time mesh  $\frac{1}{n_T}$ . The induced distribution of the fake data is given by  $G_{\theta_g}(\mathbf{Z})$ . Hence, the min-max objective of our basic version PCF-GAN is
$$
\min  _ {\theta_ {g}} \max  _ {\theta_ {M}} \operatorname {E P C F D} _ {\theta_ {M}} \left(G _ {\theta_ {g}} (\mathbf {Z}), \mathbf {X}\right).
$$
We apply mini-batch gradient descent to optimise the model parameters of the generator and discriminator in an alternative manner. In particular, to compute gradients of the discriminator parameter  $\theta_{M}$ , we use the efficient backpropagation algorithm through time introduced in [26], which effectively leverages the Lie group-valued outputs and the recurrence structure of the unitary feature. The initialisation of  $\theta_{M}$  for the optimisation is outlined in the Appendix B.4.1.
Learning time-dependent Ornstein-Uhlenbeck process Following [19], we apply the proposed PCF-GAN to the toy example of learning the distribution of synthetic time series data simulated via the time-dependent Ornstein-Uhlenbeck (OU) process. Let  $(\mathbf{X}_t)_{t\in [0,T]}$  be an  $\mathbb{R}$ -valued stochastic process described by the SDE, i.e.,  $d\mathbf{X}_t = (\mu t - \theta \mathbf{X}_t)dt + \sigma d\mathbf{B_t}$  with  $\mathbf{X_0} \sim \mathcal{N}(\mathbf{0},\mathbf{1})$ , where  $(\mathbf{B_t})_{t\in [0,\mathbf{T}]}$  is 1D Brownian motion and  $\mathcal{N}(0,1)$  is the standard normal distribution. We set  $\mu = 0.01$ ,  $\theta = 0.02$ ,  $\sigma = 0.4$  and time discretisation  $\delta t = 0.1$ . We generate 10000 samples from  $t = 0$  to  $t = 63$ , down-sampled at each integer time point. Figure 2 shows that the synthetic data generated by our GAN model, which uses the EPCFD discriminator, is visually indistinguishable from true data. Also, our model accurately captures the marginal distribution at various time points.