SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For PCF-GAN, the coefficient  $\lambda_{1}$  for the recovery loss was 50, and the number of unitary linear maps was 8.
D Supplementary results
D.1 Ablation study
An ablation study was conducted on the PCF-GAN model to evaluate the importance of its various components. Specifically, the reconstruction loss and regularization loss were disabled in order to assess their impact on model performance across benchmark datasets and various test metrics. Table 3 consistently demonstrated that the PCF-GAN model outperformed the ablated versions, confirming the significance of these two losses in the overall model performance.
Table 3: Ablation study of PCF-GAN
Dataset	Test Metrics	PCF-GAN	w/o \( L_{recovery} \)	w/o \( L_{regularization} \)	w/o \( L_{regularization} \& L_{recovery} \)
RV	Discriminative	.0108±.006	.0178±.017	.0152±.020	.0101±.007
Predictive	.0390±.000	.0389±.000	.0390±.003	.0391±.001
Sig-MMD	.0024±.001	.0037±.001	.0036 ± .002	.0027±.001
Stock	Discriminative	.0784±.028	.0963±.011	.2538±.052	.0815±.001
Predictive	.0125±.000	.0123±.000	.0127±.000	.0126±.001
Sig-MMD	.0017±.000	.0062±.002	.0024±.001	.0021±.001
Air	Discriminative	.2326±.058	.3940±.068	.4783±.029	.3875±.009
Predictive	.0237±.000	.0239±.000	.0283±.001	.0240±.000
Sig-MMD	.0126±.005	.0111±.003	.0232±.004	.0163±.004
EEG	Discriminative	.3660±.025	.4942±.010	.5000±.000	.4649±.015
Predictive	.0246±.000	.0299±.000	.0636±.007	.0248±.000
Sig-MMD	.0180±.004	.0296±.008	1.197±.234	.0278±007
Notably, the inclusion of the two additional losses significantly improved model performance on high-dimensional time series datasets, such as Air Quality and EEG, indicating that the proposed auto-encoder architecture effectively learns meaningful low-dimensional sequential embeddings.
Conversely, the exclusive use of the reconstruction loss led to a notable decrease in model performance, suggesting that the  $l^2$  samplewise distance might not be suitable for time series data. However, the additional regularization loss helped overcome this issue by ensuring that the sequential embedding space is confined to a predetermined noise space, such as the discretized Brownian motion. As a result, the regularization loss helped to mitigate the problems that arose when relying solely on the reconstruction loss.
D.2 Generated samples
In this section, we present random samples from the four benchmark datasets generated by PCGFAN, TimeGAN, RGAN, and COT-GAN. Although interpreting the sample plots of the generated time series poses a challenge, our observations reveal that PCF-GAN successfully generates time series that capture the temporal dependencies exhibited in the original time series across all datasets. Conversely, COT-GAN generates trajectories that are relatively smoother compared to the real time series samples, demonstrated on Stock and EEG datasets, by Figure 8 and Figure 10 respectively. Figure 10 shows that TimeGAN occasionally produces samples with higher oscillations than those found in the real samples.
Figure 7: Generated samples from all models on Rough volatility dataset
Figure 8: Generated samples from all models on Stock dataset
Figure 9: Generated samples from all models on Air Quality dataset
Figure 10: Generated samples from all models on EEG dataset
D.3 Reconstructed samples
In this section, we present additional reconstructed time series samples generated by PCF-GAN and TimeGAN. Figure 11 illustrates that PCF-GAN consistently outperforms TimeGAN by producing higher-quality reconstructed samples across all datasets.
Figure 11: Reconstructed samples from PCF-GAN and TimeGAN on all benchmark datasets.
D.4 Test metrics on (auto-)correlation and marginal distribution
This subsection details the supplementary test metrics in terms of fitting the autocorrelation, cross-correlation, and marginal distribution, as presented in Table 4. This table confirms that our proposed PCF-GAN consistently outperforms the benchmarking models across all datasets.
Table 4: Performance comparison of PCF-GAN and baselines on auto-correlation, cross-correlation and marginal distribution metrics. Best for each task is shown in bold.
Task	Generation
Dataset	Test Metrics	RGAN	COT-GAN	TimeGAN	PCF-GAN
RV	Auto-cor (lag 1)	.0393±.001	.0608±.001	.0031±.001	.0022±.000
Auto-cor (lag 5)	.0134±.002	.119±.002	.0035±.002	.0030±.002
Cross-cor (lag 0)	.0193±.007	.0234±.002	.0187±.011	.0264±.011
Cross-cor (lag 5)	.0222±.007	.1441±.012	.0219±.010	.0158±.011
Marginal Dist	.311±1.13	.2157±.306	.1636±.223	.1234±.126
Stock	Auto-cor (lag 1)	.127±.005	.202±.0035	.210±.005	.0123±.005
Auto-cor (lag 5)	.149±.009	.267±.006	.104±.006	.0187±.006
Cross-cor (lag 0)	.145±.031	.169±.041	.549±.034	.1815±.058
Cross-cor (lag 5)	.341±.031	.456±.053	.747±.038	.2510±.062
Marginal Dist	.3276±.044	.2826±.061	.4264±.063	.2730±.033
Air	Auto-cor (lag 1)	.1678±.010	.320±.006	.1949±.006	.0927±.003
Auto-cor (lag 5)	.3226±.016	.520±.028	.5349±.034	.4739±.023
Cross-cor (lag 0)	2.608±.106	1.942±.059	2.844±.0812	2.687±.149
Cross-cor (lag 5)	3.181±.101	2.176±.116	2.536±.112	2.115±.121
Marginal Dist	.5527±.523	.5142±.600	.6229±.595	.5066±.572
EEG	Auto-cor (lag 1)	5.918±.116	6.202±.111	5.754±.083	5.668±.079
Auto-cor (lag 5)	4.285±.074	5.911±.107	5.265±.083	4.467±.127
Cross-cor (lag 0)	51.16±.508	24.12±.702	26.84±.638	22.27±.550
Cross-cor (lag 5)	47.97±.354	31.31±.920	25.95±.466	19.43±.412
Marginal Dist	15.18±21.94	8.518±13.6	13.35±21.7	10.09±16.6