# PCF-GAN: generating sequential data via the characteristic function of measures on the path space

arXiv: https://arxiv.org/abs/2305.12511 · HTML: https://mlsanity.com/paper/2305.12511/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 55 references that are in the map.
- First posted on arXiv 21 May 2023.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs](https://mlsanity.com/paper/1706.02633/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning](https://mlsanity.com/paper/1705.07761/) (2017)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Unusual Effectiveness of Averaging in GAN Training](https://mlsanity.com/paper/1806.04498/) (2018)
- [COT-GAN: Generating Sequential Data via Causal Optimal Transport](https://mlsanity.com/paper/2006.08571/) (2020)
- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [GT-GAN: General Purpose Time Series Synthesis with Generative Adversarial Networks](https://mlsanity.com/paper/2210.02040/) (2022)
- [A Characteristic Function Approach to Deep Implicit Generative Modeling](https://mlsanity.com/paper/1909.07425/) (2019)
- [Generating multivariate time series with COmmon Source CoordInated GAN (COSCI-GAN)](https://mlsanity.com/paper/2205.13741/) (2022)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [Path Development Network with Finite-dimensional Lie Group Representation](https://mlsanity.com/paper/2204.00740/) (2022)
- [Reciprocal Adversarial Learning via Characteristic Functions](https://mlsanity.com/paper/2006.08413/) (2020)
- [Conditional Loss and Deep Euler Scheme for Time Series Generation](https://mlsanity.com/paper/2102.05313/) (2021)

## Built on by (most cited first, 15 of 15)

- [Structured Agentic Workflows for Financial Time-Series Modeling with LLMs and Reflective Feedback](https://mlsanity.com/paper/2508.13915/) (2025)
- [Time-Causal VAE: Robust Financial Time Series Generator](https://mlsanity.com/paper/2411.02947/) (2024)
- [Generative Modelling of L'evy Area for High Order SDE Simulation](https://mlsanity.com/paper/2308.02452/) (2023)
- [GCN-DevLSTM: Path Development for Skeleton-Based Action Recognition](https://mlsanity.com/paper/2403.15212/) (2024)
- [High Rank Path Development: an approach of learning the filtration of stochastic processes](https://mlsanity.com/paper/2405.14913/) (2024)
- [MOSAIC: Modular Orchestration for Structured Agentic Intelligence and Composition](https://mlsanity.com/paper/2606.00708/) (2026)
- [E4GEN: Event-level Explainable Extreme-Enhanced Time-series Generation](https://mlsanity.com/paper/2606.01634/) (2026)
- [OpenFinGym: A Verifiable Multi-Task Gym Environment for Evaluating Quant Agents](https://mlsanity.com/paper/2606.26350/) (2026)
- [Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection](https://mlsanity.com/paper/2407.11873/) (2024)
- [STEB: In Search of the Best Evaluation Approach for Synthetic Time Series](https://mlsanity.com/paper/2505.21160/) (2025)
- [Learning Adaptive Distribution Alignment with Neural Characteristic Function for Graph Domain Adaptation](https://mlsanity.com/paper/2602.10489/) (2026)
- [Generating Financial Time Series by Matching Random Convolutional Features](https://mlsanity.com/paper/2606.05138/) (2026)
- [Diffusion Models for Adaptive Sequential Data Generation](https://mlsanity.com/paper/2606.06007/) (2026)
- [Diffusion-Based Synthetic Data Pretraining for Enhancing Activity Recognition](https://mlsanity.com/paper/2610.02292/) (2026)
- [Cross-View Time-Frequency Diffusion for Extreme-Aware Time-Series Generation](https://mlsanity.com/paper/2610.17407/) (2026)

## Abstract (arXiv)

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. Code is available at https://github.com/DeepIntoStreams/PCF-GAN.