  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.
