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.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced in [Lavenant et al., 2021], and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.

We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals.

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.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced in [Lavenant et al., 2021], and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator.

We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-art Nuclear Norm and OptSpace methods.