Robust time series generation via Schrödinger Bridge: a comprehensive evaluation

TL;DR

This work systematically evaluates the Schrödinger Bridge for time-series generation (SBTS), recasting synthesis as entropic optimal interpolation between a reference path measure and a target distribution to yield a finite-horizon SDE that captures temporal dynamics. It formalizes SBTS on discrete grids with a kernel-based drift estimator, introduces bandwidth and Markov-order considerations for long series, and benchmarks SBTS against state-of-the-art GAN- and diffusion-based methods across multiple real and toy datasets. The study also presents a robustness framework using parametric stochastic processes and proposes a practical scaling procedure to align variance with theoretical diffusion assumptions. Overall, SBTS often outperforms GAN-based models and is competitive with diffusion models, while offering fast sample generation and minimal pre-training, albeit with bandwidth sensitivity and a constant-variance limitation that motivates future work toward stochastic volatility and neural drift estimation.

Abstract

We investigate the generative capabilities of the Schrödinger Bridge (SB) approach for time series. The SB framework formulates time series synthesis as an entropic optimal interpolation transport problem between a reference probability measure on path space and a target joint distribution. This results in a stochastic differential equation over a finite horizon that accurately captures the temporal dynamics of the target time series. While the SB approach has been largely explored in fields like image generation, there is a scarcity of studies for its application to time series. In this work, we bridge this gap by conducting a comprehensive evaluation of the SB method's robustness and generative performance. We benchmark it against state-of-the-art (SOTA) time series generation methods across diverse datasets, assessing its strengths, limitations, and capacity to model complex temporal dependencies. Our results offer valuable insights into the SB framework's potential as a versatile and robust tool for time series generation.

Figures (8)

• Figure 1: Generation of a Markovian GARCH model of order 2 (see \ref{['sec:appendix_dataset']}), with $h=0.02$ (left) and $h=0.2$ (right). We can clearly see the effect of a too small $h$, here $0.02$, as the left plot exhibits undesirable behavior, whereas the right one displays the desired outcome.
• Figure 2: Bandwidth selection for GARCH data defined as in \ref{['sec:appendix_dataset']} and Apple close daily return of length $252$, using different values of $k$. For both time series, we see that a larger $h$ is needed when using the whole time series ($k=252$). Otherwise, a smaller $h$ is sufficient, and the Markovianity even improved the $MSE$ for GARCH data, as it is really Markovian.
• Figure 3: Comparison of original and SBTS data for Stock and AR datasets.
• Figure 4: Distribution of estimated Ornstein-Uhlenbeck parameters using MLE. We show in orange, blue and green the density respectively from the SBTS samples, data samples, and real range.
• Figure 5: Distribution of estimated Heston parameters using MLE. We show in orange, blue and green the density respectively from the SBTS samples, data samples, and real range.

Theorems & Definitions (1)

• theorem 1