# COT-GAN: Generating Sequential Data via Causal Optimal Transport

arXiv: https://arxiv.org/abs/2006.08571 · HTML: https://mlsanity.com/paper/2006.08571/

## Key facts

- Cited by 71 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2020.

## Built on by (most cited first, 30 of 71)

- [Diffusion Models for Time Series Applications: A Survey](https://mlsanity.com/paper/2305.00624/) (2023)
- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Towards Generating Real-World Time Series Data](https://mlsanity.com/paper/2111.08386/) (2021)
- [TSGBench: Time Series Generation Benchmark](https://mlsanity.com/paper/2309.03755/) (2023)
- [Generating multivariate time series with COmmon Source CoordInated GAN (COSCI-GAN)](https://mlsanity.com/paper/2205.13741/) (2022)
- [Regular Time-series Generation using SGM](https://mlsanity.com/paper/2301.08518/) (2023)
- [FlowTS: Time Series Generation via Rectified Flow](https://mlsanity.com/paper/2411.07506/) (2024)
- [Generative modeling for time series via Schr"odinger bridge](https://mlsanity.com/paper/2304.05093/) (2023)
- [PCF-GAN: generating sequential data via the characteristic function of measures on the path space](https://mlsanity.com/paper/2305.12511/) (2023)
- [SPATE-GAN: Improved Generative Modeling of Dynamic Spatio-Temporal Patterns with an Autoregressive Embedding Loss](https://mlsanity.com/paper/2109.15044/) (2021)
- [TransFusion: Generating Long, High Fidelity Time Series using Diffusion Models with Transformers](https://mlsanity.com/paper/2307.12667/) (2023)
- [A Multi-scale Time-series Dataset with Benchmark for Machine Learning in Decarbonized Energy Grids](https://mlsanity.com/paper/2110.06324/) (2021)
- [Time-series Transformer Generative Adversarial Networks](https://mlsanity.com/paper/2205.11164/) (2022)
- [Emerging Synergies in Causality and Deep Generative Models: A Survey](https://mlsanity.com/paper/2301.12351/) (2023)
- [TimeLDM: Latent Diffusion Model for Unconditional Time Series Generation](https://mlsanity.com/paper/2407.04211/) (2024)
- [Time-series Generation by Contrastive Imitation](https://mlsanity.com/paper/2311.01388/) (2023)
- [Existence and Estimation of Critical Batch Size for Training Generative Adversarial Networks with Two Time-Scale Update Rule](https://mlsanity.com/paper/2201.11989/) (2022)
- [Time-Causal VAE: Robust Financial Time Series Generator](https://mlsanity.com/paper/2411.02947/) (2024)
- [Robust time series generation via Schr"odinger Bridge: a comprehensive evaluation](https://mlsanity.com/paper/2503.02943/) (2025)
- [The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs](https://mlsanity.com/paper/2302.00713/) (2023)
- [Generative modeling of spatio-temporal weather patterns with extreme event conditioning](https://mlsanity.com/paper/2104.12469/) (2021)
- [Bisimulation Metrics are Optimal Transport Distances, and Can be Computed Efficiently](https://mlsanity.com/paper/2406.04056/) (2024)
- [TimeBridge: Better Diffusion Prior Design with Bridge Models for Time Series Generation](https://mlsanity.com/paper/2408.06672/) (2024)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [Conditional Loss and Deep Euler Scheme for Time Series Generation](https://mlsanity.com/paper/2102.05313/) (2021)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [WaveStitch: Flexible and Fast Conditional Time Series Generation with Diffusion Models](https://mlsanity.com/paper/2503.06231/) (2025)
- [Conditional COT-GAN for Video Prediction with Kernel Smoothing](https://mlsanity.com/paper/2106.05658/) (2021)
- [High Rank Path Development: an approach of learning the filtration of stochastic processes](https://mlsanity.com/paper/2405.14913/) (2024)
- [MSDformer: Multi-scale Discrete Transformer For Time Series Generation](https://mlsanity.com/paper/2505.14202/) (2025)

## Abstract (arXiv)

We introduce COT-GAN, an adversarial algorithm to train implicit generative models optimized for producing sequential data. The loss function of this algorithm is formulated using ideas from Causal Optimal Transport (COT), which combines classic optimal transport methods with an additional temporal causality constraint. Remarkably, we find that this causality condition provides a natural framework to parameterize the cost function that is learned by the discriminator as a robust (worst-case) distance, and an ideal mechanism for learning time dependent data distributions. Following Genevay et al. (2018), we also include an entropic penalization term which allows for the use of the Sinkhorn algorithm when computing the optimal transport cost. Our experiments show effectiveness and stability of COT-GAN when generating both low- and high-dimensional time series data. The success of the algorithm also relies on a new, improved version of the Sinkhorn divergence which demonstrates less bias in learning.