# Generative modeling of time-dependent densities via optimal transport and projection pursuit

arXiv: https://arxiv.org/abs/2304.09663 · HTML: https://mlsanity.com/paper/2304.09663/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 42 references that are in the map.
- First posted on arXiv 19 Apr 2023.

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

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Quant GANs: Deep Generation of Financial Time Series](https://mlsanity.com/paper/1907.06673/) (2019)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [FloMo: Tractable Motion Prediction with Normalizing Flows](https://mlsanity.com/paper/2103.03614/) (2021)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Large-scale optimal transport map estimation using projection pursuit](https://mlsanity.com/paper/2106.05838/) (2021)
- [Continuous Latent Process Flows](https://mlsanity.com/paper/2106.15580/) (2021)
- [Temporal Normalizing Flows](https://mlsanity.com/paper/1912.09092/) (2019)

## Built on by (most cited first, 2 of 2)

- [MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation](https://mlsanity.com/paper/2404.15731/) (2024)
- [Measure-Theoretic Time-Delay Embedding](https://mlsanity.com/paper/2409.08768/) (2024)

## Abstract (arXiv)

Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative which requires minimal hyperparameter tuning and scales favorably to high dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al., 2019] to join successive samples and subsequently use transport splines [Chewi et al., 2020] to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are thus computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can thus be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities.