Adapting Noise to Data: Generative Flows from Learned 1D Processes

The default Gaussian latent in flow-based generative models poses challenges when learning certain distributions such as heavy-tailed ones. We introduce a general framework for learning data-adaptive parametric prior distributions (latent noise) using one-dimensional quantile functions, optimized via the Wasserstein distance between noise and data. The quantile-based prior parameterization naturally adapts to both heavy-tailed and compactly supported distributions and shortens transport paths. Numerical results on heavy-tailed weather and image datasets confirm the method’s flexibility and effectiveness achieved with negligible computational overhead.

Modern generative models create new data, such as images or weather fields, by starting from an easy-to-sample random distribution and repeatedly refining it until the result resembles examples in a dataset. Most methods use Gaussian noise as this starting point because it is simple and well understood. However, this choice can make learning harder when the data look very different from a Gaussian distribution, especially for heavy-tailed data, where rare extreme events occur much more often than Gaussian noise would suggest. In our work, we study how changing the starting noise distribution affects generative modeling. Instead of fixing the starting distribution to be Gaussian, we learn a data-adapted distribution directly from the training samples. To keep this efficient, we restrict the learned starting distribution so that its dimensions do not directly interact, which allows us to describe it using simple one-dimensional building blocks while leaving cross-dimensional structure to the main neural network. Experiments on synthetic data, images, and heavy-tailed weather data show that choosing a better starting distribution can shorten the path the model has to learn and improve the generation of difficult tail behavior.