Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

TL;DR

This paper introduces Conditional Wasserstein Autoencoders (CWAEs), a novel framework for conditional simulation leveraging low-dimensional structures and triangular decoders, with theoretical insights and improved approximation accuracy.

Contribution

The paper develops CWAEs with triangular decoders, explores their theoretical properties, and demonstrates their superior performance over existing methods in low-dimensional conditional problems.

Findings

CWAEs achieve lower approximation error than LREnKF.

Triangular decoders enable effective exploitation of low-dimensional structure.

Numerical experiments confirm the advantages of CWAEs in conditional simulation.

Abstract

We present Conditional Wasserstein Autoencoders (CWAEs), a framework for conditional simulation that exploits low-dimensional structure in both the conditioned and the conditioning variables. The key idea is to modify a Wasserstein autoencoder to use a (block-) triangular decoder and impose an appropriate independence assumption on the latent variables. We show that the resulting model gives an autoencoder that can exploit low-dimensional structure while simultaneously the decoder can be used for conditional simulation. We explore various theoretical properties of CWAEs, including their connections to conditional optimal transport (OT) problems. We also present alternative formulations that lead to three architectural variants forming the foundation of our algorithms. We present a series of numerical experiments that demonstrate that our different CWAE variants achieve substantial reductions in approximation error relative to the low-rank ensemble Kalman filter (LREnKF), particularly in problems where the support of the conditional measures is truly low-dimensional.