Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

arXiv cs.LG / 4/6/2026

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Key Points

The paper introduces Conditional Wasserstein Autoencoders (CWAEs) as a framework for conditional simulation that leverages low-dimensional structure in both the conditioning and conditioned variables.
It adapts a Wasserstein autoencoder by using a (block-) triangular decoder structure and applying an independence assumption on latent variables to enable conditional simulation.
The authors study theoretical properties of CWAEs, including connections to conditional optimal transport (OT) formulations.
They propose multiple architectural variants (three variants) and corresponding algorithms based on alternative formulations of the CWAE idea.
Numerical experiments show substantial approximation-error reductions versus a low-rank ensemble Kalman filter (LREnKF), especially when the conditional measures’ support is genuinely low-dimensional.

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.