Enhanced Importance Sampling through Latent Space Exploration in Normalizing Flows

TL;DR

This paper tackles efficient estimation of rare-event probabilities in safety-critical simulations by performing importance sampling in the latent space of a pre-trained normalizing flow. By mapping complex target outcomes to a simpler latent distribution, the method facilitates isotropic and comprehensive exploration of failure regions, then recovers target-space samples via the invertible flow. The authors introduce a Löwner–John ellipsoid-based cost function and evaluate two IS strategies—cross-entropy and sequential IS—within latent space across three autonomous-system simulators, showing improved sample efficiency, coverage, and density over target-space IS. The approach offers a flexible, plug-and-play framework for safety validation that remains effective when failure criteria evolve while the target distribution stays fixed, with promising implications for high-fidelity robotic and aerospace simulators.

Abstract

Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

Figures (3)

• Figure 1: Importance sampling in target space (top row) versus importance sampling in a flow's latent space (bottom row). The target space proposal distribution generates many samples in the low-probability valley between the two failure modes, while the latent proposal generates samples that more closely map to the two failure regions.
• Figure 2: A normalizing flow transforms a base distribution to a target distribution.
• Figure 3: Failure regions for the nonholonomic robot shown in target space and latent space. The points in $\mathcal{X}_i$ are mapped to latent space and the Löwner--John ellipsoids are re-computed.