From Points to Sets: Set-Based Safety Verification in the Latent Space

TL;DR

This paper introduces a set-based safety verification method in the latent space using zonotopes, improving safety guarantees over point-based methods by accounting for state uncertainty.

Contribution

It extends latent safety control to set-valued states with zonotope propagation, enabling adaptive, set-based safety certificates that outperform point-based evaluations.

Findings

Set evaluation achieved 5/5 collision-free passages on a quadrotor task.

Set evaluation detects 4.1% more safety blind spots than point evaluation.

Safety gaps vary up to 12x across certificate heads, justifying adaptive set evaluation.

Abstract

We extend latent representation methods for safety control design to set-valued states. Recent work has shown that barrier functions designed in a learned latent space can transfer safety guarantees back to the original system, but these methods evaluate certificates at single state points, ignoring state uncertainty. A fixed safety margin can partially address this but cannot adapt to the anisotropic and time-varying nature of the uncertainty gap across different safety constraints. We instead represent the system state as a zonotope, propagate it through the encoder to obtain a latent zonotope, and evaluate certificates over the worst case of the entire set. On a 16-dimensional quadrotor suspended-load gate passage task, set-valued evaluation achieves 5/5 collision-free passages, compared to 1/5 for point-based evaluation and 2/5 for a fixed-margin baseline. Set evaluation reports safety in 44.4% of per-head evaluations versus 48.5% for point-based, and this greater conservatism detects 4.1% blind spots where point evaluation falsely certifies safety, enabling earlier corrective control. The safety gap between point and set evaluation varies up to $12\times$ across certificate heads, explaining why no single fixed margin suffices and confirming the need for per-head, per-timestep adaptation, which set evaluation provides by construction.