# On the Difficulty of Intersection Checking with Polynomial Zonotopes

arXiv: https://arxiv.org/abs/2305.09901 · HTML: https://mlsanity.com/paper/2305.09901/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2023.

## Built on by (most cited first, 1 of 1)

- [Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis](https://mlsanity.com/paper/2406.04476/) (2024)

## Abstract (arXiv)

Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning. Despite this widespread use, a frequently overlooked difficulty associated with polynomial zonotopes is intersection checking. Determining whether the reachable set, represented as a polynomial zonotope, intersects an unsafe set is not straightforward. In fact, we show that this fundamental operation is NP-hard, even for a simple class of polynomial zonotopes. The standard method for intersection checking with polynomial zonotopes is a two-part algorithm that overapproximates a polynomial zonotope with a regular zonotope and then, if the overapproximation error is deemed too large, splits the set and recursively tries again. Beyond the possible need for a large number of splits, we identify two sources of concern related to this algorithm: (1) overapproximating a polynomial zonotope with a zonotope has unbounded error, and (2) after splitting a polynomial zonotope, the overapproximation error can actually increase. Taken together, this implies there may be a possibility that the algorithm does not always terminate.We perform a rigorous analysis of the method and detail necessary conditions for the union of overapproximations to provably converge to the original polynomial zonotope.