Theoretical bounds on data requirements for the ray-based classification

TL;DR

This paper establishes theoretical bounds on the minimum number of rays needed for convex shape classification in high-dimensional spaces, improving data efficiency over traditional volumetric methods.

Contribution

It provides the first theoretical bounds on ray-based shape classification, relating the number of rays to geometric parameters for arbitrary convex shapes.

Findings

Derived lower bounds for 2D shapes based on length, diameter, and angles.

Generalized bounds for convex polytopes in higher dimensions.

Enables shape estimation with fewer data elements than volumetric methods.

Abstract

The problem of classifying high-dimensional shapes in real-world data grows in complexity as the dimension of the space increases. For the case of identifying convex shapes of different geometries, a new classification framework has recently been proposed in which the intersections of a set of one-dimensional representations, called rays, with the boundaries of the shape are used to identify the specific geometry. This ray-based classification (RBC) has been empirically verified using a synthetic dataset of two- and three-dimensional shapes (Zwolak et al. in Proceedings of Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020), Vancouver, Canada [December 11, 2020], arXiv:2010.00500, 2020) and, more recently, has also been validated experimentally (Zwolak et al., PRX Quantum 2:020335, 2021). Here, we establish a bound on the number of rays necessary for shape classification, defined by key angular metrics, for arbitrary convex shapes. For two dimensions, we derive a lower bound on the number of rays in terms of the shape's length, diameter, and exterior angles. For convex polytopes in $\mathbb{R}^N$, we generalize this result to a similar bound given as a function of the dihedral angle and the geometrical parameters of polygonal faces. This result enables a different approach for estimating high-dimensional shapes using substantially fewer data elements than volumetric or surface-based approaches.