Theoretical bounds on data requirements for the ray-based classification
This work provides a theoretical foundation for shape classification that could reduce data requirements compared to volumetric or surface-based methods, but it is incremental as it builds on prior empirical and experimental validations of the ray-based framework.
The paper tackles the problem of classifying high-dimensional convex shapes by establishing theoretical bounds on the number of rays needed for ray-based classification, deriving lower bounds in terms of geometric parameters like length, diameter, and angles for 2D shapes and generalizing to convex polytopes in higher dimensions.
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.