Theoretical Bounds on Data Requirements for the Ray-Based Classification
Brian Weber, Sandesh Kalantre, Thomas McJunkin, Jacob Taylor, Justyna Zwolak
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 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.
, Kalantre, S.
, McJunkin, T.
, Taylor, J.
and Zwolak, J.
Theoretical Bounds on Data Requirements for the Ray-Based Classification, SN Computer Science, [online], https://doi.org/10.1007/s42979-021-00921-0, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=931504
(Accessed August 9, 2022)