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Theoretical Bounds on Data Requirements for the Ray-Based Classification

Published

Author(s)

Brian Weber, Sandesh Kalantre, Thomas McJunkin, Jacob Taylor, Justyna Zwolak

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 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.
Citation
SN Computer Science
Volume
3
Issue
1

Keywords

machine learning, quantum dots, high-dimensional data, classification

Citation

Weber, B. , Kalantre, S. , McJunkin, T. , Taylor, J. and Zwolak, J. (2021), 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)
Created November 10, 2021, Updated February 26, 2022