Topology versus Geometry in Higher-Dimension Data Analysis

High-dimensional data is often difficult to analyze because of the exponential growth of the size of the space in which the data lives as the dimension increases. [1] However, new methods involving higher-dimension geometries and topological data analysis may be used to overcome the challenge of effectively analyzing such data sets. My objective is to compare the ability of topological and geometrical structures in extracting information from higher-dimensional data sets.

The Reuleaux triangle has the smallest volume of a shape of constant diameter in two dimensions, meaning that the distance from its center to any point on the boundary is always constant. In a recent mathematical study, [2,3] researchers have generalized the algorithm for constructing Reuleaux triangles to higher dimensions. With this new algorithm, one can find constant-diameter shapes in any dimension. More importantly, the researchers found that their volumes are a factor of 0.9^n times the volume of an equivalent n-th dimensional ball. Thus, the Reuleaux construction’s volume decreases exponentially as dimension increases.

For my project, I plan to analyze higher-dimensional data sets using both topological data analysis (TDA) and these newly-discovered geometrical shapes. This could impact such studies as machine learning [4] and natural language processing. [5]

1. “Curse of Dimensionality,” Eamonn Keogh & Abdullah Mueen, Enclyclopedia of Machine Learning and Data Mining, 2017.

2. “Small bodies of constant width,” Andrii Arman, et al., arXiv:2405.18501.

3. “Convex bodies of constant width with exponential illumination number,” Andrii Arman, et al., Discrete and Computational Geometry, 2024.

4. “Mathematicians discover new shapes to solve decades-old geometry problem,” Gregory Barber, Quanta Magazine, 2024.

5. “Persistent homology on phonological data: a preliminary study,” Catherine Wolfram, https://math.uchicago.edu/~may/REU2017/REUPapers/Wolfram.pdf.