My research deals with the investigation of useful properties of metric spaces through the lens of topological data analysis and magnitude. An isometric invariant of metric spaces, magnitude has been shown to encode a number of other valuable invariants, such as dimension and curvature. In particular, Magnitude is known to be strongly connected to Minkowski dimension for positive definite compact metric spaces. It stands to reason that magnitude could be leveraged to estimate the dimensions of compact metric spaces from which point clouds are sampled. However, the computational complexity of magnitude renders this prospect nearly impossible to realize for metric spaces of sufficient size. In recent work with Sara Kalisnik and Nina Otter, we identify alpha magnitude as a method through which topological data analysis can be applied to provide a potential solution to this problem, at substantially reduced computational complexity.
In forthcoming work, we establish the continuity of magnitude for finite spaces of strictly negative type, expanding the class of spaces to which magnitude may be applied. While magnitude is difficult to compute for spaces of high cardinality, spaces of low cardinality but high dimension still provide an opportunity for magnitude to provide insight.
- Alpha Magnitude. Miguel O’Malley, Sara Kališnik, Nina Otter. 5/19/2022. arXiv:2205.09521. To appear in Journal of Pure and Applied Algebra.
- Github for Alpha Magnitude:
