Magnitude, alpha magnitude, and applications.
Miguel O'Malley · 2022
Magnitude, an isometric invariant of metric spaces, is known to bear rich connections to other desirable invariants, such as dimension, volume, and curvature.Connections between magnitude and persistent homology, a method to observe topological features in datasets, are well studied and fruitful.We leverage one such connection, persistent magnitude, to introduce alpha magnitude, a new invariant which bears many of the same properties of magnitude.We show in particular a strong connection to the Minkowski dimensions of compact subspaces of R n and conjecture the connection exists in general.We further provide a new proof for the stability of the extended magnitude function for finite spaces of strictly negative type, and construct hierarchical clustering methods for the implementation of both magnitude and alpha magnitude in data science.iii 5.4.4.Feigenbaum attractor 6. Clustering analysis 6.1.Background on clustering 6.2.Magnitude clustering 6.3.Alpha Magnitude clustering 7.