Miguel O'Malley

GitHub

TDA and Alpha Magnitude

In the course of data analysis, most often methods which determine significance are purely statistical, observing groupings of occurences, behaviors, or characteristics that indicate the presence of interesting qualities. The study of topological data analysis is done under the premise that there is important information not just within the statistically measurable qualities of data, but indeed the shape of the dataset. The presence of holes, embeddability into manifolds, and merge/divergence points are qualities which we find deeply interesting in TDA, and ones which we are able to use for the purposes of finding information otherwise overlooked by purely statistical methods.

One benefit of TDA is that often, the shape of a point cloud can be efficiently computed. This is not so for magnitude, an isometric invariant of metric spaces of some interest. Magnitude offers the ability to evaluate the "effective number" of points in a space. That is to say, often, given a population, there may be a large number of distinct observations which all seem similar, or a lesser number of distinct observations that truly are quite different. Magnitude offers a real number invariant quantifying the difference between these two scenarios. However, the computation of magnitude is expensive, as it requires in the simplest case the inversion of a matrix derived from the distance matrix of a given point cloud. It is thus desirable to develop an invariant which contains the same information magnitude provides, while being signficantly more computationally efficient.

Towards this end, we introduce alpha magnitude, an invariant similar to magnitude derived from the alpha complex on a real point cloud. Alpha magnitude exhibits many of the same desirable qualities magnitude does, while being almost linearly computable in low dimensional cases. We demonstrate a number of these desirable qualities, examine alpha magnitudes for various point clouds, and extend alpha magnitude to the infinite setting in a manner commensurate with that followed for magnitude. Further, we show that an invariant we derive from alpha magnitude and similar to one derived from magnitude, alpha magnitude dimension, is identical to the box-counting dimension for some key point clouds, and we further conjecture that the alpha magnitude dimension and the box counting dimension are identical, when both are defined.

My research statement.