The directional transform: how to look at your data from every direction at once

The field of topological data analysis (TDA) has emerged as a robust method for measuring the shape of data.  This field of research takes ideas from algebraic topology, in concert with ideas from algorithms, statistics, category theory, and others, to provide methods for quantifying the shape of data. This data can either be cases where shape is obvious, such as studying the shape of a plant seed, or not-so-obvious, such as studying the shape of a reconstructed attractor from a time series.

In this talk, we will discuss an idea that has been weaving its way through more and more of the developing TDA methodology; that is, the directional transform. The idea is that since most methods in TDA require input consisting of a topological space with a real valued function, however it is not always obvious what that function should be. In particular, when we are in the setting where we have access to a region of Euclidean space, we can probe the shape with a function and compute a representation such as a persistence diagram, but which direction to choose? The directional transform, first introduced by Turner et al, takes as representation the topological signature for all directions at once. The result is a summary which can be used from theoretical analysis to computational experiments such as being an input to machine learning models. In this talk, we will discuss some uses of the directional transform, including for understanding structures such as X-Ray CT scans of barley seeds as well as for comparing embedded graphs.