Persistence Landscapes of Fractals
Proving that the persistence landscape of a fractal is itself self-similar.
A persistence landscape is a vectorization of a persistence diagram as a sequence of piecewise-linear functions. Unlike the diagram itself, landscapes live in a Hilbert space, which means you can average them, compute norms, and apply statistical tests. This makes them a powerful representation of topological features for use in machine learning pipelines.
On the other hand, a fractal is a set defined as the fixed point of an iterated function system (IFS). Start with any compact set, repeatedly apply a finite collection of contracting maps, and the sequence converges to a self-similar limit. The Cantor set, Sierpinski triangle, and Koch snowflake are classical examples. Because these sets are self-similar by definition, a natural question is whether their topological summaries inherit that self-similarity.
This project answers that question affirmatively for persistence landscapes. We show that for an affine IFS, there is an affine transformation on the space of persistence landscapes that intertwines the action of the IFS. In other words, the persistence landscape of the fractal satisfies a fixed-point equation of its own. Thus, in a precise sense, the persistence landscape of a fractal is a fractal in the space of landscapes. Rather than computing the landscape of the fractal directly (which would require an infinite iteration), this result implies that we can compute the landscapes of finitely many iterations and use the intertwining map to extrapolate to the limit.
This project is joint work with Lee Przybylski and Eric Weber in (Catanzaro et al., 2022).
