Topological deep learning (TDL) is an emerging area that combines the principles of
Topological data analysis (TDA) with deep learning techniques. TDA provides insight into …

Persistence modules are a central algebraic object arising in topological data analysis. The
notion of interleaving provides a natural way to measure distances between persistence …

We use weights on objects in an abelian category to define what we call a path metric. We
introduce three special classes of weight: those compatible with short exact sequences; …

Building on the notion of normed category as suggested by Lawvere, we introduce notions
of Cauchy convergence and cocompleteness which differ from proposals in previous works …

The interleaving distance, although originally developed for persistent homology, has been
generalized to measure the distance between functors modeled on many posets or even …

This paper makes mathematically precise the idea that conditional probabilities are
analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category …

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call
truncated smoothing; this in turn allows us to define a new family of metrics which generalize …

The mapper graph is a popular tool from topological data analysis that provides a graphical
summary of point cloud data. It has been used to study data from cancer research, sports …

Optimal transport is widely used in pure and applied mathematics to find probabilistic
solutions to hard combinatorial matching problems. We extend the Wasserstein metric and …

We present a new language for persistent homology in terms of Galois connections. This
language has two main advantages over traditional approaches. First, it simplifies and …