Topological Data Analisis is a booming research field with a lot of approaches to be studied from. In this case, using some basic knowledge about finite simplicial complexes and maps between them, we see that we can create a special type of neural networks that can deal with classification problems. We also show proposals for improving the computational efficiency of this method.

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. We shows that both the dimension vector and the rank invariant are homological invariants. We then provide an explicit example of a homological invariant which is finer than the rank invariant. This is based on joint work with Benjamin Blanchette and Thomas Brüstle.

The kth fold filtration at parameter r, whose nerve is the kth order Čech filtration at r, is formed by the intersections of k balls with radius r. We construct an approximation for the kth fold filtration, inspired by sparsification techniques for the case k = 1, that extends to a sparsification of the higher order Čech complex. The sparsified filtration has size linear on the number of input points and arbitrary approximation parameter.

We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), a tool from TDA that allows one to algorithmically detect geometric voids in a data set and quantify the persistence of these voids. We construct an efficient filtered complex such that the voids in our filtered complex are in one-to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (an approach for visualizing PH), to track how the locations of the anomalies change with time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code at a single point in time. Second, we study a year-long data set of COVID-19 case rates in neighborhoods in the city of Los Angeles.

Motivated by computational aspects of persistent homology for Vietoris–Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris–Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We introduce the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the Rips filtration. We further show that for finite tree metrics the Vietoris–Rips complexes collapse to their corresponding subtrees. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data.

Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d simplicial complexes, has witnessed a great attention in recent years. However, there has not been enough effort to extend representation learning to higher dimensional simplicial objects due to the additional complexity these objects hold, especially when it comes to the entire-simplicial complex representation learning. In this work, we propose a method for simplicial complex-level representation learning that embeds a simplicial complex to a universal embedding space in a way that complex-to-complex proximity is preserved. Our proposed method uses novel geometric message passing schemes (GMPSs) to learn an entire simplicial complex representation in an end-to-end fashion. We demonstrate the proposed method on a publicly available mesh dataset. To the best of our knowledge, this work presents the first method for learning simplicial complex-level representation.