In Algebraic Topology, persistent homology is a methodology to compute the homology groups of a simplicial complexes endowed with a filtration. The simplicial complex is usually constructed from a topological space and homology groups persisting over a large range of the filtration can be interpreted as meaningful features of the underlying topological space.

Datasets often have interesting structure, but due to high dimensionality and the discrete nature this structure is difficult to quantify. Through the construction of a Vietoris-Rips complex and a computation of the persistent homology one is able to make qualitative statements on the structure, or loosly speaking shape, of the underlying topological space from which the original dataset was sampled. The resulting so-called persistence diagram can be interpreted as the number of connected components, loops and voids in the dataset at various scales, providing valuable information to practitioners.

We aim to study how the topological information of datasets can be exploited to create deep learning architectures that are both sparser and more expressive. We hope to apply this to expand and combine on the work done and apply results to biomedical data.