Discrete signature tensors: bridging topological data analysis and machine-learning

As the scale and complexity of biological data increases, problems such as interpretation, classification and quantification of biological structures require novel mathematical methods for investigation. Persistent homology provides a multiscale geometric descriptor of data that is functorial, stable to perturbations and interpretable: it takes in a filtered simplicial complex that describes the shape of data, and outputs a persistence module that can be visualized as a barcode. A limitation of persistence homology is that persistence modules lack structure that is necessary for statistical analysis and machine-learning integration. To this effect, an active pursuit in topological data analysis is to find convenient vectorizations, i.e. maps from persistent homology to mathematical objects that are suitable for machine-learning methods. 

In this talk, I will introduce an approach for vectorization based on (a discrete version of) path signatures. I will argue that this approach is natural and describe its discriminative power. Time permitting, I will present an application to knotted proteins and show how the discrete signature captures fundamental structural information such as the protein's knot depth and structural homology class in a statistically significant way.