Topology and geometry for data

Introduction

Topology and geometry has recently provided a suite of powerful and robust tools for nonlinear data analysis. Of these, the most mature is persistent homology, which complements traditional statistical methods by providing noise-tolerant summaries which describe the shape of any given dataset.

Our ability to compute these statistics for datasets has improved dramatically over the last decade, scaling up in size from thousands to tens-of-billions, even as hardware improvements have remained modest.

Inherent to these remarkable advances are optimisations sourced from many diverse fields of applied mathematics and computer science, spanning sparse linear algebra, discrete Morse theory, matroids and highly specialised data structures.