Computing multiparameter persistent homology through a discrete Morse-based approach

Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations

A global reduction method for multidimensional size graphs

This paper introduces the concept of discrete multidimensional size function, a mathematical tool that studies particular graphs called size graphs. A global method for reducing size graphs and a theorem, stating that discrete multidimensional size functions are invariant with respect to this reduction method, are shown. This result allows us to easly and fast compute discrete multidimensional size functions for applications