Florian Russold

Tamal K. Dey, Florian Russold

A family of simplicial complexes connected by simplicial maps and indexed by a finite poset $P$ is called a poset tower. Poset towers subsume multi-parameter filtrations, zigzag filtrations, and one-parameter simplicial towers, while allowing arbitrary finite posets and simplicial maps. The homology of a poset tower is a $P$-persistence module. To compute it globally over $P$, we consider the chain complex segment of $P$-persistence modules $C_{\ell-1}\xleftarrow{\partial_{\ell}}C_\ell \xleftarrow{\partial_{\ell+1}}C_{\ell+1}$ induced by the simplices of the tower. Unlike in one-critical multi-filtrations, the chain modules $C_\ell$ need not be projective and may have a complicated structure. We address the problem of replacing this segment by projective modules and $P$-graded matrices while preserving homology. The resulting projective implicit representation (PiRep) plays the role of the graded boundary-matrix representation in the classical persistence algorithm: it converts simplicial data into algebraic input on which persistent homology can be computed globally over $P$. In particular, a PiRep can be used as input to algorithms for computing minimal presentations of persistent homology. We give an efficient algorithm to compute a PiRep from a poset tower. It constructs degreewise minimal presentations and asymptotically minimal second terms of projective resolutions of the chain modules $C_\ell$, lifts the boundary maps $\partial_\ell$ to these resolutions, and assembles the resulting data into a PiRep using an additional correction term. The method is tailored to chain complexes induced by poset towers and computes the required algebraic data combinatorially, exploiting their special structure and avoiding general-purpose algebraic reduction. In the context of poset towers, it is fully general and can serve as a foundation for efficient algorithms on specific posets.

Michael Kerber, Florian Russold

We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries are usually based on complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield an informative and interpretable summary and can be computed as efficient as one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-art approaches on various datasets.