An entropy-based persistence barcode

• We define the notion of “minimal” barcodes in terms of entropy.
• Given a simplicial complex, an algorithm for computing a proper filter F is detailed.
• F preserves the partial ordering imposed by the filtration.
• F achieves a persistence barcode with small entropy.
• Examples demonstrating the utility of computing such a proper filter are given.

In persistent homology, the persistence barcode encodes pairs of simplices meaning birth and death of homology classes. Persistence barcodes depend on the ordering of the simplices (called a filter) of the given simplicial complex. In this paper, we define the notion of “minimal” barcodes in terms of entropy. Starting from a given filtration of a simplicial complex K, an algorithm for computing a “proper” filter (a total ordering of the simplices preserving the partial ordering imposed by the filtration as well as achieving a persistence barcode with small entropy) is detailed, by way of computation, and subsequent modification, of maximum matchings on subgraphs of the Hasse diagram associated to K. Examples demonstrating the utility of computing such a proper ordering on the simplices are given.