A-infinity Persistence
This addresses a theoretical limitation in topological data analysis for researchers by offering a more nuanced tool for noise detection, though it appears incremental as an extension of persistence homology.
The paper introduces A-infinity persistence, a family of homological invariants for topological space filtrations that provides information beyond standard persistent Betti numbers, potentially helping detect noise in both simplicial structure and geometric properties. It characterizes these invariants in terms of barcodes based on zigzag module classification.
We introduce and study A-infinity persistence of a given homology filtration of topological spaces. This is a family, one for each n > 0, of homological invariants which provide information not readily available by the (persistent) Betti numbers of the given filtration. This may help to detect noise, not just in the simplicial structure of the filtration but in further geometrical properties in which the higher codiagonals of the A-infinity structure are translated. Based in the classification of zigzag modules, a characterization of the A-infinity persistence in terms of its associated barcode is given.