The Ring of Algebraic Functions on Persistence Bar Codes

Persistent homology ([3], [13]) is a fundamental tool in the area of computational topology. It can be used to infer topological structure in data sets (see [1], [4]), but variations on the method can be applied to study aspects of the shape of point clouds which are not overtly topological ([5], [8]). The methodology assigns to any finite metric space (such as are typically obtained in experimental data of various kinds) and non-negative integer k a bar code, by which we will mean a finite collection of intervals with endpoints on the real line. The integer k specifies a dimension of a feature (zero-dimensional for a cluster, one-dimensional for a loop, etc.), and an interval represents a feature which is “born” at the value of a parameter (the persistence parameter) given by the left hand endpoint of the interval, and which “dies” at the value given by the right hand endpoint. These barcodes have been demonstrated to identify structure in spaces of image patches in [1] and [4], and have been demonstrated to distinguish between handdrawn letters in [8]. Because of the unusual structure of the invariant, i.e. as a collection of intervals rather than numerical quantities, the method currently requires substantial knowledge of topological methods. It would clearly be useful to assign and interpret various numerical quantities attached to bar codes, so that these outputs could be used as input to standard algorithms within machine learning, cluster analysis, and other methods. It is the purpose of this