Algebraic Data Structure for Decomposing Multipersistence Modules

Single-parameter persistent homology techniques in topological data analysis have seen increasing usage in recent years. These techniques have found particular success because of the existence of a complete, discrete, efficiently computable invariant to describe persistence modules in the single-parameter case: the barcode. Attempts to develop an equally robust theory of multiparameter persistent homology, however, have been slow to progress because there is no natural multiparameter analogue to the barcode. Relatively little is known about the structure of decompositions of multiparameter persistence (multipersistence) modules or how to classify their indecomposables. In fact, even for the problem of computing decompositions, there currently is no generalization to multiple parameters of the decomposition algorithm from single-parameter persistent homology. In this paper, we define a new algebraic data structure, the QR code, which was first proposed in https://arxiv.org/abs/1709.08155 but was formulated somewhat erroneously. Additionally, we prove a theorem stating that the QR code recovers all the information of the module it encodes. We suggest that this new data structure, which seeks to encode a module using births and deaths rather than births and relations, may be the correct language in which to solve the problem of decomposing arbitrary finitely generated multipersistence modules.