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We first introduce the notion of metarank for a 2parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the metadiagram of a 2parameter persistence module to be the MÃ¶bius inversion of the metarank, resulting in a function that takes values from signed 1parameter persistence modules. We show that the metarank and metadiagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the metarank and metadiagram of a 2parameter module M indexed by a bifiltration of n simplices in O(n^3) time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has O(n^4) time complexity. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of M from O(n^4) to O(n^3). In addition, we define notions of erosion distance between metaranks and between metadiagrams, and show that under these distances, metaranks and metadiagrams are stable with respect to the interleaving distance. Lastly, the metadiagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the wellunderstood persistent diagram in the 1parameter setting.more » « less

Guelen, A ; Memoli, F ; Wan, Z ; Wang, Y ( , Intl. Sympos. Comput. Geom. (SoCG))

Hang, H ; Memoli, F ; Mio, W. ( , Journal of applied and computational topology)We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multiscale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel.more » « less