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In this paper, we study how basis-independent partial matchings induced by morphisms between persistence modules (also called ladder modules) can be defined. Besides, we extend the notion of basis-independent partial matchings to the situation of a pair of morphisms with same target persistence module. The relation with the state-of-the-art methods is also given. Apart form the basis-independent property, another important property that makes our partial matchings different to the state-of-the-art ones is their linearity with respect to ladder modules.
The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching $mathcal{G}_f$, and a partial matching $mathcal{M}_f$, induced by a
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf
The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a ne
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace the noeth