Structural Features of Persistent Homology and their Algorithmic Transformations

Friday, April 30, 2021 - 11:00am
Zoom presentation
Andrei Pavlichenko
(MU Math)


We re-examine the theory and orthodox methods that underlie the study of persistent homology, particularly in its calculation of homological cycle representatives that are associated to persistence diagrams. A common background to the subject covers several aspects: schemes to process input data (embedding it in a low-dimensional manifold), categorical descriptions of persistence objects, and algorithms by which the barcode summarizing the homology is found.

We overview these aspects, focusing on filtered simplicial complexes, traditional computation of persistent homology, and the stability theorem for barcodes. By reformulating these notions in the language of category theory, we can speak more plainly on some recurring notions that are relevant to our discussion. This ultimately sets up for vector space filtrations that prove to be suitable tools for codifying the homology of a complex.

The main body of work then presents an alternative approach to persistent homology, based on filtrations of vector spaces. We elaborate with an interesting example whose persistent homology is readily computed using our variant algorithm; in the process, we produce an appropriate basis of homological cycles, a step that is often overlooked in existing literature. The example also illustrates the complexity of (co)kernels and (co)images associated to morphisms of persistence objects; while there exist algorithms to compute the barcodes of these, such methods are not easily generalizable. Instead, we compute appropriate homological cycles and use a certain algorithmic matching scheme that both implies the usual barcode matching and attempts to better interpret this interesting behavior.

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