Structural features of persistent homology and their algorithmic transformations

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 altered 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 complexes, including the (co)images and (co)kernels arising from morphisms of complexes. 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 as a quotient of appropriate filtrations; in the process, we produce a representative basis of homological cycles, a step that is often overlooked in existing literature. The proposed algorithm is also notable in that it easily handles the calculation of (co)images and (co)kernels for persistent morphisms, supplying us with the same level of detail; while other algorithms do exist for computing the barcodes of these universal objects, such methods are not easily generalizable. Finally, 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.