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Measuring and Localing Homology Classes
We develop a method for measuring and localizing homology classes. This
involves two problems. First, we define relevant notions of size for both a
homology class and a homology group basis, using ideas from relative homology.
Second, we propose an algorithm to compute the optimal homology basis, using
techniques from persistent homology and finite field algebra. Classes of the
computed optimal basis are localized with cycles conveying their sizes. The
algorithm runs in time, where is the size of the
simplicial complex and is the Betti number of the homology group.Comment: References 8 and 22 are fixe