Summary of Persistent Homology

Singapore Maths Tuition

We summarize the work so far and relate it to previous results. Our input is a filtered complex $latex K$ and we wish to find its $latex k$th homology $latex H_k$. In each dimension the homology of complex $latex K^i$ becomes a vector space over a field, described fully by its rank $latex beta_k^i$. (Over a field $latex F$, $latex H_k$ is a $latex F$-module which is a vector space.)

We need to choose compatible bases across the filtration (compatible bases for $latex H_k^i$ and $latex H_k^{i+p}$) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module $latex mathscr{M}$ corresponding to $latex K$, which is a direct sum of these vector spaces ($latex alpha(mathscr{M})=bigoplus M^i$). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.

Specifically, each $latex mathcal{P}$-interval $latex (i,j)$ describes a basis element…

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