Pseudo-Boolean function

See homology for an introduction to the notation.

Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters.

To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets.

Formally, consider a real-valued function on a simplicial complex $f:K\rightarrow \mathbb {R}$ that is non-decreasing on increasing sequences of faces, so $f(\sigma )\leq f(\tau )$ whenever $\sigma$ is a face of $\tau$ in $K$. Then for every $a\in \mathbb {R}$ the sublevel set $K(a)=f^{-1}(-\infty ,a]$ is a subcomplex of K, and the ordering of the values of $f$ on the simplices in $K$ (which is in practice always finite) induces an ordering on the sublevel complexes that defines the filtration

$\emptyset =K_{0}\subseteq K_{1}\subseteq \ldots \subseteq K_{n}=K$

When $0\leq i\leq j\leq n$, the inclusion $K_{i}\hookrightarrow K_{j}$ induces a homomorphism $f_{p}^{i,j}:H_{p}(K_{i})\rightarrow H_{p}(K_{j})$ on the simplicial homology groups for each dimension $p$. The $p^{th}$ persistent homology groups are the images of these homomorphisms, and the $p^{th}$ persistent Betti numbers $\beta _{p}^{i,j}$ are the ranks of those groups.

There are various software packages for computing persistence intervals of a finite filtration, such as javaPlex, Dionysus, Perseus, PHAT, and the phom R package.

References

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