Persistent Homology (MATH 528)

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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 spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters.[1]

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 a filtration{\displaystyle \emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \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 {\displaystyle p^{\text{th))} persistent homology groups are the images of these homomorphisms, and the {\displaystyle p^{\text{th))} persistent Betti numbers \beta _{p}^((i,j)) are the ranks of those groups.[2] Persistent Betti numbers for p=0 coincide with the size function, a predecessor of persistent homology.[3]

Any filtered complex over a field F can be brought by a linear transformation preserving the filtration to so called canonical form, a canonically defined direct sum of filtered complexes of two types: one-dimensional complexes with trivial differential {\displaystyle d(e_{t_{i)))=0} and two-dimensional complexes with trivial homology {\displaystyle d(e_{s_{j}+r_{j)))=e_{r_{j))}.[4]

persistence module over a partially ordered set P is a set of vector spaces U_{t} indexed by P, with a linear map u_t^s: U_s \to U_t whenever s\leq t, with {\displaystyle u_{t}^{t)) equal to the identity and {\displaystyle u_{t}^{s}\circ u_{s}^{r}=u_{t}^{r)) for r \leq s \leq t. Equivalently, we may consider it as a functor from P considered as a category to the category of vector spaces (or -modules). There is a classification of persistence modules over a field F indexed by \mathbb {N} :

{\displaystyle U\simeq \bigoplus _{i}x^{t_{i))\cdot F[x]\oplus \left(\bigoplus _{j}x^{r_{j))\cdot (F[x]/(x^{s_{j))\cdot F[x]))\right).}

Multiplication by x corresponds to moving forward one step in the persistence module. Intuitively, the free parts on the right side correspond to the homology generators that appear at filtration level t_{i} and never disappear, while the torsion parts correspond to those that appear at filtration level r_j and last for s_{j} steps of the filtration (or equivalently, disappear at filtration level s_j+r_j).[5][4]

Each of these two theorems allows us to uniquely represent the persistent homology of a filtered simplicial complex with a barcode or persistence diagram. A barcode represents each persistent generator with a horizontal line beginning at the first filtration level where it appears, and ending at the filtration level where it disappears, while a persistence diagram plots a point for each generator with its x-coordinate the birth time and its y-coordinate the death time. Equivalently the same data is represented by Barannikov’s canonical form,[4] where each generator is represented by a segment connecting the birth and the death values plotted on separate lines for each p.