Persistent Homology (MATH 528)

Required Reading (Lectures): https://mrzv.org/

Course Website: http://graphics.stanford.edu/courses/cs468-09-fall/

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.

Definition

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

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 $p^{\text{th))$ persistent homology groups are the images of these homomorphisms, and the $p^{\text{th))$ 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 $d(e_{t_{i)))=0$ and two-dimensional complexes with trivial homology $d(e_{s_{j}+r_{j)))=e_{r_{j))$ .^[4]

A 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 $p$ .

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