User:Takina Inoue/sandbox


 * “背景知识请参见 「同调」页面. ”

持续同调 是一个在空间不同尺度下计算算拓扑特征的方法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.

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
 * $$ \emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K $$

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. Persistent Betti numbers for $$p=0$$ coincide with the size function, a predecessor of persistent homology.

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 $$u_t^t$$ equal to the identity and $$u_t^s \circ u_s^r = u^r_t$$ 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 $$R$$-modules). There is a classification of persistence modules over a field $$F$$ indexed by $$\mathbb{N}$$: $$U \simeq \bigoplus_i x^{t_i} \cdot F[x] \oplus \left(\bigoplus_j x^{r_j} \cdot (F[x]/(x^{s_j}\cdot F[x]))\right).$$ 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 $$s_j+r_j$$).

This theorem 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.

Stability
Persistent homology is stable in a precise sense, which provides robustness against noise. There is a natural metric on the space of persistence diagrams given by $$W_{\infty}(X,Y):= \inf_{\phi: X \to Y} \sup_{x \in X} \Vert x-\phi(x)\Vert _{\infty},$$ called the bottleneck distance. A small perturbation in the input filtration leads to a small perturbation of its persistence diagram in the bottleneck distance. For concreteness, consider a filtration on a space $$X$$ homeomorphic to a simplicial complex determined by the sublevel sets of a continuous tame function $$f:X\to \mathbb{R}$$. The map taking $$f$$ to the persistence diagram of its $$k$$th homology is 1-Lipschitz with respect to the $$\sup$$-metric on functions and the bottleneck distance on persistence diagrams. That is, $$W_{\infty}(D(f),D(g)) \leq \lVert f-g \rVert_{\infty}$$.

Computation
There are various software packages for computing persistence intervals of a finite filtration.