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## 数学代写|拓扑学代写Topology代考|Space Filtration

Consider a real-valued function $f: \mathbb{T} \rightarrow \mathbb{R}$ defined on a topological space T. Let $\mathbb{T}{a}=f^{-1}(-\infty, a$ ] denote the sublevel set for the function value $a$. Certainly, we have inclusions: $$\mathbb{T}{a} \subset \mathbb{T}{b} \text { for } a \leq b \text {. }$$ Now consider a sequence of reals $a{1} \leq a_{2} \leq \cdots \leq a_{n}$ which are often chosen to be critical values where the homology group of the sublevel sets change as illustrated in Figure 3.2. Considering the sublevel sets at these values and a dummy value $a_{0}=-\infty$ with $\mathbb{T}{a{0}}=\varnothing$, we obtain a nested sequence of subspaces of $\mathbb{T}$ connected by inclusions which gives a filtration $\mathfrak{F}{f}$ : $$\mathcal{F}{f}: \varnothing=\mathbb{T}{a{0}} \hookrightarrow \mathbb{T}{a{1}} \hookrightarrow \mathbb{T}{a{2}} \hookrightarrow \cdots \hookrightarrow \mathbb{T}{a{n}} .$$
Figure $3.2$ shows an example of the inclusions of the sublevel sets. The inclusions in a filtration induce linear maps in the singular homology groups of the subspaces involved. So, if $\iota: \mathbb{T}{a{i}} \rightarrow \mathbb{T}{a{j}}, i \leq j$, denotes the inclusion map $x \mapsto x$, we have an induced homomorphism
$$h_{p}^{i, j}=\iota_{*}: \mathrm{H}{p}\left(\mathbb{T}{a_{i}}\right) \rightarrow \mathrm{H}{p}\left(\mathbb{T}{a_{j}}\right)$$
for all $p \geq 0$ and $0 \leq i \leq j \leq n$. Therefore, we have a sequence of homomorphisms induced by inclusions forming what we call a homology module:
$$0=\mathrm{H}{p}\left(\mathbb{T}{a_{0}}\right) \rightarrow \mathrm{H}{p}\left(\mathbb{T}{a_{1}}\right) \rightarrow \mathrm{H}{p}\left(\mathbb{T}{a_{2}}\right) \rightarrow \cdots \rightarrow \mathrm{H}{p}\left(\mathbb{T}{a_{n}}\right) .$$

## 数学代写|拓扑学代写Topology代考|Simplicial Filtrations and Persistence

Persistence on topological spaces involves computing singular homology groups for sublevel sets. Computationally, this is cumbersome. So, we take refuge in the discrete analogue of the topological persistence. This involves two important adaptations: first, the topological space is replaced with a simplicial complex; second, singular homology groups are replaced with simplicial homology groups. This means that the topological space $\mathbb{I}$ considered before is replaced with one of its triangulations, as Figure $3.4$ illustrates. For point cloud data, the union of balls can be replaced by their nerve, the Cech complex or its cousin Vietoris-Rips complex introduced in Section 2.2. Figure $3.5$ illustrates this conversion for example in Figure 3.3. Of course, these replacements need to preserve the original persistence in some sense, which is addressed in general by the notion of stability introduced in Section 3.4.

The nested sequence of topological spaces that arise with growing sublevel sets translates into a nested sequence of simplicial complexes in the discrete analogue. This brings in the concept of filtration of simplicial complexes that allows defining the persistence using simplicial homology groups.

## 数学代写|拓扑学代写Topology代考|Persistence Diagram

Fact $3.3$ provides a qualitative characterization of the pairing of births and deaths of classes. Now we give a quantitative characterization which helps to draw a visual representation of this pairing called a persistence diagram; see Figure $3.8$ (a). Consider the extended plane $\overline{\mathbb{R}}^{2}:=(\mathbb{R} \cup{\pm \infty})^{2}$ on which we represent the birth at $a_{i}$ paired with the death at $a_{j}$ as a point $\left(a_{i}, a_{j}\right)$. This pairing uses a persistence pairing function $\mu_{p}^{l, J}$ defined below. Strictly positive valuēs of this function corres̄pond to multip̄licities of points in the peresisténce diagram (Definition 3.8). In what follows, to account for classes that never die, we extend the induced module in Eq. (3.3) on the right end by assuming that $\mathrm{H}{p}\left(X{n+1}\right)=0$
Definition 3.6. For $0<i<j \leq n+1$, define
$$\mu_{p}^{l, j}=\left(\beta_{p}^{l, j-1}-\beta_{p}^{l, j}\right)-\left(\beta_{p}^{l-1, j-1}-\beta_{p}^{l-1, j}\right)$$
The first difference on the right-hand side counts the number of independent classes that are born at or before $X_{i}$ and die entering $X_{j}$. The second difference counts the number of independent classes that are born at or before $X_{i-1}$ and die entering $X_{j}$. The difference between the two differences thus counts the number of independent classes that are born at $X_{i}$ and die entering $X_{j}$. When $j=n+1, \mu_{p}^{i, n+1}$ counts the number of independent classes that are born at $X_{i}$ and die entering $X_{n+1}$. They remain alive till the end in the original filtration without extension, or we say that they never die. To emphasize that classes which exist in $X_{n}$ actually never die, we equate $n+1$ with $\infty$ and take $a_{n+1}=$ $a_{\infty}=\infty$. Observe that, with this assumption, we have $\beta^{i, n+1}=\beta^{i, \infty}=0$ for every $i \leq n$.

## 数学代写|拓扑学代写Topology代考|Space Filtration

$$\mathbb{T} a \subset \mathbb{T} b \text { for } a \leq b .$$

$$\mathcal{F} f: \varnothing=\mathbb{T} a 0 \hookrightarrow \mathbb{T} a 1 \hookrightarrow \mathbb{T} a 2 \hookrightarrow \cdots \hookrightarrow \mathbb{T} a n .$$

$$h_{p}^{i, j}=\iota_{*}: \mathrm{H} p\left(\mathbb{T} a_{i}\right) \rightarrow \mathrm{H} p\left(\mathbb{T} a_{j}\right)$$

$$0=\mathrm{H} p\left(\mathbb{T} a_{0}\right) \rightarrow \mathrm{H} p\left(\mathbb{T} a_{1}\right) \rightarrow \mathrm{H} p\left(\mathbb{T} a_{2}\right) \rightarrow \cdots \rightarrow \mathrm{H} p\left(\mathbb{T} a_{n}\right) .$$

## 数学代写|拓扑学代写Topology代考|Persistence Diagram

(3.3) 在右端假设 $\mathrm{H} p(X n+1)=0$

$$\mu_{p}^{l, j}=\left(\beta_{p}^{l, j-1}-\beta_{p}^{l, j}\right)-\left(\beta_{p}^{l-1, j-1}-\beta_{p}^{l-1, j}\right)$$

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