### 数学代写|表示论代写Representation theory代考|MATH4314

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|表示论代写Representation theory代考|Stability of Persistence Diagrams

A persistence diagram $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$, as a set of points in the extended plane $\overline{\mathbb{R}}^2$, summarizes certain topological information of a simplicial complex (space) in relation to the function $f$ that induces the filtration $\mathcal{F}_f$. However, this is not useful in practice unless we can be certain that a slight change in $f$ does not change this diagram dramatically. In practice $f$ is seldom measured accurately, and if its persistence diagram can be approximated from a slightly perturbed version, it becomes useful. Fortunately, persistence diagrams are stable. To formulate this stability, we need a notion of distance between persistence diagrams.

Let $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$ be two persistence diagrams for two functions $f$ and $g$. We want to consider bijections between points from $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$. However, they may have different cardinality for off-diagonal points. Recall that persistence diagrams include the points on the diagonal $\Delta$ each with infinite multiplicity. This addition allows us to borrow points from the diagonal when necessary to define the bijections. Note that we are considering only filtrations of finite complexes which also make each homology group finite.

Definition 3.9. (Bottleneck distance) Let $\Pi=\left{\pi: \operatorname{Dgm}p\left(\mathcal{F}_f\right) \rightarrow\right.$ $\left.\operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right}$ denote the set of all bijections. Consider the distance between two points $x=\left(x_1, x_2\right)$ and $y=\left(y_1, y_2\right)$ in $L{\infty}$-norm $|x-y|_{\infty}=$ $\max \left{\left|x_1-x_2\right|,\left|y_1-y_2\right|\right}$ with the assumption that $\infty-\infty=0$. The bottleneck distance between the two diagrams (see Figure $3.10$ ) is
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf {\pi \in \Pi} \sup {x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)}|x-\pi(x)|{\infty} .$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

Let $A$ and $B$ be the nondiagonal points in two persistence diagrams $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$, respectively. For a point $a \in A$, let $\bar{a}$ denote the nearest point of $a$ on the diagonal. Define $\bar{b}$ for every point $b \in B$ similarly. Let $\bar{A}={\bar{a}}$ and $\bar{B}={\bar{b}}$. Let $\tilde{A}=A \cup \bar{B}$ and $\tilde{B}=B \cup \bar{A}$. We want to bijectively match points in $\tilde{A}$ and $\tilde{B}$. Let $\Pi={\pi}$ denote such a matching. It follows from the definition that
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\min {\pi \in \Pi} \sup {a \in \tilde{A}, \pi(a) \in \tilde{B}}|a-\pi(a)|{\infty} .$$
Then, the bottleneck distance we want to compute must be $L_{\infty}$ distance $\max \left{\left|x_a-x_b\right|,\left|y_a-y_b\right|\right}$ for two points $a \in \tilde{A}$ and $b \in \tilde{B}$. We do a binary search on all such possible $O\left(n^2\right)$ distances where $|\tilde{A}|=|\tilde{B}|=n$. Let $\delta_0, \delta_1, \ldots, \delta_{n^{\prime}}$ be the sorted sequence of these distances in a nondecreasing order.

Given a $\delta=\delta_i \geq 0$ where $i$ is the median of the index in the hinary search interval $[\ell, u]$, we construct a bipartite graph $G=(\tilde{A} \cup \tilde{B}, E)$ where an edge $e=(a, b){{a \in \tilde{A}, b \in \tilde{B}}}$ is in $E$ if and only if either both $a \in \bar{A}$ and $b \in \bar{B}$ (weight $(e)=0$ ) or $|a-b|{\infty} \leq \delta$ (weight $(e)=|a-b|_{\infty}$ ). A complete matching in $G$ is a set of $n$ edges so that every vertex in $\tilde{A}$ and $\tilde{B}$ is incident to exactly one edge in the set. To determine if $G$ has a complete matching, one can use an $O\left(n^{2.5}\right)$ algorithm of Hopcroft and Karp [198] for complete matching in a bipartite graph. However, exploiting the geometric embedding of the points in the persistence diagrams, we can apply an $O\left(n^{1.5}\right)$ time algorithm of Efrat et al. [154] for the purpose. If such an algorithm affirms that a complete matching exists, we do the following: If $\ell=u$ we output $\delta$, otherwise we set $u=i$ and repeat. If no matching exists, we set $\ell=i$ and repeat. Observe that matching has to exist for some value of $\delta$, in particular for $\delta_{n^{\prime}}$ and thus the binary search always succeeds. Algorithm 1: BoTTLENECK lays out the pseudocode for this matching. The algorithm runs in $O\left(n^{1.5} \log n\right)$ time accounting for the $O(\log n)$ probes for binary search each applying an $O\left(n^{1.5}\right)$ time matching algorithm. However, to achieve this complexity, we have to avoid sorting $n^{\prime}=O\left(n^2\right)$ values taking $O\left(n^2 \log n\right)$ time. Again, using the geometric embedding of the points, one can perform the binary probes without incurring the cost for sorting. For details and an efficient implementation of this algorithm, seee [209].

## 数学代写|表示论代写Representation theory代考|Stability of Persistence Diagrams

$$\mathrm{d} b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf \pi \in \Pi \sup x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)|x-\pi(x)| \infty$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

$(e)=0$ ) 要么 $|a-b| \infty \leq \delta$ (重量 $(e)=|a-b|{\infty}$ ). 一个完整的匹配 $G$ 是一组 $n$ 边使得每个顶点在 $\tilde{A}$ 和 $\tilde{B}$ 恰好与集合中的一条边相关。确定是否 $G$ 有一个完整的匹配，一个可以使用 $O\left(n^{2.5}\right)$ Hopcroft 和 Karp [198] 的算法用于二分图中的完全匹配。然而，利用持久性图中点的几何嵌入，我们可以应用 $O\left(n^{1.5}\right)$ Efrat 等人的时间算法。[154] 的目的。如果这样的算法确认存在完全匹配，我们将执行以下操 作: 如果 $\ell=u$ 我们输出 $\delta$ ，否则我们设置 $u=i$ 并重复。如果不存在匹配项，我们设置 $\ell=i$ 并重复。观 察到对于某些值必须存在匹配 $\delta$ ，特别是对于 $\delta{n^{\prime}}$ 因此二分查找总是成功的。算法 1: BOTTLENECK 列出 了此匹配的伪代码。该算法运行于 $O\left(n^{1.5} \log n\right)$ 时间占 $O(\log n)$ 用于二进制搜索的探针每个应用一个 $O\left(n^{1.5}\right)$ 时间匹配算法。然而，为了实现这种复杂性，我们必须避免排序 $n^{\prime}=O\left(n^2\right)$ 取值
$O\left(n^2 \log n\right)$ 时间。同样，使用点的几何嵌入，可以在不产生排序成本的情况下执行二元探测。有关此 算法的详细信息和有效实现，请参阅 [209]。

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## MATLAB代写

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