### 数学代写|拓扑学代写Topology代考|MTH 3002

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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

In what follows, for simplicity of presentation, we assume that we consider smooth ( $C^{\infty}$-continuous) functions and smooth manifolds embedded in $\mathbb{R}^{d}$, even though often we only require the functions (resp. manifolds) to be $C^{2}$ continuous (resp. $C^{2}$-smooth).

To provide intuition, let us start with a smooth scalar function defined on the real line, $f: \mathbb{R} \rightarrow \mathbb{R}$; the graph of such a function is shown in Figure $1.8(\mathrm{~b})$. Recall that the derivative of a function at a point $x \in \mathbb{R}$ is defined as
$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t}$$

The value $D f(x)$ gives the rate of change of the value of $f$ at $x$. This can be visualized as the slope of the tangent line of the graph of $f$ at $(x, f(x))$. The critical points of $f$ are the set of points $x$ such that $D f(x)-0$. For a function defined on the real line, there are two types of critical points in the generic case: maxima and minima, as marked in Figure $1.8(b)$.

Now suppose we have a smooth function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ defined on $\mathbb{R}^{d}$. Fix an arbitrary point $x \in \mathbb{R}^{d}$. As we move a little around $x$ within its local neighborhood, the rate of change of $f$ differs depending on which direction we move. This gives rise to the directional derivative $D_{v} f(x)$ at $x$ in direction (i.e., a unit vector) $v \in \mathbb{S}^{d-1}$, where $\mathbb{S}^{d-1}$ is the unit $(d-1)$-sphere, defined as
$$D_{v} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t \cdot v)-f(x)}{t}$$
The gradient vector of $f$ at $x \in \mathbb{R}^{d}$ intuitively captures the direction of steepest increase of the function $f$. More precisely, we have the following.

Definition 1.25. (Gradient for functions on $\mathbb{R}^{d}$ ) Given a smooth function $f$ : $\mathbb{R}^{d} \rightarrow \mathbb{R}$, the gradient vector field $\nabla f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is defined as follows: for any $x \in \mathbb{R}^{d}$
$$\nabla f(x)=\left[\frac{\partial f}{\partial x_{1}}(x), \frac{\partial f}{\partial x_{2}}(x), \ldots, \frac{\partial f}{\partial x_{d}}(x)\right]^{\mathrm{T}}$$
where $\left(x_{1}, x_{2}, \ldots, x_{d}\right)$ represents an orthonormal coordinate system for $\mathbb{R}^{d}$. The vector $\nabla f(x) \in \mathbb{R}^{d}$ is called the gradient vector of $f$ at $x$. A point $x \in \mathbb{R}^{d}$ is a critical point if $\nabla f(x)=\left[\begin{array}{llll}0 & 0 & \ldots\end{array}\right]^{\mathrm{T}}$; otherwise, $x$ is regular.

## 数学代写|拓扑学代写Topology代考|Connection to Topology

We now characterize how critical points influence the topology of $M$ induced by the scalar function $f: M \rightarrow \mathbb{R}$.

Detinition 1.29. (Interval, sublevel, and superlevel sets) Given $f: M \rightarrow \mathbb{K}$ and $I \subseteq \mathbb{R}$, the interval levelset of $f$ with respect to $I$ is defined as
$$M_{I}=f^{-1}(I)={x \in M \mid f(x) \in I}$$
The case for $I=(-\infty, a]$ is also referred to as the sublevel set $M_{\leq a}:=$ $f^{-1}((-\infty, a])$ of $f$, while $M_{\geq a}:=f^{-1}([a, \infty))$ is called the superlevel set; and $f^{-1}(a)$ is called the levelset of $f$ at $a \in \mathbb{R} .$

Given $f: M \rightarrow \mathbb{R}$, imagine sweeping $M$ with increasing function values of $f$. It turns out that the topology of the sublevel sets can only change when we sweep through critical values of $f$. More precisely, we have the following classical result, where a diffeomorphism is a homeomorphism that is smooth in both directions.

Theorem 1.3. (Homotopy type of sublevel sets) Let $f: M \rightarrow \mathbb{R}$ be a smooth function defined on a manifold $M$. Given $a<b$, suppose the interval levelset $M_{[a, b]}=f^{-1}([a, b])$ is compact and contains no critical points of $f$. Then $M_{\leq a}$ is diffeomorphic to $M_{\leq b}$.

Furthermore, $M_{\leq a}$ is a deformation retract of $M_{\leq b}$, and the inclusion map $i: M_{\leq a} \hookrightarrow M_{\leq b}$ is a homotopy equivalence .

As an illustration, consider the example of height function $f: M \rightarrow \mathbb{R}$ defined on a vertical torus as shown in Figure $1.10(a)$. There are four critical points for the height function $f, u$ (minimum), $v, w$ (saddles), and $z$ (maximum). We have that $M_{\leq a}$ is: (i) empty for $af(z)$.
Theorem $1.3$ states that the homotopy type of the sublevel set remains the same until it passes a critical point. For Morse functions, we can also characterize the homotopy type of sublevel sets around critical points, captured by attaching $k$-cells.

## 数学代写|拓扑学代写Topology代考|Complexes and Homology Groups

This chapter introduces two very basic tools on which topological data analysis (TDA) is built. One is simplicial complexes and the other is homology groups. Data supplied as a discrete set of points do not have an interesting topology. Usually, we construct a scaffold on top of the data which is commonly taken as a simplicial complex. It consists of vertices at the data points, edges connecting them, and triangles, tetrahedra, and their higher-dimensional analogues that establish higher-order connectivity. Section $2.1$ formalizes this construction. There are different kinds of simplicial complexes. Some are easier to compute, but take more space. Others are more sparse, but take more time to compute. Section $2.2$ presents an important construction called the nerve and a complex called the Cech complex which is defined on this construction. This section also presents a commonly used complex in topological data analysis called the Vietoris-Rips complex that interleaves with the Cech complexes in terms of containment. In Section 2.3, we introduce some of the complexes which are sparser in size than the Vietoris-Rips or Čech complexes.

The second topic of this chapter, the homology groups of a simplicial complex, are the essential algebraic structures with which TDA analyzes data. Homology groups of a topological space capture the space of cycles up to those called boundaries that bound “higher-dimensional” subsets. For simplicity, we introduce the concept in the context of simplicial complexes instead of topological spaces. This is called simplicial homology. The essential entities for defining the homology groups are chains, cycles, and boundaries which we cover in Section 2.4. For simplicity and also for relevance in TDA, we define these structures under $\mathbb{Z}_{2}$-additions.

Section $2.5$ defines the simplicial homology group of a simplicial complex as the quotient space of the cycles with respect to the boundaries. Some of the concepts related to homology groups, such as induced homology under a map, singular homology groups for general topological spaces, relative homology groups of a complex with respect to a subcomplex, and the dual concept of homology groups, called cohomology groups are also introduced in this section.

## 拓扑学代考

$$D f(x)=\frac{d}{d x} f(x)=\lim {t \rightarrow 0} \frac{f(x+t)-f(x)}{t}$$ 价值 $D f(x)$ 给出值的变化率 $f$ 在 $x$. 这可以可视化为图形的切线的斜率 $f$ 在 $(x, f(x))$. 的关键点 $f$ 是点的集合 $x$ 这样 $D f(x)-0$. 对于定义在实线上的函数，一般情况下有两种临界点：极大值和极小值，如图所示 $1.8(b)$. 现在假设我们有一个平滑函数 $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ 定义于 $\mathbb{R}^{d}$. 修复任意点 $x \in \mathbb{R}^{d}$. 当我们稍微移动一下 $x$ 在其当地社区 内，变化率 $f$ 根据我们移动的方向而有所不同。这产生了方向导数 $D{v} f(x)$ 在 $x$ 在方向 (即，单位向量) $v \in \mathbb{S}^{d-1}$ ，在哪里S ${ }^{d-1}$ 是单位 $(d-1)$-sphere，定义为
$$D_{v} f(x)=\lim {t \rightarrow 0} \frac{f(x+t \cdot v)-f(x)}{t}$$ 的梯度向量 $f$ 在 $x \in \mathbb{R}^{d}$ 直观地捕捉到函数增长最陡的方向 $f$. 更准确地说，我们有以下内容。 定义 1.25。(函数的梯度 $\mathbb{R}^{d}$ ) 给定一个平滑函数 $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, 梯度向量场 $\nabla f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ 定义如下: 对于任何 $x \in \mathbb{R}^{d}$ $$\nabla f(x)=\left[\frac{\partial f}{\partial x{1}}(x), \frac{\partial f}{\partial x_{2}}(x), \ldots, \frac{\partial f}{\partial x_{d}}(x)\right]^{\mathrm{T}}$$

## 数学代写|拓扑学代写Topology代考|Connection to Topology

$$M_{I}=f^{-1}(I)=x \in M \mid f(x) \in I$$

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