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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|凸优化作业代写Convex Optimization代考|Affine sets and affine hulls

A set $C$ is said to be an affine set if for any $\mathbf{x}{1}, \mathbf{x}{2} \in C$ and for any $\theta_{1}, \theta_{2} \in \mathbb{R}$ such that $\theta_{1}+\theta_{2}=1$, the point $\theta_{1} \mathbf{x}{1}+\theta{2} \mathbf{x}_{2}$ also belongs to the set $C$. For instance, the line defined in (2.1) is an affine set. This concept can be extended to more than two points, as illustrated in the following example.

Example 2.1 If a set $C$ is affine with $\mathbf{x}{1}, \mathbf{x}{2}, \ldots, \mathbf{x}{k} \in C$, then $\sum{i=1}^{k} \theta_{i} \mathbf{x}{i} \in C$ for every $\boldsymbol{\theta}=\left[\theta{1}, \ldots, \theta_{k}\right]^{T} \in \mathbb{R}^{k}$ satisfying $\sum_{i=1}^{k} \theta_{i}=1$.
Proof: Assume that $\mathbf{x}{1}, \mathbf{x}{2}$, and $\mathbf{x}{3} \in C$. Then, it is true that $$\boldsymbol{x}{2}=\theta_{2} \mathbf{x}{2}+\left(1-\theta{2}\right) \mathbf{x}{3} \in C, \theta{2} \in \mathbb{R},$$
and $\mathbf{x}=\theta_{1} \mathbf{x}{1}+\left(1-\theta{1}\right) \boldsymbol{x}{2} \in C, \theta{1} \in \mathbb{R}$. Hence
\begin{aligned} \mathbf{x} &=\theta_{1} \mathbf{x}{1}+\left(1-\theta{1}\right) \theta_{2} \mathbf{x}{2}+\left(1-\theta{1}\right)\left(1-\theta_{2}\right) \mathbf{x}{3} \in C \ &=\alpha{1} \mathbf{x}{1}+\alpha{2} \mathbf{x}{2}+\alpha{3} \mathbf{x}{3} \end{aligned} where $\alpha{1}=\theta_{1}, \alpha_{2}=\left(1-\theta_{1}\right) \theta_{2}$, and $\alpha_{3}=\left(1-\theta_{1}\right)\left(1-\theta_{2}\right) \in \mathbb{R}$, and $\alpha_{1}+\alpha_{2}+$ $\alpha_{3}=1$. This can be extended to any $k$ points $\in C$ by induction. Thus we have completed the proof.

That is to say that the affine set contains all the affine combinations (linear combinations of points with sum of the real coefficients equal to one) of the points in it. It is worthwhile to mention that the affine set need not contain the origin, whereas a subspace must contain the origin. In fact, it is very straightforward to show that the following set is a subspace:
$$V=C-\left{\mathbf{x}{0}\right}=\left{\mathbf{x}-\mathbf{x}{0} \mid \mathbf{x} \in C\right} \text { (cf. (1.22) and (1.23)), }$$
where $C$ is an affine set and $\mathbf{x}{0} \in C$. Also note that $V$ and $C$ have the same dimension (as illustrated in Figure 2.1). Hence the dimension of the affine set $C$, denoted as affdim $(C)$, is defined as the dimension of the associated subspace $V$ given by (2.4), i.e., $$\operatorname{affdim}(C) \triangleq \operatorname{dim}(V) .$$ Given a set of vectors $\left{\mathbf{s}{1}, \ldots, \mathbf{s}{n}\right} \subset \mathbb{R}^{\ell}$, the affine hull of this set is defined as $$\operatorname{aff}\left{\mathbf{s}{1}, \ldots, \mathbf{s}{n}\right}=\left{\mathbf{x}=\sum{i=1}^{n} \theta_{i} \mathbf{s}{i} \mid\left(\theta{1}, \ldots, \theta_{n}\right) \in \mathbb{R}^{n}, \sum_{i=1}^{n} \theta_{i}=1\right}$$

数学代写|凸优化作业代写Convex Optimization代考|Relative interior and relative boundary

Affine hull defined in (2.13) and affine dimension of a set defined in (2.14) play an essential role in convex geometric analysis, and have been applied to dimension reduction in many signal processing applications such as blind separation (or unmixing) of biomedical and hyperspectral image signals (to be introduced in Chapter 6). To further illustrate their characteristics, it would be useful to address the interior and the boundary of a set w.r.t. its affine hull, which are, respectively, termed as relative interior and relative boundary, and are defined below.

The relative interior of $C \subseteq \mathbb{R}^{n}$ is defined as
\begin{aligned} \text { relint } C &={\mathbf{x} \in C \mid B(\mathbf{x}, r) \cap \text { aff } C \subseteq C, \text { for some } r>0} \ &=\operatorname{int} C \text { if aff } C=\mathbb{R}^{n} \quad(c f .(1.20)) \end{aligned}
where $B(\mathbf{x}, r)$ is a 2 -norm ball with center at $\mathbf{x}$ and radius $r$. It can be inferred from (2.16) that
$$\text { int } C= \begin{cases}\text { relint } C, & \text { if affdim } C=n \ \emptyset, & \text { otherwise. }\end{cases}$$
The relative boundary of a set $C$ is defined as
\begin{aligned} \operatorname{relbd} C &=\mathbf{c l} C \backslash \operatorname{relint} C \ &=\mathbf{b d} C, \text { if int } C \neq \emptyset(\text { by }(2.17)) \end{aligned}

数学代写|凸优化作业代写Convex Optimization代考| Affine sets and affine hulls

$$x 2=\theta_{2} \mathbf{x} 2+(1-\theta 2) \mathbf{x} 3 \in C, \theta 2 \in \mathbb{R}$$

$$\mathbf{x}=\theta_{1} \mathbf{x} 1+(1-\theta 1) \theta_{2} \mathbf{x} 2+(1-\theta 1)\left(1-\theta_{2}\right) \mathbf{x} 3 \in C \quad=\alpha 1 \mathbf{x} 1+\alpha 2 \mathbf{x} 2+\alpha 3 \mathbf{x} 3$$

$$\operatorname{affdim}(C) \triangleq \operatorname{dim}(V) .$$

，则此集合的仿射壳定义为

数学代写|凸优化作业代写Convex Optimization代考| Relative interior and relative boundary

(2.13) 中定义的仿射壳体和 (2.14) 中定义的集合的仿射维数在凸几何分析中起着至关重要的作用，并且已应用于许多信号处理应用中的降维，例如生物医学 和高光谱图像信号的盲分离 (或解混) (将在第 6 章中介绍)。为了进一步说明它们的特征，解决一个集合的内部和边界将是有用的，w.r.t.它的仿射船体，它们 分别被称为相对内部和相对边界，并在下面定义。

$$\text { relint } C=\mathbf{x} \in C \mid B(\mathbf{x}, r) \cap \text { aff } C \subseteq C, \text { for some } r>0 \quad=\operatorname{int} C \text { if aff } C=\mathbb{R}^{n} \quad(c f .(1.20))$$

$$\text { int } C={\text { relint } C, \quad \text { if affdim } C=n \emptyset, \quad \text { otherwise. }$$

$$\operatorname{relbd} C=\mathbf{c l} C \backslash \operatorname{relint} C \quad=\operatorname{bd} C, \text { if } \operatorname{int} C \neq \emptyset(\text { by }(2.17))$$

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assignmentutor™您的专属作业导师
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