text: assignmentutor™您的专属作业导师
  
  assignmentutor-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。
  
  • 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|凸优化作业代写Convex Optimization代考|Singular value decomposition
  
  The singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and communications. Applications which employ the SVD include computation of pseudo-inverse, least-squares fitting of data, matrix approximation, and determination of rank, range space, and null space of a matrix and so on and so forth.
  Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(\mathbf{A})=r$. The SVD of $\mathbf{A}$ is expressed in the form
  $$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T} \text {. }$$
  In the full SVD (1.107) for $\mathbf{A}, \mathbf{U}=\left[\mathbf{U}{r}, \mathbf{U}^{\prime}\right] \in \mathbb{R}^{m \times m}$ and $\mathbf{V}=\left[\mathbf{V}{r}, \mathbf{V}^{\prime}\right] \in \mathbb{R}^{n \times n}$ are orthogonal matrices in which $\mathbf{U}{r}=\left[\mathbf{u}{1}, \ldots, \mathbf{u}{r}\right]$ (consisting of $r$ left singular vectors) is an $m \times r$ semi-unitary matrix (cf. (1.75)) due to $$\mathbf{U}{r}^{T} \mathbf{U}{r}=\mathbf{I}{r}$$
  and $\mathbf{V}{r}=\left[\mathbf{v}{1}, \ldots, \mathbf{v}{r}\right]$ (consisting of $r$ right singular vectors) is an $n \times r$ semiunitary matrix (i.e., $\mathbf{V}{r}^{T} \mathbf{V}{r}=\mathbf{I}{r}$ ), and
  $$\boldsymbol{\Sigma}=\left[\begin{array}{c|c} \operatorname{Diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right) & 0_{r \times(n-r)} \ \hline 0_{(m-r) \times r} & 0_{(m-r) \times(n-r)} \end{array} \in \mathbb{R}^{m \times n}\right.$$
  is a rectangular matrix with $r$ positive singular values (supposedly arranged in nonincreasing order), denoted as $\sigma_{i}$, as the first $r$ diagonal elements and zeros elsewhere. Moreover, the range space of $\mathbf{A}$ and that of $\mathbf{U}{r}$ are identical, i.e., $$\mathcal{R}(\mathbf{A})=\mathcal{R}\left(\mathbf{U}{r}\right)$$
  The thin $S V D$ of an $m \times n$ matrix $\mathbf{A}$ with rank $r$ is given by
  $$\mathbf{A}=\mathbf{U}{r} \boldsymbol{\Sigma}{r} \mathbf{V}{r}^{T}=\sum{i=1}^{r} \sigma_{i} \mathbf{u}{i} \mathbf{v}{i}^{T}$$
  
  ## 数学代写|凸优化作业代写Convex Optimization代考|Least-squares approximation
  
  The method of least squares is extensively used to approximately solve for the unknown variables of a linear system with the given set of noisy measurements. Least squares can be interpreted as a method of data fitting. The best fit, between modeled and observed data, in the LS sense is that the sum of squared residuals reaches the least value, where a residual is the difference between an observed value and the value computed from the model.
  Consider a system characterized by a set of linear equations,
  $$\mathbf{b}=\mathbf{A x}+\boldsymbol{\epsilon},$$
  where $\mathbf{A} \in \mathbb{R}^{m \times n}$ is the given system matrix, b is the given data vector, and $\epsilon \in$ $\mathbb{R}^{m}$ is the measurement noise vector. The LS problem is to find an optimal $\mathbf{x} \in \mathbb{R}^{n}$ by minimizing $|\mathbf{A x}-\mathbf{b}|_{2}^{2}$. The LS solution, denoted as $\mathbf{x}{\mathrm{LS}}$, that minimizes $|\mathbf{A x}-\mathbf{b}|{2}^{2}$, is known as
  $$\mathbf{x}{\mathrm{LS}} \triangleq \arg \min {\mathbf{x} \in \mathbb{R}^{n}}\left{|\mathbf{A} \mathbf{x}-\mathbf{b}|_{2}^{2}\right}=\mathbf{A}^{\dagger} \mathbf{b}+\mathbf{v}, \mathbf{v} \in \mathcal{N}(\mathbf{A})$$
  which is actually an unconstrained optimization problem and the solution (with the same form as the solution (1.119) to the linear equations (1.118)) may not be unique.
  
  As $m \geq n$, the system (1.123) is an over-determined system (i.e., more equations than unknowns), otherwise an under-determined system (i.e., more unknowns than equations). Suppose that $\mathbf{A}$ is of full column rank for the overdetermined case $(m \geq n)$, and thus
  $$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I}{n}$$ Then the optimal $\mathbf{x}{\mathrm{LS}}=\mathbf{A}^{\dagger} \mathbf{b}$ is unique with the approximation error
  $$\left|\mathbf{A} \mathbf{x}{\mathrm{LS}}-\mathbf{b}\right|{2}^{2}=\left|\mathbf{P}{\mathbf{A}}^{\perp} \mathbf{b}\right|{2}^{2}>0 \text {, if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
  In other words, $\mathbf{A} \mathbf{x}_{\mathrm{LS}}$ is the image vector of $\mathbf{b}$ projected on the range space $\mathcal{R}(\mathbf{A})$, and so the approximation error is equal to zero only if $\mathbf{b} \in \mathcal{R}(\mathbf{A})$.
  
  ## 数学代写|凸优化作业代写Convex Optimization代考| Singular value decomposition
  
  $$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T}$$
  
  $$\mathbf{U} r^{T} \mathbf{U} r=\mathbf{I} r$$
  
  $$\mathcal{R}(\mathbf{A})=\mathcal{R}(\mathbf{U} r)$$
  
  $$\mathbf{A}=\mathbf{U} r \boldsymbol{\Sigma} r \mathbf{V} r^{T}=\sum i=1^{r} \sigma_{i} \mathbf{u} i \mathbf{v} i^{T}$$
  
  ## 数学代写|凸优化作业代写Convex Optimization代考| Least-squares approximation
  
  $$\mathbf{b}=\mathbf{A} \mathbf{x}+\boldsymbol{\epsilon}$$
  
  $$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I} n$$
  
  $$|\mathbf{A x L S}-\mathbf{b}| 2^{2}=\left|\mathbf{P A}^{\perp} \mathbf{b}\right| 2^{2}>0, \text { if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
  
  ## 有限元方法代写
  
  assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
  
  ## MATLAB代写
  
  MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
  
  assignmentutor™您的专属作业导师
  assignmentutor™您的专属作业导师
explanation: The provided text is a technical excerpt from a textbook or lecture notes on convex optimization, specifically discussing the singular value decomposition (SVD) and least-squares approximation. The text does not contain a question or an answer to a question. It is purely expository, explaining the concepts and providing mathematical formulations.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: assignmentutor™您的专属作业导师
  
  assignmentutor-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。
  
  • 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|凸优化作业代写Convex Optimization代考|Singular value decomposition
  
  The singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and communications. Applications which employ the SVD include computation of pseudo-inverse, least-squares fitting of data, matrix approximation, and determination of rank, range space, and null space of a matrix and so on and so forth.
  Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(\mathbf{A})=r$. The SVD of $\mathbf{A}$ is expressed in the form
  $$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T} \text {. }$$
  In the full SVD (1.107) for $\mathbf{A}, \mathbf{U}=\left[\mathbf{U}{r}, \mathbf{U}^{\prime}\right] \in \mathbb{R}^{m \times m}$ and $\mathbf{V}=\left[\mathbf{V}{r}, \mathbf{V}^{\prime}\right] \in \mathbb{R}^{n \times n}$ are orthogonal matrices in which $\mathbf{U}{r}=\left[\mathbf{u}{1}, \ldots, \mathbf{u}{r}\right]$ (consisting of $r$ left singular vectors) is an $m \times r$ semi-unitary matrix (cf. (1.75)) due to $$\mathbf{U}{r}^{T} \mathbf{U}{r}=\mathbf{I}{r}$$
  and $\mathbf{V}{r}=\left[\mathbf{v}{1}, \ldots, \mathbf{v}{r}\right]$ (consisting of $r$ right singular vectors) is an $n \times r$ semiunitary matrix (i.e., $\mathbf{V}{r}^{T} \mathbf{V}{r}=\mathbf{I}{r}$ ), and
  $$\boldsymbol{\Sigma}=\left[\begin{array}{c|c} \operatorname{Diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right) & 0_{r \times(n-r)} \ \hline 0_{(m-r) \times r} & 0_{(m-r) \times(n-r)} \end{array} \in \mathbb{R}^{m \times n}\right.$$
  is a rectangular matrix with $r$ positive singular values (supposedly arranged in nonincreasing order), denoted as $\sigma_{i}$, as the first $r$ diagonal elements and zeros elsewhere. Moreover, the range space of $\mathbf{A}$ and that of $\mathbf{U}{r}$ are identical, i.e., $$\mathcal{R}(\mathbf{A})=\mathcal{R}\left(\mathbf{U}{r}\right)$$
  The thin $S V D$ of an $m \times n$ matrix $\mathbf{A}$ with rank $r$ is given by
  $$\mathbf{A}=\mathbf{U}{r} \boldsymbol{\Sigma}{r} \mathbf{V}{r}^{T}=\sum{i=1}^{r} \sigma_{i} \mathbf{u}{i} \mathbf{v}{i}^{T}$$
  
  ## 数学代写|凸优化作业代写Convex Optimization代考|Least-squares approximation
  
  The method of least squares is extensively used to approximately solve for the unknown variables of a linear system with the given set of noisy measurements. Least squares can be interpreted as a method of data fitting. The best fit, between modeled and observed data, in the LS sense is that the sum of squared residuals reaches the least value, where a residual is the difference between an observed value and the value computed from the model.
  Consider a system characterized by a set of linear equations,
  $$\mathbf{b}=\mathbf{A x}+\boldsymbol{\epsilon},$$
  where $\mathbf{A} \in \mathbb{R}^{m \times n}$ is the given system matrix, b is the given data vector, and $\epsilon \in$ $\mathbb{R}^{m}$ is the measurement noise vector. The LS problem is to find an optimal $\mathbf{x} \in \mathbb{R}^{n}$ by minimizing $|\mathbf{A x}-\mathbf{b}|_{2}^{2}$. The LS solution, denoted as $\mathbf{x}{\mathrm{LS}}$, that minimizes $|\mathbf{A x}-\mathbf{b}|{2}^{2}$, is known as
  $$\mathbf{x}{\mathrm{LS}} \triangleq \arg \min {\mathbf{x} \in \mathbb{R}^{n}}\left{|\mathbf{A} \mathbf{x}-\mathbf{b}|_{2}^{2}\right}=\mathbf{A}^{\dagger} \mathbf{b}+\mathbf{v}, \mathbf{v} \in \mathcal{N}(\mathbf{A})$$
  which is actually an unconstrained optimization problem and the solution (with the same form as the solution (1.119) to the linear equations (1.118)) may not be unique.
  
  As $m \geq n$, the system (1.123) is an over-determined system (i.e., more equations than unknowns), otherwise an under-determined system (i.e., more unknowns than equations). Suppose that $\mathbf{A}$ is of full column rank for the overdetermined case $(m \geq n)$, and thus
  $$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I}{n}$$ Then the optimal $\mathbf{x}{\mathrm{LS}}=\mathbf{A}^{\dagger} \mathbf{b}$ is unique with the approximation error
  $$\left|\mathbf{A} \mathbf{x}{\mathrm{LS}}-\mathbf{b}\right|{2}^{2}=\left|\mathbf{P}{\mathbf{A}}^{\perp} \mathbf{b}\right|{2}^{2}>0 \text {, if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
  In other words, $\mathbf{A} \mathbf{x}_{\mathrm{LS}}$ is the image vector of $\mathbf{b}$ projected on the range space $\mathcal{R}(\mathbf{A})$, and so the approximation error is equal to zero only if $\mathbf{b} \in \mathcal{R}(\mathbf{A})$.
  
  ## 数学代写|凸优化作业代写Convex Optimization代考| Singular value decomposition
  
  $$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T}$$
  
  $$\mathbf{U} r^{T} \mathbf{U} r=\mathbf{I} r$$
  
  $$\mathcal{R}(\mathbf{A})=\mathcal{R}(\mathbf{U} r)$$
  
  $$\mathbf{A}=\mathbf{U} r \boldsymbol{\Sigma} r \mathbf{V} r^{T}=\sum i=1^{r} \sigma_{i} \mathbf{u} i \mathbf{v} i^{T}$$
  
  ## 数学代写|凸优化作业代写Convex Optimization代考| Least-squares approximation
  
  $$\mathbf{b}=\mathbf{A} \mathbf{x}+\boldsymbol{\epsilon}$$
  
  $$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I} n$$
  
  $$|\mathbf{A x L S}-\mathbf{b}| 2^{2}=\left|\mathbf{P A}^{\perp} \mathbf{b}\right| 2^{2}>0, \text { if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
  
  ## 有限元方法代写
  
  assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
  
  ## MATLAB代写
  
  MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
  
  assignmentutor™您的专属作业导师
  assignmentutor™您的专属作业导师
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a technical excerpt from a textbook or lecture notes on convex optimization, specifically discussing the singular value decomposition (SVD) and least-squares approximation. The text does not contain a question or an answer to a question. It is purely expository, explaining the concepts and providing mathematical formulations.
  
  Label: no-question