## 计算机代写|机器学习代写machine learning代考|COMP4702

2022年12月24日

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## 计算机代写|机器学习代写machine learning代考|Normalizing a Two-Way Factorization

The aforementioned optimization model factorizes $D$ into two matrices $U$ and $V$. One can immediately notice that the factorization is not unique. For example, if we multiply each

entry of $U$ by 2 , then we can divide each entry of $V$ by 2 to get the same product $U V^T$. Furthermore, we can apply this trick to just a particular (say, $r$ th) column of each of $U$ and $V$ to get the same result. In other words, different normalization factors for the columns of $U$ and $V$ lead to the same product.

Therefore, some forms of dimensionality reduction convert the two-way matrix factorization into a three-way matrix factorization in which each of the matrices satisfies certain normalization conventions. This additional matrix is typically a $k \times k$ diagonal matrix of nonnegative entries, in which the $(r, r)$ th entry contains a scaling factor for the $r$ th column. Specifically, for any two-way matrix factorization $D \approx U V^T$ into $n \times k$ and $d \times k$ matrices $U$ and $V$, respectively, we can convert it into a unique ${ }^2$ three-way matrix factorization of the following form:
$$D \approx Q \Sigma P^T$$
Here, $Q$ is a normalized $n \times k$ matrix (derived from $U$ ), $P$ is a normalized $d \times k$ matrix (derived from $V$ ), and $\Sigma$ is a $k \times k$ diagonal matrix in which the diagonal entries contain the nonnegative normalization factors for the $k$ concepts. Each of the columns of $Q$ and $P$ satisfy the constraint that its $L_2$-norm (or $L_1$-norm) is one unit. It is common to use $L_2$-normalization in methods like singular value decomposition and $L_1$-normalization in methods like probabilistic latent semantic analysis. For the purpose of discussion, let us assume that we use $L_2$-normalization. Then, the conversion from two-way factorization to three-way factorization can be achieved as follows:

1. For each $r \in{1 \ldots k}$, divide the $r$ th column $\overline{U_r}$ of $U$ with its $L_2$-norm $\left|\overline{U_r}\right|$. The resulting matrix is denoted by $Q$.
2. For each $r \in{1 \ldots k}$, divide the $r$ th column $\overline{V_r}$ of $V$ with its $L_2$-norm $\left|\overline{V_r}\right|$. The resulting matrix is denoted by $P$.
3. Create a $k \times k$ diagonal matrix $\Sigma$, in which the $(r, r)$ th diagonal entry is the nonnegative value $\left|\overline{U_r}\right| \cdot\left|\overline{V_r}\right|$.

It is easy to show that the newly created matrices $Q, \Sigma$, and $P$ satisfy the following relationship:
$$Q \Sigma P^T=U V^T$$

## 计算机代写|机器学习代写machine learning代考|Singular Value Decomposition

Singular value decomposition (SVD) is used in all forms of multidimensional data, and its instantiation in the text domain is referred to as latent semantic analysis (LSA). Consider the simplest possible factorization of the $n \times d$ matrix $D$ into an $n \times k$ matrix $U=\left[u_{i j}\right]$ and the $d \times k$ matrix $V=\left[v_{i j}\right]$ as an unconstrained matrix factorization problem:
\begin{aligned} \text { Minimize }{U, V} & \left|D-U V^T\right|_F^2 \ & \text { subject to: } \ & \text { No constraints on } U \text { and } V \end{aligned} Here $|\cdot|_F^2$ refers to the (squared) Frobenius norm of a matrix, which is the sum of squares of its entries. The matrix $\left(D-U V^T\right)$ is also referred to as the residual matrix, because its entries contain the residual errors obtained from a low-rank factorization of the original matrix $D$. This optimization problem is the most basic form of matrix factorization with a popular objective function and no constraints. This formulation has infinitely many alternative optimal solutions (see Exercises 2 and 3). However, one ${ }^3$ of them is such that the columns of $V$ are orthonormal, which allows transformations of new documents (not included in $D$ ) with simple axis rotations (i.e., matrix multiplication). A remarkable property of the unconstrained optimization problem above is that imposing orthogonality constraints does not worsen the optimal solution. The following constrained optimization problem shares at least one optimal solution as the unconstrained version $[171,530]$ : \begin{aligned} \text { Minimize }{U, V} & \left|D-U V^T\right|_F^2 \ & \text { subject to: } \ & \text { Columns of } U \text { are mutually orthogonal } \ & \text { Columns of } V \text { are mutually orthonormal } \end{aligned}
In other words, one of the alternative optima to the unconstrained problem also satisfies orthogonality constraints. It is noteworthy that only the solution satisfying the orthogonality constraint is considered SVD because of its interesting properties, even though other optima do exist (see Exercises 2 and 3 ).

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Normalizing a Two-Way Factorization

$$D \approx Q \Sigma P^T$$

1. 对于每个 $r \in 1 \ldots k$, 划分 $r$ 第列 $\overline{U_r}$ 的 $U$ 与其 $L_2$-规范 $\left|\overline{U_r}\right|$. 结 果矩阵表示为 $Q$.
2. 对于每个 $r \in 1 \ldots k$ ，划分 $r$ 第列 $\overline{V_r}$ 的 $V$ 与其 $L_2$-规范 $\left|\overline{V_r}\right|$. 结 果矩阵表示为 $P$.
3. 创建一个 $k \times k$ 对角矩阵 $\Sigma$, 其中 $(r, r)$ 第对角线项是非负值 $\left|\overline{U_r}\right| \cdot\left|\overline{V_r}\right|$.
很容易证明新创建的矩阵 $Q, \Sigma ，$ 和 $P$ 满足以下关系:
$$Q \Sigma P^T=U V^T$$

## 计算机代写|机器学习代写machine learning代考|Singular Value Decomposition

Minimize $U, V\left|D-U V^T\right|_F^2 \quad$ subject to: No constraints

Minimize $U, V\left|D-U V^T\right|_F^2 \quad$ subject to: Columns of $U$ a

## 有限元方法代写

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## 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。