# 计算机代写|复杂网络代写complex network代考|CS60078

#### Doug I. Jones

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## 计算机代写|复杂网络代写complex network代考|Dimensionality Reduction with Minimal Squared Error

Suppose we are given a set of real valued measurements of some objects. As an example, for all boats in a marina, we measure length over all, width, height of the mast, the area of the sail, power of the engine, length of the waterline, and so forth. Let $N$ be the number of measurements, i.e., the number of boats in the marina, and let the measurements be vectors of dimension $d$, i.e., the number of things we have measured. We compile our measurements into a data matrix $\mathbf{A} \in \mathbb{R}^{N \times d}$, i.e., we write the individual measurement vectors as the rows of matrix $\mathbf{A}$. Let us further assume that we have already subtracted the mean across all measurements from each individual sample such that the columns of A sum to zero, i.e., we have centered our data.

Now we see that $\mathbf{A}^T \mathbf{A}$ is a $d \times d$ matrix describing the covariance of the individual dimensions in which we measured our data.

We now ask if we can drop some of the $d$ dimensions and still describe our data well. Naturally, we want to drop those dimensions in which our data do not vary much or we would like to replace two dimensions which are correlated by a single dimension. We can discard the unnecessary dimensions by projecting our data from the $d$-dimensional original space in a lower dimensional space of dimension $q<d$. Such a projection can be achieved by a matrix $\mathbf{V} \in \mathbb{R}^{d \times q}$. Taking measurement $\mathbf{a}{\mathbf{i}} \in \mathbb{R}^d$ from row $i$ of $\mathbf{A}$, we find the coordinates in the new space to be $\mathbf{b}{\mathbf{i}}=\mathbf{a}{\mathbf{i}} \mathbf{V}$ with $\mathbf{b}{\mathbf{i}} \in \mathbb{R}^q$. We can also use the transpose of $\mathbf{V}$ to project back into the original space of dimension $d$ via $\mathbf{a}{\mathbf{i}}^{\prime}=\mathbf{b}{\mathbf{i}} \mathbf{V}^T$. Since in the two projections we have visited a lower dimensional space, we find that generally the reconstructed data point does not coincide with the original datum $\mathbf{a}{\mathbf{i}} \mathbf{V} \mathbf{V}^T=\mathbf{a}{\mathbf{i}}^{\prime} \neq \mathbf{a}_{\mathbf{i}}$.

However, if we would have first started in the $q$-dimensional space with $\mathbf{b}{\mathbf{i}}$ and projected it into the $d$-dimensional space via $\mathbf{V}^T$ and then back again via $\mathbf{V}$ we require that our projection does not lose any information and hence $\mathbf{b}{\mathbf{i}} \mathbf{V}^T \mathbf{V}=\mathbf{b}_{\mathbf{i}}$. This means that we require $\mathbf{V}^T \mathbf{V}=\mathbb{1}$ or in other words we require that our projection matrix $\mathbf{V}$ be unitary.

The natural question is now how to find a unitary matrix such that it minimizes some kind of reconstruction error. Using the mean square error, we could write
\begin{aligned} E \propto \sum_i^N \sum_j^d\left(\mathbf{A}-\mathbf{A}^{\prime}\right){i j}^2 & =\sum_i^N \sum_j^d\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^{\mathbf{T}}\right){i j}^2 \ & =\operatorname{Tr}\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^T\right)^T\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^T\right) \end{aligned}

## 计算机代写|复杂网络代写complex network代考|Squared Error for Multivariate Data and Networks

Let us consider in the following the reconstruction of the adjacency matrix of a network $\mathbf{A} \in{0,1}^{N \times N}$ of rank $r$ by another adjacency matrix $\mathbf{B} \in{0,1}^{N \times N}$ possibly of lower rank $q<r$ as before. For the squared error we have
$$E=\sum_{i j}(\mathbf{A}-\mathbf{B})_{i j}^2 .$$
Then, there are only four different cases we need to consider in Table 3.1. The squared error gives equal value to the mismatch on the edges and missing edges in A. We could say it weighs every error by its own magnitude. While this is a perfectly legitimate approach for multivariate data, it is, however, highly problematic for networks. The first reason is that many networks are sparse. The fraction of non-zero entries in A is generally very, very small compared to the fraction of zero entries. A low rank approximation under the squared error will retain this sparsity to the point that $\mathbf{B}$ may be completely zero. Furthermore, we have seen that real networks tend to have a very heterogeneous degree distribution, i.e., the distribution of zeros and ones per row and column in $\mathbf{A}$ is also very heterogeneous. Why give every entry the same weight in the error function? Most importantly, for multivariate data, all entries of $\mathbf{A}_{i j}$ are equally important measurements in principle. For networks this is not the case: the edges are in principle more important than the missing edges. There are fewer of them and they should hence be given more importance than missing edges. Taken all of these arguments together, we see that our first goal will have to be the derivation of an error function specifically tailored for networks that does not suffer from these deficiencies.

# 复杂网络代写

## 计算机代写|复杂网络代写complex network代考|Dimensionality Reduction with Minimal Squared Error

\begin{aligned} E \propto \sum_i^N \sum_j^d\left(\mathbf{A}-\mathbf{A}^{\prime}\right){i j}^2 & =\sum_i^N \sum_j^d\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^{\mathbf{T}}\right){i j}^2 \ & =\operatorname{Tr}\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^T\right)^T\left(\mathbf{A}-\mathbf{A V} \mathbf{V}^T\right) \end{aligned}

## 计算机代写|复杂网络代写complex network代考|Squared Error for Multivariate Data and Networks

$$E=\sum_{i j}(\mathbf{A}-\mathbf{B})_{i j}^2 .$$

## 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。

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