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

2022年12月30日

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## 计算机代写|机器学习代写machine learning代考|The Case of Standard Distance Kernels

We have seen in Theorem $4.1$ that the dominant eigenvectors of $\mathbf{K}$ contain the class label information (through the indicator matrix $\mathbf{J}$ ) and can thus be used for spectral clustering. Yet, $\mathbf{K}$ has the inconvenience that its first two dominant eigenvalues scale like $O(n)$ and $O(\sqrt{n})$, and only the latter is informative, in the sense that it depends on the covariance traces $\mathbf{t}$, but not on the means $\mathbf{M}$ or covariance “shapes.” As for the matrix $\mathbf{D}-\mathbf{K}$, it can be readily seen as quite inappropriate for clustering. Indeed, while the informative spectrum of $\mathbf{K}$ is essentially of order $O(1)$ (if we exclude the little informative two dominant eigenvectors), the matrix $\mathbf{D}$ has diagonal elements
$$[\mathbf{D}]_{i i}=n f\left(\tau_p\right)+\zeta_i+O(1), \quad \zeta_i=n f^{\prime}\left(\tau_p\right)[\psi]_i=O(\sqrt{n}),$$
where the random $\zeta_i$ terms are “essentially” of zero mean and asymptotically independent across $i$. Consequently, the spectrum of $\mathbf{D}-\mathbf{K}$ is largely dominated by the noninformative diagonal elements of $\mathbf{D}$, and the dominant eigenvectors of $\mathbf{D}-\mathbf{K}$ are thus uncorrelated with the structure in $\mathbf{J}$ : this comes in stark opposition to the finitedimensional intuitions according to which the dominant (here smallest) eigenvectors of $\mathbf{D}-\mathbf{K}$ should be aligned with the vectors of classes. As such, $\mathbf{D}-\mathbf{K}$ is not appropriate for large-dimensional spectral clustering, and this is largely confirmed by empirical results (as already empirically established, but with no strong theoretical argument, in the spectral clustering literature).

The matrix $\mathbf{D}^{\frac{1}{2}} \mathbf{K D}{ }^{\frac{1}{2}}$ advocated by Ng-Weiss-Jordan is more interesting. First, sincè $d_i=\mathbf{D}_{i i}=O(n)$, it is more convenient to consider the said normalized Laplacian matrix
$$\mathbf{L}=n \mathbf{D}^{-\frac{1}{2}} \mathbf{K D}^{-\frac{1}{2}}$$
than the difference (of matrices of misaligned orders of magnitude) $\mathbf{D}-\mathbf{K} .^{14}$ In addition, note that $\mathbf{D}^{\frac{1}{2}} \mathbf{1}_n$ is an eigenvector for $\mathbf{L}$ with corresponding eigenvalue $n$, since
$$n \mathbf{D}^{-\frac{1}{2}} \mathbf{K} \mathbf{D}^{-\frac{1}{2}}\left(\mathbf{D}^{\frac{1}{2}} \mathbf{1}_n\right)=\mathbf{D}^{-\frac{1}{2}} \mathbf{K} \mathbf{1}_n=n \mathbf{D}^{-\frac{1}{2}} \mathbf{D} \mathbf{1}_n=n \mathbf{D}^{\frac{1}{2}} \mathbf{1}_n$$

## 计算机代写|机器学习代写machine learning代考|The Case of “α-β” and Properly Scaling Kernels

The previous section demonstrated that, despite the phenomenon of distance concentration, spectral clustering with the normalized Laplacian $\mathbf{L}$ remains valid under large-dimensional data assumptions, at the expense of a few unexpected outcomes (presence of noninformative isolated eigenvectors, incoherence between the number of classes and the number of informative eigenvectors, etc.). These are immediate consequences of the theoretical study performed in Section $4.2$ and were shown to adequately match the actual performance of spectral clustering on, not only Gaussian, but also real-world data.

But Section $4.2$ also argued that kernels of the type $f\left(\left|\mathbf{x}_i-\mathbf{x}_j\right|^2 / p\right)$, despite their wide popularity, are suboptimal when it comes to classifying data down to their minimal statistical discrimination rate (particularly in exploiting the covariance structures). We then proved in Section $4.2 .4$ that $\alpha-\beta$ kernels, satisfying $f\left(\tau_p\right)=O(1)$, $f^{\prime}\left(\tau_p\right)=\alpha p^{-1 / 2}$ and $f^{\prime \prime}\left(\tau_p\right)=2 \beta$ for some $\alpha, \beta=O(1)$, are more powerful in discriminating data having close (even equal) means and slightly differing covariances. In Section $4.3, \alpha-\beta$ kernels were then shown to be a special case of the family of properly scaling kernels, which yield as good performance as $\alpha-\beta$ kernels (in exploiting covariancee “shape” structure) and havè the additionál advantage of being nonsmooth and thus be computed more efficiently.
We consider the $\alpha-\beta$ and properly scaling kernels here.
Specifically, Figure $4.12$ displays the comparative performance of Gaussian versus $\alpha-\beta$ inner-product kernels in the setting of a two-class Gaussian mixture data with equal means but slightly differing covariances (thus here with $\alpha=0$ ). We observe that the Gaussian kernel is incapable of resolving the two classes, while the $\alpha$ – $\beta$ kernel is fully adapted. Figure $4.13$ then extends the analysis to a real-world EEG dataset (epileptic versus sane patients) [Andrzejak et al., 2001] specifically chosen since, being a more or less stationary zero-mean time series, the critical class-discriminating features lie more in the second-order statistics (i.e., in the covariance matrix structure) than in the first (i.e., in the structure of the means). The data vectors were appropriately centered and normalized (such that $\left|\mathbf{x}_i\right|=\sqrt{p}$ ) to specifically exploit the covariance “shape” structure. In this case, the Gaussian kernel is observed to have less discriminating power compared with the $\alpha-\beta$ kernel (chosen here again with $\alpha=0$, that is, with $f^{\prime}\left(\tau_p\right)=0$ ).

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|The Case of Standard Distance Kernels

$$[\mathbf{D}]{i i}=n f\left(\tau_p\right)+\zeta_i+O(1), \quad \zeta_i=n f^{\prime}\left(\tau_p\right)[\psi]_i$$ 随机的地方 $\zeta_i$ 项“本质上”为零均值且渐近独立 $i \ldots . .$. 因 此，Imathbf{D}-Imathbf{K}的谱 $\mathbf{D}-\mathbf{K}$ 主要由的非 信息对角线元素主导 $\mathbf{D}$, 以及的主要特征向量 $\mathbf{D}-\mathbf{K}$ 因 此与结构不相关 $\mathbf{J}$ : 这与有限维直觉截然相反，根据有 限维直觉， D-K应该与类的向量对齐。像这样， $\mathbf{D}-\mathbf{K}$ 不适用于大维谱聚类，这在很大程度上得到了 经验结果的证实（在谱聚类文献中已经凭经验建立，但 没有强有力的理论论据）。 矩阵 $\mathbf{D}^{\frac{1}{2}} \mathbf{K D}^{\frac{1}{2}}$ Ng-Weiss-Jordan 提倡的更有趣。首 先， sinced $d_i=\mathbf{D}{i i}=O(n)$, 考虑上述归一化拉普拉 斯矩阵更方便
$$\mathbf{L}=n \mathbf{D}^{-\frac{1}{2}} \mathbf{K D}^{-\frac{1}{2}}$$

$$n \mathbf{D}^{-\frac{1}{2}} \mathbf{K} \mathbf{D}^{-\frac{1}{2}}\left(\mathbf{D}^{\frac{1}{2}} \mathbf{1}_n\right)=\mathbf{D}^{-\frac{1}{2}} \mathbf{K} \mathbf{1}_n=n \mathbf{D}^{-\frac{1}{2}} \mathbf{D} \mathbf{1}_n$$

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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。