### 计算机代写|机器学习代写machine learning代考|COMP5318

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• 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

## 计算机代写|机器学习代写machine learning代考|The Pitfalls of Large-Dimensional Statistics

The big data revolution comes along with the challenging needs to parse, mine, and compress a large amount of large-dimensional and possibly heterogeneous data. In many applications, the dimension $p$ of the observations is as large as $-$ if not much larger than – their number $n$. In array processing and wireless communications, the number of antennas required for fine localization resolution or increased communication throughput may be as large (today in the order of hundreds) as the number of available independent signal observations [Li and Stoica, 2007, Lu et al., 2014]. In genomics, the identification of correlations among hundreds of thousands of genes based on a limited number of independent (and expensive) samples induces an even larger ratio $\mathrm{p} / \mathrm{n}$ [Arnold et al., 1994]. In statistical finance, portfolio optimization relies on the need to invest on a large number $p$ of assets to reduce volatility but at the same time to estimate the current (rather than past) asset statistics from a relatively small number $n$ of asset return records [Laloux et al., 2000].

As we shall demonstrate in the following section, the fact that in these problems $n$ is not much larger than $p$ annihilates most of the results from standard asymptotic statistics that assume $n$ alone is large [Vaart, 2000]. As a rule of thumb, by “much larger” we mean here that $n$ must be at least 100 times larger than $p$ for standard asymptotic statistics to be of practical convenience (see our argument in Section 1.1.2). Many algorithms in statistics, signal processing, and machine learning are precisely derived from this $n \gg p$ assumption that is no longer appropriate today. A major objective of this book is to cast some light on the resulting biases and problems incurred and to provide a systematic random matrix framework to improve these algorithms.

Possibly more importantly, we will see in this book that (small $p$ ) small-dimensional intuitions at the core of many machine learning algorithms (starting with spectral clustering [Ng et al., 2002, Luxburg, 2007]) may strikingly fail when applied in a simultaneously large $n, p$ setting. A compelling example lies in the notion of “distance” between vectors. Most classification methods in machine learning are rooted in the observation that random data vectors arising from a mixture distribution (say Gaussian) gather in “groups” of close-by vectors in the Euclidean norm. When dealing with large-dimensional data, however, concentration phenomena arise that make Euclidean distances useless, if not counterproductive: Vectors from the same mixture class may be further away in Euclidean distance than vectors arising from different classes. While classification may still be doable, it works in a rather different way from our small-dimensional intuition. The book intends to prepare the reader for the multiple traps caused by this “curse of dimensionality.”

## 计算机代写|机器学习代写machine learning代考|Sample Covariance Matrices in the Large n,p Regime

Let us consider the following example that illustrates a first elementary, yet counterintuitive, result: For simultaneously large $n, p$, the sample covariance matrix $\hat{\mathbf{C}} \in \mathbb{R}^{p \times p}$ based on $n$ samples $\mathbf{x}i \sim \mathcal{N}(\mathbf{0}, \mathbf{C})$ is an entry-wise consistent estimator of the population covariance $\mathbf{C} \in \mathbb{R}^{p \times p}$ (i.e., $|\hat{\mathbf{C}}-\mathbf{C}|{\infty} \rightarrow 0$ as $p, n \rightarrow \infty$ for $|\mathbf{A}|_{\infty} \equiv \max {i j}\left|\mathbf{A}{i j}\right|$ ) while overall being an extremely poor estimator in a (more practical) operator norm sense (i.e., $|\hat{\mathbf{C}}-\mathbf{C}| \nrightarrow 0$, with $|\cdot|$ being the operator norm here). Matrix norms are, in particular, not equivalent in the large $n, p$ scenario.

Let us detail this claim, in the simplest case where $\mathbf{C}=\mathbf{I}p$. Consider a dataset $\mathbf{X}=\left[\mathbf{x}_1, \ldots, \mathbf{x}_n\right] \in \mathbb{R}^{p \times n}$ of $n$ independent and identically distributed (i.i.d.) observations from a $p$-dimensional standard Gaussian distribution, that is, $\mathbf{x}_i \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$ for $i \in{1, \ldots, n}$. We wish to estimate the population covariance matrix $\mathbf{C}=\mathbf{I}_p$ from the $n$ available samples. The maximum likelihood estimator in this zero-mean Gaussian setting is the sample covariance matrix $\hat{\mathbf{C}}$ defined by $$\hat{\mathbf{C}}=\frac{1}{n} \sum{i=1}^n \mathbf{x}_i \mathbf{x}_i^{\top}=\frac{1}{n} \mathbf{X} \mathbf{X}^{\top}$$
By the strong law of large numbers, for fixed $p, \hat{\mathbf{C}} \rightarrow \mathbf{I}_p$ almost surely as $n \rightarrow \infty$, so that $\left|\hat{\mathbf{C}}-\mathbf{I}_p\right| \stackrel{\text { a.s. }}{\longrightarrow} 0$ holds for any standard matrix norm and in particular for the operator norm.

One must be more careful when dealing with the case $n, p \rightarrow \infty$ with the ratio $p / n \rightarrow$ $c \in(0, \infty)$ (or, from a practical standpoint, $n$ is not much larger than $p$ ). First, note that the entry-wise convergence still holds since, invoking the law of large numbers again,
$$[\hat{\mathbf{C}}]{i j}=\frac{1}{n} \sum{l=1}^n[\mathbf{X}]{i l}[\mathbf{X}]{j l} \stackrel{\text { a.s. }}{\longrightarrow} \begin{cases}1, & i=j \ 0, & i \neq j .\end{cases}$$
Besides, by a concentration inequality argument, it can even be shown that
$$\max {1 \leq i, j \leq p}\left|\left[\hat{\mathbf{C}}-\mathbf{I}_p\right]{i j}\right| \stackrel{\text { a.s. }}{\longrightarrow} 0$$

which holds as long as $p$ is no larger than a polynomial function of $n$, and thus:
$$\left|\hat{\mathbf{C}}-\mathbf{I}p\right|{\infty} \stackrel{\text { a.s. }}{\longrightarrow} 0 \text {. }$$
Consider now the case $p>n$. Since $\hat{\mathbf{C}}=\frac{1}{n} \sum_{i=1}^n \mathbf{x}_i \mathbf{x}_i^{\top}$ is the sum of $n$ rank-one matrices, the rank of $\hat{\mathbf{C}}$ is at most equal to $n$ and thus, being a $p \times p$ matrix with $p>n$, the sample covariance matrix $\hat{\mathbf{C}}$ must be a singular matrix having at least $p-n>0$ null eigenvalues. As a consequence,
$$\left|\hat{\mathbf{C}}-\mathbf{I}_p\right| \not \neg$$
for $|\cdot|$ the matrix operator (or spectral) norm.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Sample Covariance Matrices in the Large n,p Regime

$$\left|\hat{\mathbf{C}}-\mathbf{I}_p\right| \not$$

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## MATLAB代写

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