### 统计代写|主成分分析代写Principal Component Analysis代考|OLET5610

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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

## 统计代写|主成分分析代写Principal Component Analysis代考|Sparse PCA via Rotation and Truncation

SPCArt was motivated by the Eckart-Young theorem [6]. This theorem considers the problem of approximating a matrix by the product of two low-rank matrices.

Theorem 1 (Eckart-Young Theorem) Assume the $S V D$ of a matrix $A \in \mathbb{R}^{n \times p}$ is $A=U \Sigma V^{T}$, where $U \in \mathbb{R}^{n \times m}, m \leq \min {n, p}, \Sigma \in \mathbb{R}^{m \times m}$ is diagonal with

$\Sigma_{11} \geq \Sigma_{22} \geq \cdots \geq \Sigma_{m m}$, and $V \in \mathbb{R}^{p \times m}$. A rank-r $(r \leq m)$ approximation of $A$ is to solve the following problem:
$$\min {Y, X}\left|A-Y X^{T}\right|{F}^{2}, \text { s.t. } X^{T} X=I,$$
where $Y \in \mathbb{R}^{n \times r}$, and $X \in \mathbb{R}^{p \times r} .$ A solution is
$$X^{}=V_{1: r}, Y^{}=A X^{}$$ where $V_{1: r}$ is the first $r$ columns of $V$. Alternatively, the solution can be expressed as $$Y^{}=U_{1 r} \Sigma_{1: r,}, X^{}=\operatorname{Polar}\left(A^{T} Y^{}\right),$$
where Polar( ) is the orthonormal component of the polar decomposition [13]. From the SVD perspective, its equivalent definition is provided in Table $2 .$

Note that if $A$ is a mean-removed data matrix with $n$ samples, then $V_{1: r}$ are the loadings obtained by PCA. Clearly, $X^{}=V_{1: r} R$ and $Y^{}=A X^{*}=U_{1: r} \Sigma_{1: r} R$ are also a solution of (1), $\forall$ rotation matrix $R$. This implies that any rotation of the $r$ orthonormal leading eigenvectors, $V_{1: r}$, is a solution of the best rank- $r$ approximation of $A$. That is, any orthonormal basis in the corresponding eigensubspace is capable of representing the original data distribution as well as the original basis. Thus, a natural idea for sparse $\mathrm{PCA}$ is to find a rotation matrix, $R$, so that $X=V_{1 s} R$ becomes sparse, i.e.,
$$\min {R \in \mathbb{R}{r \times r}}\left|V_{1: r} R\right|_{0}, s . t . R^{T} R=I,$$
where $|$ – $|_{0}$ denotes the sum of $\ell_{0}$ (pseudo) norm of the columns of a matrix, i.e., the count of non-zeros of a matrix.

## 统计代写|主成分分析代写Principal Component Analysis代考|Objective and Optimization

Unfortunately, the above problem is difficult to solve, so SPCArt approximates it. Since $X=V_{1 r} R \Leftrightarrow V_{1: r}=X R^{T}$, SPCArt wants to find a rotation matrix, $R$, through which a sparse basis, $X$, approximates the PCA loadings. Without confusion, we use $V$ to denote $V_{1: r}$ hereafter. Let us first consider the $\ell_{1}$ version:
\begin{aligned} &\min {X, R} \frac{1}{2}|V-X R|{F}^{2}+\lambda \sum_{i}\left|X_{i}\right|_{1}, \ &\text { s.t. } \forall i,\left|X_{i}\right|_{2}=1, R^{T} R=I, \end{aligned}

where $|\cdot|_{1}$ is the $\ell_{1}$ norm of a vector, i.e., the sum of absolute values. The $\ell_{0}$ version will be introduced in the next section. It is well-known that $\ell_{1}$ norm is sparsity inducing, which is a convex surrogate of the $\ell_{0}$ norm [5]. Under this objective, the solution may not be orthogonal, and may deviate from the eigensubspace spanned by $V$. However, if the approximation is accurate enough, i.e., $V \approx X R$, then $X \approx V R^{T}$ would be nearly orthogonal and explain similar variance as $V$. Note that the above objective turns out to be a matrix approximation problem of the EckartYoung theorem. The key difference is that a sparsity penalty is added, but the solutions still share some common features.

There are no closed-form solutions for $R$ and $X$ simultaneously. To find a local minimum, SPCArt solves the problem by fixing one and optimizing the other alternately, i.e., the block coordinate descent [23]. Fortunately, both subproblems have closed-form solutions.

## 统计代写|主成分分析代写Principal Component Analysis代考|Fixing R and Solving X

When $R$ is fixed, it becomes
$$\min {X} \frac{1}{2}\left|V R^{T}-X\right|{F}^{2}+\lambda \sum_{i}\left|X_{i}\right|_{1}, \text { s.t. } \forall i,\left|X_{i}\right|_{2}=1 .$$
There are $r$ independent subproblems, one for each column: $\min {X{i}} 1 / 2\left|Z_{i}-X_{i}\right|_{2}^{2}+$ $\lambda\left|X_{i}\right|_{1}$, s.t. $\left|X_{i}\right|_{2}=1$, where $Z=V R^{T}$. It is equivalent to $\max {X{i}} Z_{i}^{T} X_{i}-\lambda$ $\left|X_{i}\right|_{1}$, s.t. $\left|X_{i}\right|_{2}=1$. The solution is $X_{i}^{*}=S_{\lambda}\left(Z_{i}\right) /\left|S_{\lambda}\left(Z_{i}\right)\right|_{2}[13]$. $S_{\lambda}(\cdot)$ is the entry-wise soft thresholding, defined in Table 2 . This is the truncation type T- $\ell_{1}$, i.e., soft thresholding.

The solution has the following physical explanations. $Z$ contains the rotated PCA loadings, so it is orthonormal. $X$ is obtained via truncating small entries of $Z$. On one hand, because of the unit length of each column in $Z$, a single threshold $0 \leq \lambda<1$ is feasible to make the sparsity distribute evenly among the columns in $X$; otherwise we have to apply different thresholds for different columns, which would be difficult to determine. On the other hand, because of the orthogonality of $Z$ and small truncations, $X$ is still possible to be nearly orthogonal. These are the most distinctive features of SPCArt, which enable simple analysis and parameter setting.

The algorithm of SPCArt is presented in Algorithm 1, where the truncation in line 7 can be any type, including $\mathrm{T}-\ell_{1}$ and the others that will be introduced in next section.

The computational complexity of SPCArt is shown in Table 1. Except for the computational cost of PCA loadings, SPCArt scales linearly with the data dimension. When the number of loadings is not too large, it is efficient.

## 统计代写|主成分分析代写Principal Component Analysis代考|Sparse PCA via Rotation and Truncation

SPCArt 受到 Eckart-Young 定理 [6] 的启发。该定理考虑了通过两个低秩矩阵的乘积来逼近矩阵的问题。

Σ11≥Σ22≥⋯≥Σ米米， 和在∈Rp×米. 秩-r(r≤米)的近似值一个是解决以下问题：

X=在1:r,是=一个X在哪里在1:r是第一个r列在. 或者，解决方案可以表示为

## 统计代写|主成分分析代写Principal Component Analysis代考|Fixing R and Solving X

SPCArt 的算法在算法 1 中给出，其中第 7 行的截断可以是任何类型，包括吨−ℓ1以及将在下一节中介绍的其他内容。

SPCArt 的计算复杂度如表 1 所示。除了 PCA 加载的计算成本外，SPCArt 与数据维度呈线性关系。当加载的数量不太大时，它是有效的。

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