Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

前言

这篇文章用的也是交替算法,不得不说,这个东西太好用了,变来变去怎么都能玩出花来。这篇论文的关键之处,我感觉是对adjusted variance的算法,比较让人信服。

文章概述

Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
\(X是中心化的样本矩阵\)
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
考虑下面的一个最优分解(F-范数)。
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
本文采取的也是一种搜索算法,每次计算一个载荷向量,所以,每次都处理的是rank-1的分解。

也就是:
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

为了稀疏化,必须加上惩罚项,一般的就是1-范数或者0-范数。
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
限定\(\|\widetilde{\mathrm{u}}\|=1\)

固定\(\widetilde{\mathrm{v}}\)

跟之前的论文差不了太多,挺好证的。
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

固定\(\widetilde{\mathrm{u}}\)

上述问题,根据下面的改写:
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
于是就可以分解为多个等价的小问题,每个子问题只需要考虑\(\widetilde{v}_j\)就可以了。
根据不同的截断手法,有不同的解:
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)
注:对于第一种方法,存疑,因为我证出来的不大一样,当然,可能是我哪里搞错了。
还有一点要注意:因为最后我们要求的其实是载荷向量\(\mathrm{v}\),它只需在最后收敛后归一化即可,看下来算法:
Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

Adjusted Variance

Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation(Adjusted Variance)

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