【机器学习】PCA推导
PCA(主成分分析)是一种线性降维方式,它基于最大重构性对数据进行降维。
记数据集为Xm×nX_{m\times n}Xm×n,其中mmm为样本数量,nnn为原始特征空间的特征维数。设线性变换矩阵为Wn×dW_{n\times d}Wn×d,其中列向量之间彼此正交,且模长为1。则降维后的样本Z=XWZ=XWZ=XW, 重构损失可以表示为 loss=∣∣X−XWWT∣∣F2loss = || X-XWW^T||_{F}^{2}loss=∣∣X−XWWT∣∣F2.
argminθ∣∣X−XWWT∣∣F2=argminθtr[(X−XWWT)(X−XWWT)T]=argminθtr[(X−XWWT)(XT−WWTXT)]=argminθtr(XXT−2XWWTXT−XWWTWWTXT)=argminθtr(XXT−2XWWTXT+XWWTXT)(因为WTW=Ed×d))=argminθtr(XXT−XWWTXT)=argminθtr(−XWWTXT)=argmaxθtr(XWWTXT)=argmaxθtr[WT(XTX)W]\begin{aligned} &\mathop{\arg\min}\limits_{\theta} || X-XWW^T||_{F}^{2}\\ =&\mathop{\arg\min}\limits_{\theta} tr[(X-XWW^T)(X-XWW^T)^T]\\ =&\mathop{\arg\min}\limits_{\theta} tr[(X-XWW^T)(X^T-WW^TX^T)]\\ =&\mathop{\arg\min}\limits_{\theta} tr(XX^T-2XWW^TX^T-XWW^TWW^TX^T)\\ =&\mathop{\arg\min}\limits_{\theta} tr(XX^T-2XWW^TX^T+XWW^TX^T)(因为W^TW=E_{d\times d}))\\ =&\mathop{\arg\min}\limits_{\theta} tr(XX^T-XWW^TX^T)\\ =&\mathop{\arg\min}\limits_{\theta} tr(-XWW^TX^T)\\ =&\mathop{\arg\max}\limits_{\theta} tr(XWW^TX^T)\\ =&\mathop{\arg\max}\limits_{\theta} tr[W^T(X^TX)W]\\ \end{aligned}========θargmin∣∣X−XWW