View source for deflation Method for Penalized Matrix Decomposition Sparse PCA

In the penalized matrix decomposition proposed by Witten, Tibshirani and Hastie<ref name="WTH2009">Daniela M. Witten, Robert Tibshirani, and Trevor Hastie. (2009) "A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis". ''Biostatistics'', 10(3):515–534.</ref>, after the penalized vectors <math>\,\textbf{v}_k</math> and <math>\,\textbf{u}_k</math> and the constant <math>\,d_k</math> have been determined, the data matrix <math>\,\textbf{X}^k</math> is deflated using the following formula:
<center><math>\textbf{X}^{k+1} = \textbf{X}^k - d_k\textbf{u}_k\textbf{v}_k^T</math></center>
The penalized matrix decomposition can be used to obtain a version of sparse PCA.  In this case,
<center><math>\,\textbf{u}_k = \frac{\textbf{X}^k\textbf{v}_k}{\|\textbf{X}^k\textbf{v}_k\|_2}</math></center>
and 
<center><math>\,\textbf{d}_k = \textbf{u}^T_k\textbf{X}^k\textbf{v}_k = \frac{\textbf{v}^T_k\textbf{X}^{kT}\textbf{X}^k\textbf{v}_k}{\|\textbf{X}^k\textbf{v}_k\|_2} = \frac{{\|\textbf{X}^k\textbf{v}_k\|^2_2}}{\|\textbf{X}^k\textbf{v}_k\|_2} = {\|\textbf{X}^k\textbf{v}_k\|_2}.</math></center>
Then,
<center><math>\textbf{X}^{k+1} = \textbf{X}^k - {\|\textbf{X}^k\textbf{v}_k\|_2}\frac{\textbf{X}^k\textbf{v}_k\textbf{v}_k^T}{\|\textbf{X}^k\textbf{v}_k\|_2}  = \textbf{X}^k - \textbf{X}^k\textbf{v}_k\textbf{v}_k^T  = \textbf{X}^k(I - \textbf{v}_k\textbf{v}_k^T).</math></center>
So if <math>\| \textbf{v}_k \|_2 = 1</math> then the deflation method begin used for the penalized sparse PCA is the projection deflation method.  

==References==
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