# 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 $\,\textbf{v}_k$ and $\,\textbf{u}_k$ and the constant $\,d_k$ have been determined, the data matrix $\,\textbf{X}^k$ is deflated using the following formula:

$\textbf{X}^{k+1} = \textbf{X}^k - d_k\textbf{u}_k\textbf{v}_k^T$

The penalized matrix decomposition can be used to obtain a version of sparse PCA. In this case,

$\,\textbf{u}_k = \frac{\textbf{X}^k\textbf{v}_k}{\|\textbf{X}^k\textbf{v}_k\|_2}$

and

$\,\textbf{d}_k = \textbf{u}^T_k\textbf{X}^k\textbf{v}_k = \frac{\textbf{v}^T_k\textbf{X}^{kT}\textbf{X}^k\textbf{v}_k}{\|\textbf{X}^k\textbf{v}_k\|_2} = \frac{{\|\textbf{X}^k\textbf{v}_k\|^2_2}}{\|\textbf{X}^k\textbf{v}_k\|_2} = {\|\textbf{X}^k\textbf{v}_k\|_2}.$

Then,

$\textbf{X}^{k+1} = \textbf{X}^k - {\|\textbf{X}^k\textbf{v}_k\|_2}\frac{\textbf{X}^k\textbf{v}_k\textbf{v}_k^T}{\|\textbf{X}^k\textbf{v}_k\|_2} = \textbf{X}^k - \textbf{X}^k\textbf{v}_k\textbf{v}_k^T = \textbf{X}^k(I - \textbf{v}_k\textbf{v}_k^T).$

So if $\| \textbf{v}_k \|_2 = 1$ then the deflation method begin used for the penalized sparse PCA is the projection deflation method.

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