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]\displaystyle{ \,\textbf{v}_k }[/math] and [math]\displaystyle{ \,\textbf{u}_k }[/math] and the constant [math]\displaystyle{ \,d_k }[/math] have been determined, the data matrix [math]\displaystyle{ \,\textbf{X}^k }[/math] is deflated using the following formula:

[math]\displaystyle{ \textbf{X}^{k+1} = \textbf{X}^k - d_k\textbf{u}_k\textbf{v}_k^T }[/math]

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

[math]\displaystyle{ \,\textbf{u}_k = \frac{\textbf{X}^k\textbf{v}_k}{\|\textbf{X}^k\textbf{v}_k\|_2} }[/math]

and

[math]\displaystyle{ \,\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]

Then,

[math]\displaystyle{ \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]

So if [math]\displaystyle{ \| \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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