proof of Theorem 1

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Let [math]\textbf{u}_k[/math] and [math]\textbf{v}_k[/math] denote column k of [math]\textbf{U}[/math] and [math]\textbf{V}[/math] respectively, We prove the theorem by expanding out the squared Frobenius norm and rearranging terms:

[math] \begin{align} \| \textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T \|^2_F & = tr((\textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T)^T(\textbf{X} - \textbf{U}\textbf{D}\textbf{V}^T)) \\ & = -2tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{X}) + tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{U}\textbf{D}\textbf{V}^T) + \|\textbf{X}\|^2_F \\ & = \sum_{k=1}^K d_k^2 - 2tr(\textbf{V}\textbf{D}\textbf{U}^T\textbf{X}) + \|\textbf{X}\|^2_F \\ & = \sum_{k=1}^K d_k^2 - \sum_{k=1}^K d_k\textbf{u}_k^T\textbf{X}\textbf{v}_k + \|\textbf{X}\|^2_F \end{align} [/math]

The above proof is the proof of Theorem 2.1 in Witten, Tibshirani and Hastie, 2009 <ref name="WTH2009"> </ref>