Lecture 10/11 (Mar 11)

Thm (Error in the low rank approximation): For any $\nu$ with $0\le \nu \le r=$rank$(A)$, let
$$
A_{\nu} = \sum_{j=1}^{\nu} \sigma_j \mathbf{u}_j \mathbf{v}_j^* \, .
$$
Then
\begin{eqnarray*}
\|A-A_{\nu}\|_2 &=& \mbox{inf}_{B\in \mathbb{R}^{M\times N}, \mbox{rank}(B)\le \nu} \|A-B\|_2 = \sigma_{\nu+1} \, , \\
\|A-A_{\nu}\|_F &=& \mbox{inf}_{B\in \mathbb{R}^{M\times N}, \mbox{rank}(B)\le \nu} \|A-B\|_F = \sqrt{\sigma_{\nu+1}+\cdots+\sigma_r^2} \, .
\end{eqnarray*}

Texture Image Inpainting using low rank matrix completion

Netflix problem: predict user rating for movies.

Reading materials

Lecture Notes: p.59-67