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\begin{document}
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\title[ Principal Component Pursuit Problem -- \insertframenumber/\inserttotalframenumber]
       {\large MA 7121: Topics in Scientific Computing I\\ Principal Component Pursuit Problem}
\author[$\copyright$ Suh-Yuh Yang (楊肅煜), Math. Dept., NCU, Taiwan]
       {
       \resizebox{2.0 cm}{1.5 cm}{\includegraphics{nculogo.pdf}}
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       \small\textcolor{purple}{Suh-Yuh Yang (楊肅煜)}
       }
\institute[NCU, Taiwan]
       {\small Department of Mathematics, National Central University \\
        Jhongli District, Taoyuan City 320317, Taiwan 
       }
\date{\small\today}
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\indent\textcolor{blue}{\bf Sparse plus low rank matrix decomposition} \\
\textcolor{orange}{\rule{110 mm}{1 mm}}
\vskip 0.15 cm

\small 
Let ${\mbf M}\in \mathbb{R}^{m\times n}$ be a given grayscale image. 
Suppose that ${\mbf M}$ is the superposition of a low-rank component ${\mbf L}$ 
and a sparse component ${\mbf S}$,   
\vskip -0.2 cm
\[
	\textcolor{purple}{{\mbf M} = {\mbf L} + {\mbf S}.}  
\]
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We are interested in finding the low-rank image ${\mbf L}$, 
which has high repeatability along horizontal or vertical directions. 
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\begin{center}
	\resizebox{6.75 cm}{1.5 cm}{\includegraphics{sparse_low_rank_gray.jpg}} \\
	\textcolor{orange}{(schematic diagram)} 
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{\it The sparse plus low rank decomposition problem} can be formulated as 
the constrained minimization problem:   
\textcolor{purple}{
\[
	\underset{{\mbf L}, {\mbf S}}{\min}\bigl({\rm rank}({\mbf L}) + \lambda\|{\mbf S}\|_0\bigr) 
	\quad\text{subject to}\quad {\mbf M} = {\mbf L} + {\mbf S}, 
\]}\vskip -0.5 cm where $\lambda>0$ is a tuning parameter and 
$\|{\mbf S}\|_0 $ denotes the number of non-zero entries in ${\mbf S}$. 
{\it The problem is not convex.} 

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\indent\textcolor{blue}{\bf The principal component pursuit problem} \\
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\small 
We approximate the sparse plus low rank decomposition problem 
by the following {\it principal component pursuit (PCP) problem:}  
\textcolor{purple}{
\[
	\underset{\mbf{L}, \mbf{S}}{\min}\bigl(\|{\mbf L}\|_* + \lambda \|{\mbf S}\|_1\bigr) 
	\quad\text{subject to}\quad {\mbf M} = {\mbf L} + {\mbf S}, 
\]}
\vskip -0.25 cm 
where $\Vert{\mbf L}\Vert_*$ is the nuclear (Ky Fan/樊(土畿)) norm of $\mbf{L}$ defined as 
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\textcolor{purple}{
\[
	\Vert{\mbf L}\Vert_* := \sum_{i = 1}^{r}\sigma_i, 
\]}
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and $r\in\mathbb{N}^+$ is the rank of ${\mbf L}$ and $\sigma_i$ are the singular values of ${\mbf L}$, 
and $\Vert{\mbf S}\Vert_1$  denotes the $\ell^1$-norm of ${\mbf S}$ 
{\it (seen as a long vector in $\mathbb{R}^{mn}$),} 
\textcolor{purple}{ 
\[
	\Vert{\mbf S}\Vert_1 := \sum_{i,j} |{\mbf S}_{ij}|. 
\]}
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\textcolor{blue}{\it $\star$ How about the existence of solution for the PCP problem?} 

\ \ \ (cf. Cand\`es-Li-Ma-Wright, J. ACM, 2011)
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\indent\textcolor{blue}{\bf Background recovering using the ALM method} \\
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\small 
\begin{center}
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_hotel.jpg}}\quad
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_hotel_result.jpg}}\quad   
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_crosswalk.jpg}}\quad
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_crosswalk_result.jpg}} \\
\textcolor{orange}{$(\lambda, \mu) = (0.0007, 0.5)$ \qquad\qquad $(\lambda, \mu) = (0.006, 5)$}  
\medskip 

\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_colorbricks.jpg}}\quad
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_colorbricks_result.jpg}}\quad   
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_building.jpg}}\quad
\resizebox{2.25 cm}{2.0 cm}{\includegraphics{ALM_building_result.jpg}} \\
\textcolor{orange}{$(\lambda, \mu) = (0.007525, 0.04)$ \qquad\qquad $(\lambda, \mu) = (0.0025, 1.5)$}  
 
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\indent\textcolor{blue}{\bf References} \\
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\small
\begin{enumerate}	
\item E. J. Cand\`es, X. Li, Y. Ma, and J. Wright, 	 
	Robust principal component analysis? 
	\textcolor{purple}{\it Journal of the ACM,} 58 (2011), Article 11. 

\item X. Ren and Z. Lin, 
	Linearized alternating direction method with adaptive penalty and warm starts 
	for fast solving transform invariant low-rank textures, 
	\textcolor{purple}{\it International Journal of Computer Vision,} 104, (2013), pp.1-14. 
\end{enumerate}

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