In this section, let us direct our attention to the selection criteria for the initial choice for

*U*, where

*U*∈

*R*^{
Mxk
}. We note that for the following cases where (i) rank

*k* = 1 approximation and (ii) rank 1 <

*k ≤ r*, then

*r =* rank (

*X*). In order to take the global data into account, a good choice of initial value of

*U* for a rank

*k* = 1 (i.e.,

*U*∈

*R*^{Mx1}) approximation may be obtained as follows. First, we compute the

*l*_{1} norm of each column vector in

*X*, and denoted this norm by

${x}_{1}^{c},{x}_{2}^{c},\dots ,{x}_{n}^{c}$. Next compute the

*l*_{1}norm of each row vector in

*X*, and denoted this norm by

${x}_{1}^{r},{x}_{2}^{r},\dots ,{x}_{m}^{r}$. Now we find the maximum value in

$\left\{{x}_{1}^{c},{x}_{2}^{c},{x}_{n}^{c},{x}_{1}^{r},{x}_{2}^{r},\dots ,{x}_{m}^{r}\right\}$. If the maximum corresponds to a column norm, say from column j, then chose that column (i.e.,

*U* =

*X* (:,

*j* )) as the initial choice for

*U*. If the maximum corresponds to a row norm, say row I, then we start with the transposed form of the criterion in (11) and we chose that row (i.e.,

*V*^{T} =

*X* (

*i*,:)) as the initial choice for

*V*^{
T
}. We can also extend the previous concept to find the initial choice for

*U* for the rank

*k* = 2. Essentially, we apply the rank one approximation twice in succession. Therefore our objective function can be expressed as:

$\begin{array}{ll}\text{min}\left|\right|{E}_{2}{\left|\right|}_{1}& =\underset{{u}_{1},{u}_{2},{v}_{1},{v}_{2}}{\underset{\u23df}{\text{min}}}\left|\right|X-\left[{u}_{1}\phantom{\rule{0.25em}{0ex}}{u}_{2}\right]{\left[{v}_{1\phantom{\rule{0.25em}{0ex}}}{v}_{2}\right]}^{T}{\left|\right|}_{1}\\ =\underset{U,V}{\underset{\u23df}{\mathit{\text{min}}}}\left|\right|X-U{V}^{T}{\left|\right|}_{1}\end{array}$

(14)