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For non-negative matrix factorization with Frobenius norm:

$$\min\limits_{U\in\mathbb{R}_+^{m\times r}, V\in\mathbb{R}_+^{r\times n}}||A-UV||_F^2, A\in\mathbb{R}_+^{m\times n}$$

$r=1$ is a very special case because the optimal solution is given by SVD and there is no need to perform successive minimization by $V$ with fixed $U$ and vice versa.

Is there anything special for $r=1$ when minimization is performed with respect to some other norm, for example, $l_1$, Kullback-Leibler divergence or any other divergence, or should it just be done by the algorithms for the general case?

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    $\begingroup$ l_1 minimisation falls into the framework of linear programing, this can be done by simplex or interior point methods (if the dimension is not too big (i.e. <10^5 or 10^6) $\endgroup$
    – robin girard
    Commented Nov 7, 2012 at 10:29

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