  Rank minimization has attracted a lot of attention due to its robustness in
data recovery. To overcome the computational difficulty, rank is often replaced
with nuclear norm. For several rank minimization problems, such a replacement
has been theoretically proven to be valid, i.e., the solution to nuclear norm
minimization problem is also the solution to rank minimization problem.
Although it is easy to believe that such a replacement may not always be valid,
no concrete example has ever been found. We argue that such a validity checking
cannot be done by numerical computation and show, by analyzing the noiseless
latent low rank representation (LatLRR) model, that even for very simple rank
minimization problems the validity may still break down. As a by-product, we
find that the solution to the nuclear norm minimization formulation of LatLRR
is non-unique. Hence the results of LatLRR reported in the literature may be
questionable.
