A Counterexample for the Validity of Using Nuclear Norm as a Convex Surrogate of Rank

Rank minimization has attracted a lot of attention in recent years due to its robustness in data recovery. To overcome the computational difficulty, rank is often replaced with its convex surrogate, nuclear norm. For several rank minimization problems (RMPs), such a replacement has been theoretically proven to be valid, i.e., the solution to nuclear norm minimization problem (NNMP) is also the solution to RMP. 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 RMPs the validity of replacing rank with nuclear norm may still break down. So a practitioner should not take for granted to replace rank with nuclear norm. As a by-product, we find that the solution to the NNMP formulation of LatLRR is non-unique. Hence the results of LatLRR reported in the literature may be questionable.