Minimizing the Maximal Rank

In computer vision, many problems can be formulated as finding a low rank approximation of a given measurement matrix. Ideally, if all elements of the measurement matrix are available, this is easily solved in the L2-norm using factorization. However, in practice this is rarely the case. Lately, this problem has been addressed using different approaches, one is to replace the rank term by the convex nuclear norm, another is to derive the convex envelope of the rank term plus a data term. In the latter case, matrices are divided into sub-matrices and the envelope is computed for each sub-block individually. In this paper a new convex envelope is derived which takes all sub-matrices into account simultaneously. This leads to a simpler formulation, using only one parameter, for applications where one seeks low rank approximations of multiple matrices with the same rank. We show in this paper how our general framework can be used for manifold denoising of several images at once, as well as just denoising one image. We get comparable results to other well-known methods and our framework can also be used for other applications such as linear shape models.