Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior information about the column and row spaces of the ground-truth low-rank matrix. In this paper, we exploit this prior information by proposing a weighted optimization problem where its objective function promotes both rank and prior subspace information. Using the recent results in conic integral geometry, we obtain the unique optimal weights that minimize the required number of measurements. As simulation results confirm, the proposed convex program with optimal weights requires substantially fewer measurements than the regular nuclear norm minimization.
In @cite_7 , a non-uniform sampling scheme is considered in which the authors propose a generalized nuclear norm which penalizes the directions in the vector space of @math non-uniformly; namely, allocates larger weights to certain directions than others.
@cite_7
2094394132
In geometric computer vision, the structure from motion (SfM) problem can be formulated as a optimization problem with a rank constraint. It is well known that the trace norm of a matrix can act as a convex proxy for a low rank constraint. Hence, in recent work [7], the trace-norm relaxation has been applied to the SfM problem. However, SfM problems often exhibit a certain structure, for example a smooth camera path. Unfortunately, the trace norm relaxation can not make use of this additional structure. This observation motivates the main contribution of this paper. We present the so-called generalized trace norm which allows to encode prior knowledge about a specific problem into a convex regularization term which enforces a low rank solution while at the same time taking the problem structure into account. While deriving the generalized trace norm and stating its different formulations, we draw interesting connections to other fields, most importantly to the field of compressive sensing. Even though the generalized trace norm is a very general concept with a wide area of potential applications we are ultimately interested in applying it to SfM problems. Therefore, we also present an efficient algorithm to optimize the resulting generalized trace norm regularized optimization problems. Results show that the generalized trace norm indeed achieves its goals in providing a problem-dependent regularization.