Structured Matrix Recovery via the Generalized Dantzig Selector
This work addresses structured matrix recovery for applications like recommender systems and computer vision, but it is incremental as it extends existing methods to a broader class of structures.
The paper tackles the problem of recovering structured matrices beyond low-rank cases, such as those characterized by unitarily invariant norms, using the generalized Dantzig selector under sub-Gaussian measurements, and shows that estimation error can be expressed in terms of geometric measures dependent on the matrix structure.
In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of generally structured matrices via the generalized Dantzig selector under generic sub-Gaussian measurements. We show that the estimation error can always be succinctly expressed in terms of a few geometric measures of suitable sets which only depend on the structure of the underlying true matrix. In addition, we derive the general bounds on these geometric measures for structures characterized by unitarily invariant norms, which is a large family covering most matrix norms of practical interest. Examples are provided to illustrate the utility of our theoretical development.