Guaranteed MinimumRank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
Abstract
The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization.
 Publication:

arXiv eprints
 Pub Date:
 June 2007
 arXiv:
 arXiv:0706.4138
 Bibcode:
 2007arXiv0706.4138R
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Statistics Theory;
 90C25;
 90C59;
 15A52
 EPrint:
 SIAM Review, Volume 52, Issue 3, pp. 471501 (2010)