Title: Rank Minimization over Affine Sets

			 Ben Recht
   Center for the Mathematics of Information
       California Institute of Technology

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 NP-hard.

In this two part talk, I will 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 a particular
matrix norm over the given affine space. The techniques used in the
analysis have strong parallels in the compressed sensing framework,
and I will discuss how affine rank minimization generalizes this
pre-existing concept and outline a dictionary relating concepts from
cardinality minimization to those of rank minimization.  In the
process of this discussion, I will review of many useful properties of
matrices and matrix norms necessary for the main results.