  This paper considers the problem of finding a low rank matrix from
observations of linear combinations of its elements. It is well known that if
the problem fulfills a restricted isometry property (RIP), convex relaxations
using the nuclear norm typically work well and come with theoretical
performance guarantees. On the other hand these formulations suffer from a
shrinking bias that can severely degrade the solution in the presence of noise.
  In this theoretical paper we study an alternative non-convex relaxation that
in contrast to the nuclear norm does not penalize the leading singular values
and thereby avoids this bias. We show that despite its non-convexity the
proposed formulation will in many cases have a single local minimizer if a RIP
holds. Our numerical tests show that our approach typically converges to a
better solution than nuclear norm based alternatives even in cases when the RIP
does not hold.
