  Low rank approximation is a commonly occurring problem in many computer
vision and machine learning applications. There are two common ways of
optimizing the resulting models. Either the set of matrices with a given rank
can be explicitly parametrized using a bilinear factorization, or low rank can
be implicitly enforced using regularization terms penalizing non-zero singular
values. While the former approach results in differentiable problems that can
be efficiently optimized using local quadratic approximation, the latter is
typically not differentiable (sometimes even discontinuous) and requires first
order subgradient or splitting methods. It is well known that gradient based
methods exhibit slow convergence for ill-conditioned problems.
  In this paper we show how many non-differentiable regularization methods can
be reformulated into smooth objectives using bilinear parameterization. This
allows us to use standard second order methods, such as Levenberg--Marquardt
(LM) and Variable Projection (VarPro), to achieve accurate solutions for
ill-conditioned cases. We show on several real and synthetic experiments that
our second order formulation converges to substantially more accurate solutions
than competing state-of-the-art methods.
