  Low-rank matrix approximation is one of the central concepts in machine
learning, with applications in dimension reduction, de-noising, multivariate
statistical methodology, and many more. A recent extension to LRMA is called
low-rank matrix completion (LRMC). It solves the LRMA problem when some
observations are missing and is especially useful for recommender systems. In
this paper, we consider an element-wise weighted generalization of LRMA. The
resulting weighted low-rank matrix approximation technique therefore covers
LRMC as a special case with binary weights. WLRMA has many applications. For
example, it is an essential component of GLM optimization algorithms, where an
exponential family is used to model the entries of a matrix, and the matrix of
natural parameters admits a low-rank structure. We propose an algorithm for
solving the weighted problem, as well as two acceleration techniques. Further,
we develop a non-SVD modification of the proposed algorithm that is able to
handle extremely high-dimensional data. We compare the performance of all the
methods on a small simulation example as well as a real-data application.
