  We consider a variant of matrix completion where entries are revealed in a
biased manner, adopting a model akin to that introduced by Ma and Chen. Instead
of treating this observation bias as a disadvantage, as is typically the case,
the goal is to exploit the shared information between the bias and the outcome
of interest to improve predictions. Towards this, we consider a natural model
where the observation pattern and outcome of interest are driven by the same
set of underlying latent or unobserved factors. This leads to a two stage
matrix completion algorithm: first, recover (distances between) the latent
factors by utilizing matrix completion for the fully observed noisy binary
matrix corresponding to the observation pattern; second, utilize the recovered
latent factors as features and sparsely observed noisy outcomes as labels to
perform non-parametric supervised learning. The finite-sample error rates
analysis suggests that, ignoring logarithmic factors, this approach is
competitive with the corresponding supervised learning parametric rates. This
implies the two-stage method has performance that is comparable to having
access to the unobserved latent factors through exploiting the shared
information between the bias and outcomes. Through empirical evaluation using a
real-world dataset, we find that with this two-stage algorithm, the estimates
have 30x smaller mean squared error compared to traditional matrix completion
methods, suggesting the utility of the model and the method proposed in this
work.
