  We introduce a flexible framework for making inferences about general linear
forms of a large matrix based on noisy observations of a subset of its entries.
In particular, under mild regularity conditions, we develop a universal
procedure to construct asymptotically normal estimators of its linear forms
through double-sample debiasing and low-rank projection whenever an entry-wise
consistent estimator of the matrix is available. These estimators allow us to
subsequently construct confidence intervals for and test hypotheses about the
linear forms. Our proposal was motivated by a careful perturbation analysis of
the empirical singular spaces under the noisy matrix completion model which
might be of independent interest. The practical merits of our proposed
inference procedure are demonstrated on both simulated and real-world data
examples.
