  We consider the problem of precision matrix estimation where, due to
extraneous confounding of the underlying precision matrix, the data are
independent but not identically distributed. While such confounding occurs in
many scientific problems, our approach is inspired by recent neuroscientific
research suggesting that brain function, as measured using functional magnetic
resonance imagine (fMRI), is susceptible to confounding by physiological noise
such as breathing and subject motion. Following the scientific motivation, we
propose a graphical model, which in turn motivates a joint nonparametric
estimator. We provide theoretical guarantees for the consistency and the
convergence rate of the proposed estimator. In addition, we demonstrate that
the optimization of the proposed estimator can be transformed into a series of
linear programming problems, and thus be efficiently solved in parallel.
Empirical results are presented using simulated and real brain imaging data,
which suggest that our approach improves precision matrix estimation, as
compared to baselines, when confounding is present.
