  Given $n$ i.i.d. observations of a random vector $(X,Z)$, where $X$ is a
high-dimensional vector and $Z$ is a low-dimensional index variable, we study
the problem of estimating the conditional inverse covariance matrix $\Omega(z)
= (E[(X-E[X \mid Z])(X-E[X \mid Z])^T \mid Z=z])^{-1}$ under the assumption
that the set of non-zero elements is small and does not depend on the index
variable. We develop a novel procedure that combines the ideas of the local
constant smoothing and the group Lasso for estimating the conditional inverse
covariance matrix. A proximal iterative smoothing algorithm is used to solve
the corresponding convex optimization problems. We prove that our procedure
recovers the conditional independence assumptions of the distribution $X \mid
Z$ with high probability. This result is established by developing a uniform
deviation bound for the high-dimensional conditional covariance matrix from its
population counterpart, which may be of independent interest. Furthermore, we
develop point-wise confidence intervals for individual elements of the
conditional inverse covariance matrix. We perform extensive simulation studies,
in which we demonstrate that our proposed procedure outperforms sensible
competitors. We illustrate our proposal on a S&P 500 stock price data set.
