  A recent literature considers causal inference using noisy proxies for
unobserved confounding factors. The proxies are divided into two sets that are
independent conditional on the confounders. One set of proxies are `negative
control treatments' and the other are `negative control outcomes'. Existing
work applies to low-dimensional settings with a fixed number of proxies and
confounders. In this work we consider linear models with many proxy controls
and possibly many confounders. A key insight is that if each group of proxies
is strictly larger than the number of confounding factors, then a matrix of
nuisance parameters has a low-rank structure and a vector of nuisance
parameters has a sparse structure. We can exploit the rank-restriction and
sparsity to reduce the number of free parameters to be estimated. The number of
unobserved confounders is not known a priori but we show that it is identified,
and we apply penalization methods to adapt to this quantity. We provide an
estimator with a closed-form as well as a doubly-robust estimator that must be
evaluated using numerical methods. We provide conditions under which our
doubly-robust estimator is uniformly root-$n$ consistent, asymptotically
centered normal, and our suggested confidence intervals have asymptotically
correct coverage. We provide simulation evidence that our methods achieve
better performance than existing approaches in high dimensions, particularly
when the number of proxies is substantially larger than the number of
confounders.
