A recent literature considers causal inference using two vectors of noisy proxies for unobserved confounding factors. In this paper we consider linear models in which the vectors of proxies are potentially high-dimensional and there may be many unobserved 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 satisfies a rank restriction. We can exploit the rank restriction 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 develop doubly-robust estimation and inference methods. We examine the asymptotic properties of these techniques and provide simulation evidence that they are effective.