Graphical models have attracted increasing attention in recent years, especially in settings involving high dimensional data. In particular Gaussian graphical models are used to model the conditional dependence structure among multiple Gaussian random variables. As a result of its computational e_ciency the graphical lasso (glasso) has become one of the most popular approaches for fitting high dimensional graphical models. In this article we extend the graphical models concept to model the conditional dependence structure among p random functions. In this setting, not only is p large, but each function is itself a high dimensional object, posing an additional level of statistical and computational complexity. We develop an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. The fglasso criterion can be optimized using an e_cient block coordinate descent algorithm. We establish the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property, i.e. with probability tending to one, the fglasso will correctly identify the true conditional dependence structure. Finally we show that the fglasso significantly outperforms possible competing methods through both simulations and an analysis of a real world EEG data set comparing alcoholic and non-alcoholic patients.
