LDP for the covariance process in fully connected Gaussian neural networks

In this work, we study large deviation properties of the covariance process in fully connected Gaussian deep neural networks. More precisely, we establish a large deviation principle (LDP) for the covariance process in a functional framework, viewing it as a process in the space of continuous functions. As key applications of our main results, we obtain posterior LDPs under Gaussian likelihood in both the infinite-width and mean-field regimes. The proof is based on an LDP for the covariance process as a Markov process valued in the space of non-negative, symmetric trace-class operators equipped with the trace norm.