Posterior contraction rates for constrained deep Gaussian processes in density estimation and classification

We provide posterior contraction rates for constrained deep Gaussian processes in non parametric density estimation and classification. The constraints are in the form of bounds on the values and on the derivatives of the Gaussian processes in the layers of the compositional structure. The contraction rates are first given in a general framework, in terms of a new concentration function that we introduce and that takes the constraints into account. Then, the general framework is applied to integrated Brownian motions, Riemann-Liouville processes, and Matern processes. In each of these examples, we can recover existing rates, both when the compositional structure is dense and sparse.