Sequential construction and dimension reduction of Gaussian processes\n under constraints

Accounting for inequality constraints, such as boundedness, monotonicity or\nconvexity, is challenging when modeling costly-to-evaluate black box functions.\nIn this regard, finite-dimensional Gaussian process (GP) regression models\nbring a valuable solution, as they guarantee that the inequality constraints\nare satisfied everywhere. Nevertheless, these models are currently restricted\nto small dimensional situations (up to dimension 5). Addressing this issue, we\nintroduce the MaxMod algorithm that sequentially inserts one-dimensional knots\nor adds active variables, thereby performing at the same time dimension\nreduction and efficient knot allocation. We prove the convergence of this\nalgorithm. In intermediary steps of the proof, we propose the notion of\nmulti-affine extension and study its properties. We also prove the convergence\nof finite-dimensional GPs, when the knots are not dense in the input space,\nextending the recent literature. With simulated and real data, we demonstrate\nthat the MaxMod algorithm remains efficient in higher dimension (at least in\ndimension 20), and needs fewer knots than other constrained GP models from the\nstate-of-the-art, to reach a given approximation error.\n