A Fully Bayesian Gradient-Free Supervised Dimension Reduction Method using Gaussian Processes

Modern day engineering problems are ubiquitously characterized by\nsophisticated computer codes that map parameters or inputs to an underlying\nphysical process. In other situations, experimental setups are used to model\nthe physical process in a laboratory, ensuring high precision while being\ncostly in materials and logistics. In both scenarios, only limited amount of\ndata can be generated by querying the expensive information source at a finite\nnumber of inputs or designs. This problem is compounded further in the presence\nof a high-dimensional input space. State-of-the-art parameter space dimension\nreduction methods, such as active subspace, aim to identify a subspace of the\noriginal input space that is sufficient to explain the output response. These\nmethods are restricted by their reliance on gradient evaluations or copious\ndata, making them inadequate to expensive problems without direct access to\ngradients. The proposed methodology is gradient-free and fully Bayesian, as it\nquantifies uncertainty in both the low-dimensional subspace and the surrogate\nmodel parameters. This enables a full quantification of epistemic uncertainty\nand robustness to limited data availability. It is validated on multiple\ndatasets from engineering and science and compared to two other\nstate-of-the-art methods based on four aspects: a) recovery of the active\nsubspace, b) deterministic prediction accuracy, c) probabilistic prediction\naccuracy, and d) training time. The comparison shows that the proposed method\nimproves the active subspace recovery and predictive accuracy, in both the\ndeterministic and probabilistic sense, when only few model observations are\navailable for training, at the cost of increased training time.\n