  We present a Bayesian approach to identify optimal transformations that map
model input points to low dimensional latent variables. The "projection"
mapping consists of an orthonormal matrix that is considered a priori unknown
and needs to be inferred jointly with the GP parameters, conditioned on the
available training data. The proposed Bayesian inference scheme relies on a
two-step iterative algorithm that samples from the marginal posteriors of the
GP parameters and the projection matrix respectively, both using Markov Chain
Monte Carlo (MCMC) sampling. In order to take into account the orthogonality
constraints imposed on the orthonormal projection matrix, a Geodesic Monte
Carlo sampling algorithm is employed, that is suitable for exploiting
probability measures on manifolds. We extend the proposed framework to
multi-fidelity models using GPs including the scenarios of training multiple
outputs together. We validate our framework on three synthetic problems with a
known lower-dimensional subspace. The benefits of our proposed framework, are
illustrated on the computationally challenging three-dimensional aerodynamic
optimization of a last-stage blade for an industrial gas turbine, where we
study the effect of an 85-dimensional airfoil shape parameterization on two
output quantities of interest, specifically on the aerodynamic efficiency and
the degree of reaction.
