Title: Bayesian optimization on the space of symmetric positive definite matrices Authors:  Federico Pavesi - University of Milan, Bicocca (Italy) [presenting]
Antonio Candelieri - University of Milano-Bicocca (Italy)
Abstract: An extension of Bayesian Optimization for functionals over symmetric positive definite matrices is presented. The proposed method leverages the log-euclidean product, which endows the manifold with a Lie group structure and a bi-invariant metric. This allows the construction of a metric-based positive definite kernel, which is a naive generalization of the well-known squared exponential kernel. Although the new kernel is non-stationary and generally unrelated to the squared exponential kernel in the Euclidean space, its simple form makes it attractive for Bayesian Optimization in non-Euclidean spaces, since its gradient can be computed in closed form. A Gaussian Process based on the new kernel approximates the functional of interest while the acquisition function is optimized via gradient-based methods. Both Expected Improvement and Lower Confidence Bound are considered in the paper. Experiments consider the minimization of simple functionals, like the Wasserstein distance, with results empirically proving that the proposed method is able to identify minima, even if the non-stationarity of the kernel negatively affects the quality of function approximation. Future research may focus on improving kernel design to enhance approximation accuracy while preserving optimization efficiency.