Title: Variational Bayes on manifolds and quantum speed-up
Authors: Minh-Ngoc Tran - University of Sydney (Australia) [presenting]
Abstract: Most of the existing Variational Bayes (VB) algorithms is generally restricted to the case where the variational parameter space is Euclidean, which hinders the potentially broad application of the VB method. As the first contribution, we extend the scope of VB to the case where the variational parameter space is a Riemannian manifold. We develop an efficient manifold-based VB algorithm that exploits both the geometric structure of the constraint parameter space and the information geometry of the manifold of VB approximating probability distributions. The natural gradient is an essential component of efficient VB estimation, but it is prohibitively computationally expensive in high dimensions. As the second contribution, we propose a regression-based stochastic approximation of the natural gradient, a computationally efficient method with provable convergence guarantees under standard assumptions. This regression formulation enables further computational speedup through the use of quantum computation, particularly quantum matrix inversion. We demonstrate that the problem setup fulfils all the conditions required for quantum matrix inversion to deliver computational efficiency.