B1175
Title: Variational Gaussian processes for linear inverse problems
Authors: Thibault Randrianarisoa - Bocconi University (Italy) [presenting]
Botond Szabo - Bocconi University (Italy)
Abstract: Despite their convenience as priors in regression settings, Gaussian processes suffer from a computational burden as they scale as $O(n^3)$, $n$, being the size of the dataset. Consequently, variational approximations have been proposed via $q/leq n$ inducing variables, reducing the cost to $O(nq^2)$. Lower bounds on q were then derived to ensure that the variational and actual posterior enjoy similar theoretical guarantees in the form of contraction rates. These focused, in particular, on two different choices of variables, coming from the eigendecomposition of either the covariance matrix or the covariance operator. These results are extended to inverse problems where the regression function is the image of another one through a compact linear operator. The parameter of interest is observed only indirectly through noisy observations, and this problem can be ill-posed. In front of this last point, Gaussian processes are known to provide some form of regularization, which makes them appropriate. Depending on the level of ill-posedness, the number of inducing variables required to obtain minimax contraction rates ranges from $O(\log n)$ to sublinear in $n$. As examples, the results are applied to the problems of finding the initial condition of the heat equation, the Volterra operator and the Radon transform.
