  This paper is concerned with regularized extensions of hierarchical
non-stationary temporal Gaussian processes (NSGPs) in which the parameters
(e.g., length-scale) are modeled as GPs. In particular, we consider two
commonly used NSGP constructions which are based on explicitly constructed
non-stationary covariance functions and stochastic differential equations,
respectively. We extend these NSGPs by including $L^1$-regularization on the
processes in order to induce sparseness. To solve the resulting regularized
NSGP (R-NSGP) regression problem we develop a method based on the alternating
direction method of multipliers (ADMM) and we also analyze its convergence
properties theoretically. We also evaluate the performance of the proposed
methods in simulated and real-world datasets.
