Currently, multi-output Gaussian process regression models either do not model nonstationarity or are associated with severe computational burdens and storage demands. Nonstationary multi-variate Gaussian process models (NMGP) use a nonstationary covariance function with an input-dependent linear model of coregionalisation to jointly model input-dependent correlation, scale, and smoothness of outputs. Variational sparse approximation relies on inducing points to enable scalable computations. Here, we take the best of both worlds: considering an inducing variable framework on the underlying latent functions in NMGP, we propose a novel model called the collaborative nonstationary Gaussian process model(CNMGP). For CNMGP, we derive computationally tractable variational bounds amenable to doubly stochastic variational inference. Together, this allows us to model data in which outputs do not share a common input set, with a computational complexity that is independent of the size of the inputs and outputs. We illustrate the performance of our method on synthetic data and three real datasets and show that our model generally pro-vides better predictive performance than the state-of-the-art, and also provides estimates of time-varying correlations that differ across outputs.
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