  Multi-task Gaussian process (MTGP) is a well-known non-parametric Bayesian
model for learning correlated tasks effectively by transferring knowledge
across tasks. But current MTGPs are usually limited to the multi-task scenario
defined in the same input domain, leaving no space for tackling the
heterogeneous case, i.e., the features of input domains vary over tasks. To
this end, this paper presents a novel heterogeneous stochastic variational
linear model of coregionalization (HSVLMC) model for simultaneously learning
the tasks with varied input domains. Particularly, we develop the stochastic
variational framework with Bayesian calibration that (i) takes into account the
effect of dimensionality reduction raised by domain mappings in order to
achieve effective input alignment; and (ii) employs a residual modeling
strategy to leverage the inductive bias brought by prior domain mappings for
better model inference. Finally, the superiority of the proposed model against
existing LMC models has been extensively verified on diverse heterogeneous
multi-task cases and a practical multi-fidelity steam turbine exhaust problem.
