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This paper introduces a factor graph representation for multi-output Gaussian process regression that significantly improves scalability by reducing computational complexity to \(\mathcal{O}(C(DL^2 + L^3))\). By employing a nearest-neighbor chain to order inputs and utilizing latent Mat茅rn processes, the method effectively handles missing observations without requiring covariance-matrix restructuring. Experimental results demonstrate that the factor-graph approach closely tracks the performance of exact kernel-matrix methods at low input dimensions and matches the accuracy of various baselines in forecasting tasks while maintaining linear scaling with data points.
Achieving linear scalability in multi-output Gaussian process regression could revolutionize forecasting in high-dimensional settings where traditional methods falter.
Multi-output Gaussian process regression scales cubically in the number of observations times outputs, and dense kernel-matrix methods need bespoke handling whenever different outputs are observed at different inputs. We express multi-output Gaussian process regression as a Forney-style factor graph in which a nearest-neighbor chain orders a fixed candidate set of $C$ inputs into a one-dimensional sequence. Along this chain, latent Mat\'ern processes evolve through linear-Gaussian transition factors, while the linear model of coregionalization mixes $L$ latent processes into $D$ outputs through a deterministic mixing factor and per-output scalar observation factors. Posterior computation reduces to exact Gaussian message passing on the chain at cost $\mathcal{O}(C(DL^2 + L^3))$ after chain construction, and missing observations omit their local factor without any covariance-matrix restructuring. The formulation therefore scales in the number of data samples and in the rate of missing observations, while remaining best suited to candidate sets in low input dimension.We compare the factor-graph formulation against an exact kernel-matrix baseline, a sparse-variational inducing-point baseline, and a nearest-neighbor baseline on a synthetic input-dimension sweep and on electricity time series forecasting. At low input dimension the factor-graph posterior tracks the exact kernel-matrix posterior closely, and the gap grows gradually as input dimension increases while staying competitive with both approximate baselines. On the electricity time series our factor-graph formulation matches all three baselines in forecast accuracy while scaling linearly in the number of data points, where the exact kernel-matrix method becomes infeasible and the inducing-point baseline remains substantially slower.
