We address the multi-task Gaussian process (GP) regression problem with the goal of decomposing input effects on outputs into components shared across or specific to tasks and samples. We propose a family of mixed-effects GPs, including doubly and translated mixed-effects GPs, that performs such a decomposition, while also modeling the complex task relationships. Instead of the tensor product widely used in multi-task GPs, we use the direct sum and Kronecker sum for Cartesian product to combine task and sample covariance functions. With this kernel, the overall input effects on outputs decompose into four components: fixed effects shared across tasks and across samples and random effects specific to each task and to each sample. We describe an efficient stochastic variational inference method for our proposed models that also significantly reduces the cost of inference for the existing mixed-effects GPs. On simulated and real-world data, we demonstrate that our approach provides higher test accuracy and interpretable decomposition.

In many real-world data, complex dependencies are present both among samples and among features. The Kronecker sum or the Cartesian product of two graphs, each modeling dependencies across features and across samples, has been used as an inverse covariance matrix for a matrix-variate Gaussian distribution as an alternative to Kronecker-product inverse covariance matrix due to its more intuitive sparse structure. However, the existing methods for sparse Kronecker-sum inverse covariance estimation are limited in that they do not scale to more than a few hundred features and samples and that unidentifiable parameters pose challenges in estimation. In this paper, we introduce EiGLasso, a highly scalable method for sparse Kronecker-sum inverse covariance estimation, based on Newton's method combined with eigendecomposition of the sample and feature graphs to exploit the Kronecker-sum structure. EiGLasso further reduces computation time by approximating the Hessian matrix, based on the eigendecomposition of the two graphs. EiGLasso achieves quadratic convergence with the exact Hessian and linear convergence with the approximate Hessian. We describe a simple new approach to estimating the unidentifiable parameters that generalizes the existing methods. On simulated and real-world data, we demonstrate that EiGLasso achieves two to three orders-of-magnitude speed-up, compared to the existing methods.

In this paper, we address the problem of jointly estimating dependencies across samples and dependencies across multiple features, where each set of dependencies is modeled as an inverse covariance matrix. In particular, we study a matrix-variate Gaussian distribution with the Kronecker-sum of sample-wise and feature-wise inverse covariances. While this Kronecker-sum model has been studied as an intuitively more appealing convex alternative to the Kronecker-product of two inverse covariance matrices, the existing methods do not scale to large datasets. We introduce a highly-efficient optimization method for estimating the Kronecker-sum structured inverse covariance matrix from matrix-variate data. In addition, we describe an alternative simpler approach for handling the non-identifiability of parameters than the strategies proposed in previous works. Using simulated and real data, we demonstrate our approach leads to one or two orders-of-magnitude speedup of the previous methods.