Efficient Exact Gaussian Process Inference via Incremental Block-Matrix Identity Updates

Exact gaussian process (GP) inference scales cubically in data size, posing a severe bottleneck for applications that require repeated GP kernel inversions, such as time series models like gaussian process factor analysis (GPFA), where data size can vary across trials. To address this, we introduce an algorithmic variant of GPFA that shares computations across trials of different lengths. It does so by leveraging block-matrix identities to incrementally reuse inverses and determinants from smaller GP kernel matrices to accelerate computations on larger ones without sacrificing numerical precision. By casting successive trial-length extensions as Schur-complement updates, our method reduces the cost of each new inversion to depend only on the size difference among matrices. We demonstrate that this approach yields more than two times speed-ups over direct and persymmetric inversion baselines on both synthetic data sets and data from neural recordings. In an additional comparison to sparse inducing-point GPFA, sparse inference reduced inference time for small inducing sets at the expense of worse model fits, while optimizing inducing locations was slower than our exact GPFA variant when the optimization cost was included. We also provide a scikit-learn-compatible Python GPFA package, BlockInverseGPFA, and introduce a variance-explained metric for model evaluation.