Schur Complement for Gaussian Processes
From the book
Chapter 10: Matrix Block Partitioning. In the chapter mind map this icon labels GP App: Schur Complement in Gaussian Processes. The discussion below is excerpted and lightly edited from § Schur Complement in Gaussian Processes in Mathematics for AI and Machine Learning.
In Gaussian process regression, the joint distribution of training points $\mathbf f$ and test point $f_*$ is
where $K \in \mathbb{R}^{N \times N}$ is the covariance matrix of training points, $\mathbf k_* \in \mathbb{R}^{N}$ is the covariance between training and test points, and $k_{**} \in \mathbb{R}$ is the variance of the test point.
- Mean: $\mathbf k_*^\top K^{-1} \mathbf f$
- Variance: $k_{**} - \mathbf k_^\top K^{-1} \mathbf k_$ (this is the Schur complement)
3. Interpret the Schur complement $k_{**} - \mathbf k_^\top K^{-1} \mathbf k_$ as the reduction in uncertainty about $f_*$ given observations $\mathbf f$.
What this drawing shows
What you see. Represents block Gaussian conditioning where a Schur complement gives posterior covariance or marginal uncertainty.
In the mind map. Chapter 10 — GP App: Schur Complement in Gaussian Processes. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 10: Matrix Block Partitioning — see the mind-map node GP App: Schur Complement in Gaussian Processes.