In separate conversations with Hou and with Foreman-Mackey, I found myself discouraging each of them from looking into serious sparse methods for Gaussian processes. Both students are potentially matrix-inversion-limited (or determinant-computation-limited). We could definitely benefit from having matrix respresentations (even if it means approximations) that have sparse inverses (or analytic inverses). But going there is (at this point) a research project in sparse methods, not an astronomy project. So I am going to hold off until we really need it. That said, there is a huge literature now of very promising techniques.
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