Learning Gaussian processes which map functions to functions

1 Universal Kriging

Does universal kriging fit in this notebook? (Menafoglio, Secchi, and Rosa 2013) In this setting, our observations are function-valued and we wish to spatially interpolate them. TBC. Keyword: Hilbert-Kriging.

See Júlio Hoffimann’s Hilbert-Kriging lecture.

2 Hilbert-space valued GPs

TBC

3 Incoming

Thank Rafa for the tips about the following:

• Quang (2022):

  In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences are continuous in the Hilbert-Schmidt norm, which enables us to apply laws of large numbers for Hilbert space-valued random variables. As a consequence of this, we show that, in both settings, the infinite-dimensional divergences can be consistently and efficiently estimated from their finite-dimensional versions, using finite-dimensional Gram matrices/Gaussian measures and finite sample data, with {} sample complexities in all cases. RKHS methodology plays a central role in the theoretical analysis in both settings. The mathematical formulation is illustrated by numerical experiments.

• Feldman–Hájek theorem

  In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures \(\mu\) and \(\nu\) on a locally convex space \(X\) are either equivalent measures or else mutually singular: there is no possibility of an intermediate situation in which, for example, \(\mu\) has a density with respect to \(\nu\) but not vice versa.

  Interesting corollary for our purposes: the reminder that Gaussian measures over infinite-dimensional Hilbert spaces do things that we would not suspect from their finite-dimensional analogues:

  dilating a Gaussian measure on an infinite-dimensional Hilbert space \(X\) (i.e. taking \(C_\nu = s C_\mu\) for some scale factor \(s \geq 0\)) always yields two mutually singular Gaussian measures, except for the trivial dilation with \(s = 1,\) since \((s^2 - 1) I\) is Hilbert–Schmidt only when \(s = 1.\)

4 References