Snoek et al, have a recent paper "Input Warping for Bayesian Optimization of Non-Stationary Functions" (http://arxiv.org/abs/1402.0929) which mentions "stationary functions".

I understand what a stationary kernel or process is, but I can't find a definition of a stationary function. I kind of suspect there is no such thing, and they are using the term informally to mean something along the lines of "a function such that any epsilon-ball of the function has non-zero probability according to some stationary process". But the top example in figure 1 (sine wave) doesn't seem to fit this description to me, since the distance between the image of two points themselves a fixed distance apart changes dramatically as you move the points.

I guess you can maybe define a process whose value is fixed at the peaks and troughs of the sine wave, and then this might be a stationary function under that process?

Or am I totally wrong about the definition?



I emailed Kevin Swersky, one of the authors, and he confirmed that they were not referencing a technical definition (that he was aware of).

He proposed essentially the same definition that I did (without the necessary technicality of epsilon balls).

I've since realized that my objection to the sine wave can (I believe) be overcome. If the process is conditional, and the peaks and troughs of the wave is fixed, then this is a "stationary function" as defined above. Somewhat surprisingly, the epsilon ball of the pre-warped function does not appear to have non-zero probability, even when the peaks and troughs are specified... it will overshoot the more widely spaced points, and they do not end up being the peaks or troughs of functions generated from that conditional process.

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