A Gaussian process gives a probability distribution over functions that pass through the data points. Is there a way to parameterize the Gaussian process to give a probability distribution over closed curves instead of functions? Can the same approach be generalized to give a distribution over surfaces without boundary?


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If you take 2 Gaussian processes, $X(t)$ and $Y(t)$ on the unit interval, and with the property that $X(0)=X(1)$ almost surely (similarly for $Y$), then $(X(t),Y(t))$ will be a random curve in $R^2$.

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  • $\begingroup$ How would you construct a process like this? Would you just condition on the process going through those points? (Would be great to know if there is another way of doing this.) $\endgroup$
    – user27182
    Jan 6, 2018 at 17:02
  • 1
    $\begingroup$ A simple example is the Brownian bridge with covariance kernel $k(x,y)=min(x,y)-xy$. You can also use Green's Functions to build models with boundary constraints. See page 134 of Rasmussen and Williams "Gaussian Processes for Machine Learning". It's available online. $\endgroup$
    – Placidia
    Jan 9, 2018 at 2:02

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