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Suppose we have an Euclidiean space $\mathbb{R}^p$ and Gaussian process with covariance function of the form $k(x_1, x_2) = k(\|x_1 - x_2\|), x_1, x_2 \in \mathbb{R}^p$.

I am looking for a covariance function from the family described above such that

  • This function is three times continuously differentiable with respect to covariance function parameters and
  • There exists $h$ such that $k(d) = 0$ if $d > h$ ($h$ can depend on covariance function parameters).

Do you know such a covariance function?

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I unfortunately don't know of a covariance function with compact support that is continuously differentiable an odd (= 3) number of times, but the GPML book describes some piecewise polynomial CFs that are continuously differentiable an even number of times on page 88 in chapter 4. Note, however, that these CFs need to be parameterized over the dimensionality of your inputs. Maybe the references quoted in the book will provide more help.

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  • $\begingroup$ You are right, I have to look to GPML book first. $\endgroup$ – Alexey Zaytsev Dec 3 '15 at 19:43
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The Kernel property, i.e. being positive definite (p.d.), is quite subtle. For example a $k$ such as above will not exist for every $p$.

The most thorough discussion of compactly supported positive definitive functions I know of can be found in Chapter 9 of "Scattered Data Approximation" by HOLGER WENDLAND. He shows (Theorem 9.2) that $k$ which is continuous, non-vanishing and p.d. for all $p$ will have $k(||x_1-x_2||)\ne 0$.

In Theorem 9.10 Wendland shows that piecewise polynomials with compact support on the unit disk which are p.d. for a certain $p$ possess $2n$ derivatives at zero and $2n + m + \lfloor \frac{p}{2}\rfloor$ derivatives at one. Chapter 9.5. contains examples for non-polynomial kernels with compact support.

By the way and only out of curiosity, what kind of application would need $C^3$?

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  • $\begingroup$ It is not an application, but the theorem which requires such assumptions for the covariance function. $\endgroup$ – Alexey Zaytsev Dec 4 '15 at 12:49

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