Literature¶
Paper I¶
Linear Operators and Stochastic Partial Differential Equations in GPR - Simo Särkkä - PDF
Expresses derivatives of GPs as operators
He looks at ths special case where we have a GP with a mean function zero and a covariance matrix K defined as: $$ \mathbb{E}[f(\mathbf{x})f^\top(\mathbf{x'})] = K_{ff}(\mathbf{x,x'}) $$ So in GP terminology: $$ f(\mathbf(x)) \sim \mathcal{GP}(\mathbf{0}, K_{ff}(\mathbf{x,x'})) $$ We use the rulse for linear transformations of GPs to obtain the different transformations of the kernel matrix.
Let's define the notation for the derivative of a kernel matrix. Let g(\cdot) be the derivative operator on a function f(\cdot). So: $$ g(\mathbf{x}) = \mathcal{L}_x f(\mathbf{x}) $$
So now, we want to define the cross operators between the derivative g(\cdot) and the function f(\cdot).
Example: He draws a distinction between the two operators with an example of how this works in practice. So let's take the linear operator \mathcal{L}_{x}=(1, \frac{\partial}{\partial x}). This operator:
- acts on a scalar GP f(x)
- a scalar input x
- a covariance function k_{ff}(x,x')
- outputs a scalar value y
We can get the following transformations: $$ \begin{aligned} K_{gf}(\mathbf{x,x'}) &= \mathcal{L}x f(\mathbf{x}) f(\mathbf{x}) = \mathcal{L}_xK(\mathbf{x,x'}) \ K_{fg}(\mathbf{x,x'}) &= f(\mathbf{x}) f(\mathbf{x'}) \mathcal{L}{x'} = K(\mathbf{x,x'})\mathcal{L}{x'} \ K(\mathbf{x,x'}) &= \mathcal{L}x f(\mathbf{x}) f(\mathbf{x'}) \mathcal{L} = \mathcal{L}xK(\mathbf{x,x'})\mathcal{L}_{x'}^\top \ \end{aligned} $$
Example: The Cross-Covariance term K_{fg}(\mathbf{x,x'})
We can calculate the cross-covariance term K_{fg}(\mathbf{x,x}). We apply the following operation