+ \left( \Sigma_{\epsilon} s(\mathbf{x})- \Sigma_{\epsilon} s(\mathbf{y}) \right)^\top \left( \Sigma_{\mathbf{w}_0} s(\mathbf{x})- \Sigma_{\mathbf{w}_0} s(\mathbf{y}) \right) 
\\ &\, \, \, \, \, \, \, + \left( \Sigma_{\mathbf{w}_0} s(\mathbf{x})- \Sigma_{\mathbf{w}_0} s(\mathbf{y}) \right)^\top \left( \Sigma_{\epsilon} s(\mathbf{x})- \Sigma_{\epsilon} y \right)
+ \mathcal{O}(\epsilon^2) \\
&= d_{\mathbf{w}_0}(s(\mathbf{x}),s(\mathbf{y})) + A + A^\top + \mathcal{O}(\epsilon^2),\end{aligned}$$ where $A = \left( \Sigma_{\mathbf{w}_0} s(\mathbf{x})- \Sigma_{\mathbf{w}_0} s(\mathbf{y}) \right)^\top \left( \Sigma_{\epsilon} s(\mathbf{x})- \Sigma_{\epsilon} s(\mathbf{y}) \right) $ which is linear in $\epsilon$. Suppose we order the pseudo-data such that $\mathbf{x}^{j*}$ is the $j$th closest point to the observed data based on $d_{\mathbf{w}_0}$. Provided that $$\label{jump_condition}
d_{\mathbf{w}_0}(s(\mathbf{x}^{n*}),s(\mathbf{y})) - d_{\mathbf{w}_0}(s(\mathbf{x}^{(n+1)*}),s(\mathbf{y})) > A + A^\top + \mathcal{O}(\epsilon^2),$$ then making this perturbation in $\mathbf{w}$ will not change which parameter samples are selected, as the same pseudo data will remain closest to the observed data, $\mathbf{y}$. If the same parameter samples are selected, then the value of $L(\mathbf{w})$ will remain constant as a function of $\mathbf{w}$ under the perturbation $\mathbf{w}=\mathbf{w}_0+\epsilon$.

In cases where does not hold, there will be a jump discontinuity in $L(\mathbf{w})$ as different parameter samples are selected. This will occur finitely many times corresponding to the finite number, $N$, of points in the sample of pseudo data, $\{\mathbf{x}^{i*}\}_{i=1}^N$.

We confirm computationally that $L(\mathbf{w})$ is piecewise constant for the 