Consider the standard normal distribution $Z(0,1)$. Assume $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots, y_n)$ are two samples of size $n$ from $Z$. What are ``sensible'' ways (if any) to interpolate between $x$ and $y$? Of course one could use linear interpolation, $u_\alpha =(1-\alpha)x+\alpha y$, but I am looking for interpolants that are close to what might be samples of $Z$, while also creating a path of short length from $x$ to $y$.
Edit:
Here is a possibility:
Suppose the interpolating points are $P=\{p^1,\cdots, p^k\}$. Each can be put to a normality test to see how close to a sample of size $n$ of $Z$ they are. We get a set of abnormality scores $ a_1,\cdots, a_k$, with a total $A=a_1+\cdots+a_k$.
We also calculate the length of the path $L=d(x,p^1)+d(p^1,p^2)+\cdots+d(p^k,y)$. And perhaps a measure $R$ of the roughness of the path. Now we have a total $T=A+L+R$ (perhaps after a normalization to make the components comparable). The set of $P$'s that minimizes $T$ will be the interpolating set of points.
Is there a practical way of approximating $P$?