Heuristic idea that works with geodesics: For every set of triples, x1,x2,x3, and value of p, we can view the pairwise distances as the sides of a triangle with angles a(x2,x3), a(x1,x3), and a(x2,x3) and say the error(x1,x2,x3,p) is some norm of the difference between the p-distance value for the longest side and the sum of the p-distance value for the other sides times the cosine of the relevant angle. For any set of many triples, we can say which p minimizes the error norm over that set. This can easily be sampled over a weighted selection of the distances we expect to care about.

Conceptual data from humans that is described as “similarity” can be toxically non-geometric.
For instance sim(a,c) < (sim(a,b)+sim(b,c))/ ConstantC_ArbitrarilyLargerThan_2

Effects like that can arise as a result of focusing. Any naive data set can be pre-processed for force it into the form of some sort of distance metric. But we presume that data from closely related relationships are likely to be more meaningful in conceptual spaces than data from far away relationships, while naive geometric forcing might produce the opposite weighting.

Our pre-processing algorithm for geodesics should try to force some distance metric which emphasizes preserving the rank ordering of near neighbor values (similarity or distance).

On the general topic of non-parametric regression – it is rational and not a mistake to view confidence intervals at some locations to be much wider than others, at an intuitive level. But the existence of confidence levels pre-supposes the acceptance of some assumptions that are difficult to verify – they are not generally part of the data – e.g. we didn’t just suddenly experience a historically discontinous shift. So regional confidence only exists with some kind of smoothness assumption – based on what, from where???

All of the data analyses should be thought of like:

a) for this analytical step, we’ll take x,y,z as true & work with the consequences.

b) somewhere else, at least implicitly, we confirm why we support x,y,z in that setup & view it as relevant to this other one there: