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Perhaps a different culture would always present data on a sphere and would use geodesic distance as the default. I don't know! There should no reason whatsoever to expect an arbitrary encoding and an arbitrary distance to result in an interpolative problem space.
The correct encoding and distance to use is of course the ones that are natural to the data -- that is to say, the latent manifold of the data. Which *is* an interpolative space for many problems (hence why deep learning models are able to generalize).
Saying "I expect the problem to be linearly interpolative in the original encoding space" is equivalent to saying "linear regression with *no* feature engineering is enough to solve any prediction problem." Which we have known to be nonsensical since long before computers existed
And why mathematics in physical sciences is unreasonably effective:)
There is an additional factor: while the function used to describe the data should allow a natural representation of the latent space, the function needs also be efficient to discover, construct, parameterize and represent. (That's why Euclidean representations are ubiquitous.)
Absolutely! You can't leverage the latent manifold (or rather, an approximation of it) unless you can represent it (easy part) and learn it (hard part).
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