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The part about “is it in the convex hull?” was an important part of the question. It seems to me that if it isn’t in the convex hull, it could be more fitting
by drdeca 1y ago
The part about “is it in the convex hull?” was an important part of the question.
It seems to me that if it isn’t in the convex hull, it could be more fitting to describe it as extrapolation, rather than interpolation?
In general, my question does apply to the task of predicting how a time series of vectors continues: Given a dataset of time series, where the dimension of each vector in the series is such and such, the length of each series is yea long, and there are N series in the training set, should we expect series in the test set or validation set to be in the convex hull of the ones in the training set?
I would think that the number of series in the training set, N, while large, might not be all that large compared to the dimensionality of a whole series?
Hm, are there efficient techniques for evaluating whether a high dimensional vector is in the convex hull of a large number of other high dimensional vectors?
- bunderbunder 1y agoJust shooting from the hip, LLMs operate out on a frontier where the curse of dimensionality removes a large chunk of the practical value from the concept of a convex hull. Especially in a case like this where the vector embedding process places hard limits on the range of possible magnitudes and directions for any single vector.
- drdeca 1y agoOutside of the context of a convex hull, I don’t know how to make a distinction between interpolation and extrapolation. This is the core of my question. What precisely is it that you mean when you say that it is interpolating rather than extrapolating? In the only definition that I know, the one based on convex hulls, I believe it would be extrapolating rather than interpolating. But people often say it is interpolating rather than extrapolating, and I don’t know what they mean.
- bunderbunder 1y agoI doubt they're really thinking about it in a mathematical sense when they say that. I'm guessing, for example, that "extrapolate" is meant in the more colloquial sense, which is maybe closer to "deduce" in practice.