Description
In current impl, following Lemma is used in KMeans:
0, Let x be a point, let b be a center and o be the origin, then d(x,c) >= (d(x,o)  d(c,o)) = normnorm(c)
this can be applied in EuclideanDistance, but not in CosineDistance
According to [Using the Triangle Inequality to Accelerate KMeanshttps://www.aaai.org/Papers/ICML/2003/ICML03022.pdf], we can go futher, and there are another two Lemmas can be used:
1, Let x be a point, and let b and c be centers. If d(b,c)>=2d(x,b) then d(x,c) >= d(x,b);
this can be applied in EuclideanDistance, but not in CosineDistance.
However, luckily for CosineDistance we can get a variant in the space of radian/angle.
2, Let x be a point, and let b and c be centers. Then d(x,c) >= max{0, d(x,b)d(b,c)};
this can be applied in EuclideanDistance, but not in CosineDistance
The application of Lemma 2 is a little complex: It need to cache/update the distance/lower bounds to previous centers, and thus can be only applied in training, not usable in prediction.
So this ticket is mainly for Lemma 1. Its idea is quite simple, if point x is close to center b enough (less than a precomputed radius), then we can say point x belong to center c without computing the distances between x and other centers. It can be used in both training and predction.
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SPARK36553 KMeans fails with NegativeArraySizeException for K = 50000 after issue #27758 was introduced
 Resolved
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