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zhengruifeng commented on SPARK-31007: -------------------------------------- [~srowen] This optimization needs an array of size val packedValues = Array.ofDim[Double](k * (k + 1) / 2) can may cause failure when k is large. https://issues.apache.org/jira/browse/SPARK-36553 k=50000 should we: 1, revert this optimization 2, or enable it only for small k? (but the impl will be more complex) > KMeans optimization based on triangle-inequality > ------------------------------------------------ > > Key: SPARK-31007 > URL: https://issues.apache.org/jira/browse/SPARK-31007 > Project: Spark > Issue Type: Improvement > Components: ML > Affects Versions: 3.1.0 > Reporter: zhengruifeng > Assignee: zhengruifeng > Priority: Major > Fix For: 3.1.0 > > Attachments: ICML03-022.pdf > > > 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))| = |norm(x)-norm(c)| > this can be applied in {{EuclideanDistance}}, but not in {{CosineDistance}} > According to [Using the Triangle Inequality to Accelerate > K-Means|[https://www.aaai.org/Papers/ICML/2003/ICML03-022.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 pre-computed 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. -- This message was sent by Atlassian Jira (v8.20.1#820001) --------------------------------------------------------------------- To unsubscribe, e-mail: issues-unsubscr...@spark.apache.org For additional commands, e-mail: issues-h...@spark.apache.org