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https://issues.apache.org/jira/browse/SPARK-31007?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Sean R. Owen resolved SPARK-31007.
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Fix Version/s: 3.1.0
Resolution: Fixed
Resolved by https://github.com/apache/spark/pull/27758
> 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.
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