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From "zhengruifeng (Jira)" <j...@apache.org>
Subject [jira] [Updated] (SPARK-31007) KMeans optimization based on triangle-inequality
Date Mon, 02 Mar 2020 10:03:00 GMT

     [ https://issues.apache.org/jira/browse/SPARK-31007?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
]

zhengruifeng updated SPARK-31007:
---------------------------------
    Attachment: ICML03-022.pdf

> 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
>         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 predction.
> 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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