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- W2156089267 abstract "Suppose $k$ centers are fit to $m$ points by heuristically minimizing the $k$-means cost; what is the corresponding fit over the source distribution? This question is resolved here for distributions with $pgeq 4$ bounded moments; in particular, the difference between the sample cost and distribution cost decays with $m$ and $p$ as $m^{min{-1/4, -1/2+2/p}}$. The essential technical contribution is a mechanism to uniformly control deviations in the face of unbounded parameter sets, cost functions, and source distributions. To further demonstrate this mechanism, a soft clustering variant of $k$-means cost is also considered, namely the log likelihood of a Gaussian mixture, subject to the constraint that all covariance matrices have bounded spectrum. Lastly, a rate with refined constants is provided for $k$-means instances possessing some cluster structure." @default.
- W2156089267 created "2016-06-24" @default.
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- W2156089267 date "2013-11-08" @default.
- W2156089267 modified "2023-09-27" @default.
- W2156089267 title "Moment-based Uniform Deviation Bounds for $k$-means and Friends" @default.
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