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Moment-based Uniform Deviation Bounds for $k$-means and Friends
Matus J Telgarsky · Sanjoy Dasgupta

Sat Dec 07 07:00 PM -- 11:59 PM (PST) @ Harrah's Special Events Center, 2nd Floor
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 $p\geq 4$ 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.

Author Information

Matus J Telgarsky (UIUC)
Sanjoy Dasgupta (UC San Diego)

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