Worst-Case and Smoothed Analysis of k-Means Clustering with Bregman Divergences
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Smoothed Analysis of the k-Means Method
Journal of the ACM (JACM)
Smoothed Analysis of Moore-Penrose Inversion
SIAM Journal on Matrix Analysis and Applications
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We show a worst-case lower bound and a smoothed upper bound on the number of iterations performed by the Iterative Closest Point (ICP) algorithm. First proposed by Besl and McKay, the algorithm is widely used in computational geometry, where it is known for its simplicity and its observed speed. The theoretical study of ICP was initiated by Ezra, Sharir, and Efrat, who showed that the worst-case running time to align two sets of $n$ points in $\mathbb{R}^d$ is between $\Omega(n\log n)$ and $O(n^2d)^d$. We substantially tighten this gap by improving the lower bound to $\Omega(n/d)^{d+1}$. To help reconcile this bound with the algorithm's observed speed, we also show that the smoothed complexity of ICP is polynomial, independent of the dimensionality of the data. Using similar methods, we improve the best known smoothed upper bound for the popular k-means method to $n^{O(k)}$, once again independent of the dimension.