The k-means++ algorithm due to Arthur and Vassilvitskii [David Arthur and Sergei Vassilvitskii, 2007] has become the most popular seeding method for Lloyd’s algorithm. It samples the first center uniformly at random from the data set and the other k-1 centers iteratively according to D²-sampling, i.e., the probability that a data point becomes the next center is proportional to its squared distance to the closest center chosen so far. k-means++ is known to achieve an approximation factor of 𝒪(log k) in expectation. Already in the original paper on k-means++, Arthur and Vassilvitskii suggested a variation called greedy k-means++ algorithm in which in each iteration multiple possible centers are sampled according to D²-sampling and only the one that decreases the objective the most is chosen as a center for that iteration. It is stated as an open question whether this also leads to an 𝒪(log k)-approximation (or even better). We show that this is not the case by presenting a family of instances on which greedy k-means++ yields only an Ω(𝓁⋅log k)-approximation in expectation where 𝓁 is the number of possible centers that are sampled in each iteration. Inspired by the negative results, we study a variation of greedy k-means++ which we call noisy k-means++ algorithm. In this variation only one center is sampled in every iteration but not exactly by D²-sampling. Instead in each iteration an adversary is allowed to change the probabilities arising from D²-sampling individually for each point by a factor between 1-ε₁ and 1+ε₂ for parameters ε₁ ∈ [0,1) and ε₂ ≥ 0. We prove that noisy k-means++ computes an 𝒪(log² k)-approximation in expectation. We use the analysis of noisy k-means++ to design a moderately greedy k-means++ algorithm.
@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2020.18, author = {Bhattacharya, Anup and Eube, Jan and R\"{o}glin, Heiko and Schmidt, Melanie}, title = {{Noisy, Greedy and Not so Greedy k-Means++}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.18}, URN = {urn:nbn:de:0030-drops-128848}, doi = {10.4230/LIPIcs.ESA.2020.18}, annote = {Keywords: k-means++, greedy, adaptive sampling} }
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