[{"type":"conference","status":"public","date_created":"2018-04-17T11:46:07Z","citation":{"short":"J. Blömer, S. Brauer, K. Bujna, in: 2016 IEEE 16th International Conference on Data Mining (ICDM), IEEE, 2016, pp. 805–810.","bibtex":"@inproceedings{Blömer_Brauer_Bujna_2016, title={A Theoretical Analysis of the Fuzzy K-Means Problem}, DOI={10.1109/icdm.2016.0094}, booktitle={2016 IEEE 16th International Conference on Data Mining (ICDM)}, publisher={IEEE}, author={Blömer, Johannes and Brauer, Sascha and Bujna, Kathrin}, year={2016}, pages={805–810} }","ieee":"J. Blömer, S. Brauer, and K. Bujna, “A Theoretical Analysis of the Fuzzy K-Means Problem,” in *2016 IEEE 16th International Conference on Data Mining (ICDM)*, Barcelona, Spain, 2016, pp. 805–810.","chicago":"Blömer, Johannes, Sascha Brauer, and Kathrin Bujna. “A Theoretical Analysis of the Fuzzy K-Means Problem.” In *2016 IEEE 16th International Conference on Data Mining (ICDM)*, 805–10. IEEE, 2016. https://doi.org/10.1109/icdm.2016.0094.","mla":"Blömer, Johannes, et al. “A Theoretical Analysis of the Fuzzy K-Means Problem.” *2016 IEEE 16th International Conference on Data Mining (ICDM)*, IEEE, 2016, pp. 805–10, doi:10.1109/icdm.2016.0094.","apa":"Blömer, J., Brauer, S., & Bujna, K. (2016). A Theoretical Analysis of the Fuzzy K-Means Problem. In *2016 IEEE 16th International Conference on Data Mining (ICDM)* (pp. 805–810). Barcelona, Spain: IEEE. https://doi.org/10.1109/icdm.2016.0094","ama":"Blömer J, Brauer S, Bujna K. A Theoretical Analysis of the Fuzzy K-Means Problem. In: *2016 IEEE 16th International Conference on Data Mining (ICDM)*. IEEE; 2016:805-810. doi:10.1109/icdm.2016.0094"},"user_id":"25078","page":"805-810","department":[{"_id":"64"}],"year":"2016","publisher":"IEEE","conference":{"name":"IEEE 16th International Conference on Data Mining (ICDM)","start_date":"2016-12-12","location":"Barcelona, Spain","end_date":"2016-12-15"},"doi":"10.1109/icdm.2016.0094","author":[{"first_name":"Johannes","last_name":"Blömer","id":"23","full_name":"Blömer, Johannes"},{"full_name":"Brauer, Sascha","id":"13291","first_name":"Sascha","last_name":"Brauer"},{"first_name":"Kathrin","last_name":"Bujna","full_name":"Bujna, Kathrin"}],"language":[{"iso":"eng"}],"publication":"2016 IEEE 16th International Conference on Data Mining (ICDM)","date_updated":"2019-01-03T13:13:06Z","publication_identifier":{"isbn":["9781509054732"]},"abstract":[{"lang":"eng","text":"One of the most popular fuzzy clustering techniques is the fuzzy K-means algorithm (also known as fuzzy-c-means or FCM algorithm). In contrast to the K-means and K-median problem, the underlying fuzzy K-means problem has not been studied from a theoretical point of view. In particular, there are no algorithms with approximation guarantees similar to the famous K-means++ algorithm known for the fuzzy K-means problem. This work initiates the study of the fuzzy K-means problem from an algorithmic and complexity theoretic perspective. We show that optimal solutions for the fuzzy K-means problem cannot, in general, be expressed by radicals over the input points. Surprisingly, this already holds for simple inputs in one-dimensional space. Hence, one cannot expect to compute optimal solutions exactly. We give the first (1+eps)-approximation algorithms for the fuzzy K-means problem. First, we present a deterministic approximation algorithm whose runtime is polynomial in N and linear in the dimension D of the input set, given that K is constant, i.e. a polynomial time approximation scheme (PTAS) for fixed K. We achieve this result by showing that for each soft clustering there exists a hard clustering with similar properties. Second, by using techniques known from coreset constructions for the K-means problem, we develop a deterministic approximation algorithm that runs in time almost linear in N but exponential in the dimension D. We complement these results with a randomized algorithm which imposes some natural restrictions on the sought solution and whose runtime is comparable to some of the most efficient approximation algorithms for K-means, i.e. linear in the number of points and the dimension, but exponential in the number of clusters."}],"publication_status":"published","_id":"2367","title":"A Theoretical Analysis of the Fuzzy K-Means Problem","keyword":["unsolvability by radicals","clustering","fuzzy k-means","probabilistic method","approximation algorithms","randomized algorithms"]}]