--- _id: '2367' 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. author: - first_name: Johannes full_name: Blömer, Johannes id: '23' last_name: Blömer - first_name: Sascha full_name: Brauer, Sascha id: '13291' last_name: Brauer - first_name: Kathrin full_name: Bujna, Kathrin last_name: Bujna citation: 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' 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' 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} }' 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. 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. 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. short: 'J. Blömer, S. Brauer, K. Bujna, in: 2016 IEEE 16th International Conference on Data Mining (ICDM), IEEE, 2016, pp. 805–810.' conference: end_date: 2016-12-15 location: Barcelona, Spain name: IEEE 16th International Conference on Data Mining (ICDM) start_date: 2016-12-12 date_created: 2018-04-17T11:46:07Z date_updated: 2022-01-06T06:55:58Z department: - _id: '64' doi: 10.1109/icdm.2016.0094 keyword: - unsolvability by radicals - clustering - fuzzy k-means - probabilistic method - approximation algorithms - randomized algorithms language: - iso: eng page: 805-810 publication: 2016 IEEE 16th International Conference on Data Mining (ICDM) publication_identifier: isbn: - '9781509054732' publication_status: published publisher: IEEE status: public title: A Theoretical Analysis of the Fuzzy K-Means Problem type: conference user_id: '25078' year: '2016' ...