---
_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'
...