Stronger Convergence Results for the Center-Based Fuzzy Clustering With Convex Divergence Measure

We present a novel alternative convergence theory of the fuzzy <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula>-means (FCM) clustering algorithm with a super-class of the so-called "distance like functions" which emerged from the...

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Bibliographic Details
Published in:IEEE transactions on cybernetics Vol. 49; no. 12; pp. 4229 - 4242
Main Authors: Saha, Arkajyoti, Das, Swagatam
Format: Journal Article
Language:English
Published: United States IEEE 01.12.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Closed graph theorem
Clustering
Clustering algorithms
Continuity (mathematics)
Convergence
convergence analysis
convex analysis
Cybernetics
distance like functions (DLFs)
Distance measurement
Divergence
Euclidean distance
Euclidean geometry
Iterative algorithms
Iterative methods
Linear programming
Mathematical analysis
Operators (mathematics)
partitional clustering
Partitioning algorithms
unsupervised learning
ISSN:2168-2267, 2168-2275, 2168-2275
Online Access:Get full text
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Summary:We present a novel alternative convergence theory of the fuzzy <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula>-means (FCM) clustering algorithm with a super-class of the so-called "distance like functions" which emerged from the earlier attempts of unifying the theories of center-based clustering methods. This super-class does not assume the existence of double derivative of the distance measure with respect to the coordinate of the cluster representative (first coordinate in this formulation). The convergence result does not require the separability of the distance measures. Moreover, it provides us with a stronger convergence property comparable (same to be precise, but in terms of the generalized distance measure) to that of the classical FCM with squared Euclidean distance. The crux of the convergence analysis lies in the development of a fundamentally novel mathematical proof of the continuity of the clustering operator even in absence of the closed form upgrading rule, without necessitating the separability and double differentiability of the distance function and still providing us with a convergence result comparable to that of the classical FCM. The implication of our novel proof technique goes way beyond the realm of FCM and provides a general setup for convergence analysis of the similar iterative algorithms.
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ISSN:2168-2267
2168-2275
2168-2275
DOI:10.1109/TCYB.2018.2861211