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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| Vydané v: | IEEE transactions on cybernetics Ročník 49; číslo 12; s. 4229 - 4242 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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United States
IEEE
01.12.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2168-2267, 2168-2275, 2168-2275 |
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| Abstract | 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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| AbstractList | We present a novel alternative convergence theory of the fuzzy C -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.We present a novel alternative convergence theory of the fuzzy C -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. We present a novel alternative convergence theory of the fuzzy [Formula Omitted]-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. 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. We present a novel alternative convergence theory of the fuzzy C -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. |
| Author | Saha, Arkajyoti Das, Swagatam |
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| Snippet | We present a novel alternative convergence theory of the fuzzy <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula>-means (FCM)... We present a novel alternative convergence theory of the fuzzy C -means (FCM) clustering algorithm with a super-class of the so-called "distance like... We present a novel alternative convergence theory of the fuzzy [Formula Omitted]-means (FCM) clustering algorithm with a super-class of the so-called “distance... |
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| SubjectTerms | 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 |
| Title | Stronger Convergence Results for the Center-Based Fuzzy Clustering With Convex Divergence Measure |
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