A conditionally positive definite kernel function for possibilistic clustering

2017 ◽  
Vol 6 (1) ◽  
pp. 75
Author(s):  
Meena Tushir ◽  
Dinesh Rai ◽  
Jyotsna Nigam
Keyword(s):  
Kernel Function ◽  
Positive Definite ◽  
Positive Definite Kernel ◽  
Conditionally Positive Definite ◽  
Possibilistic Clustering

Hard and Fuzzy c-Means Clustering with Conditionally Positive Definite Kernel

2012 ◽  
Vol 16 (7) ◽  
pp. 825-830
Author(s):  
Yuchi Kanzawa ◽  
◽  
Yasunori Endo ◽  
Sadaaki Miyamoto ◽  
Keyword(s):  
Optimization Problems ◽  
Clustering Algorithms ◽  
Feature Space ◽  
Positive Definite ◽  
Cluster Center ◽  
Positive Definite Kernel ◽  
Conditionally Positive Definite ◽  
Fuzzy C Means Clustering ◽  
The Relationship ◽  
Squared Euclidean Distance

In this paper, we investigate three types of c-means clustering algorithms with a conditionally positive definite (cpd) kernel. One is based on hard c-means and two are based on standard and entropy-regularized fuzzy c-means. First, based on a cpd kernel describing a squared Euclidean distance between data in feature space, these algorithms are derived from revised optimization problems of the conventional kernel c-means. Next, based on the relationship between the positive definite (pd) kernel and cpd kernel, the revised dissimilarity between a datum and a cluster center in the feature space is shown. Finally, it is shown that a cpd kernel c-means algorithm and a kernel c-means algorithm with a pd kernel derived from the cpd kernel are essentially identical to each other. Explicit mapping for a cpd kernel is also described geometrically.

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