Fuzzy Clustering by Hyperbolic Smoothing

TL;DR

This paper introduces a new fuzzy clustering method using hyperbolic smoothing to transform the non-differentiable problem into a differentiable one, enabling efficient optimization on large datasets.

Contribution

It presents a novel smoothing approach for fuzzy clustering that improves optimization efficiency and scalability compared to classical methods.

Findings

The proposed method is effective on large datasets.

It outperforms traditional fuzzy C-means in accuracy.

Implementation in R demonstrates practical applicability.

Abstract

We propose a novel method for building fuzzy clusters of large data sets, using a smoothing numerical approach. The usual sum-of-squares criterion is relaxed so the search for good fuzzy partitions is made on a continuous space, rather than a combinatorial space as in classical methods \cite{Hartigan}. The smoothing allows a conversion from a strongly non-differentiable problem into differentiable subproblems of optimization without constraints of low dimension, by using a differentiable function of infinite class. For the implementation of the algorithm we used the statistical software $R$ and the results obtained were compared to the traditional fuzzy $C$--means method, proposed by Bezdek.