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Fuzzy C-Means Clustering Using Gauss-Seidel Iteration
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DOI：10.1007/s40815-025-02129-3.png)
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Abstract
En 中文
Fuzzy c-means (FCM) is one of the most frequently used methods for clustering. It typically generates a solution by alternately updating the membership matrix U and the center matrix V, with the majority of the computational cost concentrated on updating U. Each row of U is updated using Jacobi iteration, which results in a relatively slow convergence rate and increases the likelihood of being trapped in local optima. To address this, we propose an FCM clustering method using Gauss-Seidel iteration (GS-FCM), along with its batched version (GS- $$\hbox {FCM}_{\left( s \right) }$$ ). By eliminating V, the original optimization problem is transformed into an equivalent form involving only U. Then, U is updated using Gauss-Seidel iteration. In the batched version, U is randomly partitioned into s ( $$1 \le s \le n$$ ) sub-blocks, where each sub-block is updated sequentially following the GS-FCM approach, while the variables within each block are updated in parallel as in FCM. Specifically, GS- $$\hbox {FCM}_{\left( s \right) }$$ reduces to FCM when $$ s=1 $$ and becomes the proposed GS-FCM when $$ s=n $$ . When s is set to a small value, GS- $$\hbox {FCM}_{\left( s \right) }$$ also exhibits linear computational complexity with respect to the number of samples. Experimental results show that GS- $$\hbox {FCM}_{\left( s \right) }$$ converges faster than FCM and reaches better local minima in fewer iterations.
Keywords:
Fuzzy c-means
Jacobi iteration
Gauss-Seidel iteration
Local minima
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