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.
