Fuzzy c-means (FCM) and its variants, including entropy-regularized FCM and Tsallis entropy-based FCM (TFCM), are widely used in fuzzy clustering. Although these algorithms are effective, they cannot model the cluster-specific subspaces that are often present in high-dimensional data. To address this limitation, fuzzy c-varieties (FCV) represent each cluster as a low-dimensional variety and iteratively alternate between dimension reduction and clustering, thereby capturing intrinsic cluster structures. Building on this, we propose a Tsallis entropy-regularized FCV (TFCV), which generalizes both standard FCV (SFCV) and entropy-regularized FCV (EFCV) by regularizing the SFCV objective with Tsallis entropy. TFCV inherits the enhanced flexibility of TFCM, leading to improved clustering performance. Theoretical analysis of the fuzzy classification functions confirmed that TFCV generalizes SFCV and EFCV. Empirical evaluations on 14 real datasets demonstrated that TFCV achieves a higher clustering accuracy than the existing methods with statistical significance, establishing its effectiveness as a robust approach that integrates dimension reduction and fuzzy clustering.

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