An Asymmetric Approach to Three-Way Approximation of Fuzzy Sets

The three-way approximation of fuzzy sets represents membership values using a three-valued set $\lbrace \mathbf{1}, \mathbf{m}, \mathbf{0}\rbrace$, where 1 indicates total belongingness, 0 total nonbelongingness, and m an intermediate state. This approach elevates values of membership function above a threshold $\alpha$ to 1, reduces those below $\beta$ to 0, and assigns the remaining ones to an intermediate value m. A key challenge lies in determining the thresholds $\alpha$ and $\beta$ and selecting the value of m, as existing models often lack analytical solutions and fail to fully explore the relationship between m and membership structures. This study introduces an asymmetric three-way approximation model for fuzzy sets, removing the constraint $\alpha + \beta = 1$. Analytical formulas are derived for the thresholds $ \alpha $ and $ \beta $ by minimizing information loss, and the relationship between m and membership structures is thoroughly examined. An adaptive optimizer is proposed to learn the approximate optimal value of m by minimizing the information loss. The experimental results show that information loss decreases initially before increasing as m grows. Besides, our model achieves the best classification across most datasets.