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Takagi-Sugeno-Kang (TSK) fuzzy system with Gaussian membership functions (MFs) is one of the most widely used fuzzy systems in machine learning. However, it usually has difficulty handling high-dimensional datasets. This paper explores why TSK fuzzy systems with Gaussian MFs may fail on high-dimensional inputs. After transforming defuzzification to an equivalent form of softmax function, we find that the poor performance is due to the saturation of softmax. We show that two defuzzification operations, LogTSK and HTSK, the latter of which is first proposed in this paper, can avoid the saturation. Experimental results on datasets with various dimensionalities validated our analysis and demonstrated the effectiveness of LogTSK and HTSK.

Curse of Dimensionality for TSK Fuzzy Neural Networks: Explanation and Solutions