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

arXiv: https://arxiv.org/abs/2102.04271 · HTML: https://mlsanity.com/paper/2102.04271/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [Relief-Based Feature Selection: Introduction and Review](https://mlsanity.com/paper/1711.08421/) (2017)
- [Noisy Softmax: Improving the Generalization Ability of DCNN via Postponing the Early Softmax Saturation](https://mlsanity.com/paper/1708.03769/) (2017)
- [Optimize TSK Fuzzy Systems for Regression Problems: Mini-Batch Gradient Descent with Regularization, DropRule and AdaBound (MBGD-RDA)](https://mlsanity.com/paper/1903.10951/) (2019)
- [Recommendations on Designing Practical Interval Type-2 Fuzzy Systems](https://mlsanity.com/paper/1907.01697/) (2019)
- [On the Functional Equivalence of TSK Fuzzy Systems to Neural Networks, Mixture of Experts, CART, and Stacking Ensemble Regression](https://mlsanity.com/paper/1903.10572/) (2019)
- [Distilling a Deep Neural Network into a Takagi-Sugeno-Kang Fuzzy Inference System](https://mlsanity.com/paper/2010.04974/) (2020)
- [Concise Fuzzy System Modeling Integrating Soft Subspace Clustering and Sparse Learning](https://mlsanity.com/paper/1904.10683/) (2019)
- [Supervised Enhanced Soft Subspace Clustering (SESSC) for TSK Fuzzy Classifiers](https://mlsanity.com/paper/2002.12404/) (2020)

## Built on by (most cited first, 12 of 12)

- [An Adaptive Neuro-Fuzzy System with Integrated Feature Selection and Rule Extraction for High-Dimensional Classification Problems](https://mlsanity.com/paper/2201.03187/) (2022)
- [FAME: Introducing Fuzzy Additive Models for Explainable AI](https://mlsanity.com/paper/2504.07011/) (2025)
- [PyTSK: A Python Toolbox for TSK Fuzzy Systems](https://mlsanity.com/paper/2206.03310/) (2022)
- [Semantic-Preserving Feature Partitioning for Multi-View Ensemble Learning](https://mlsanity.com/paper/2401.06251/) (2024)
- [Enhancing Interval Type-2 Fuzzy Logic Systems: Learning for Precision and Prediction Intervals](https://mlsanity.com/paper/2404.12802/) (2024)
- [Zadeh's Type-2 Fuzzy Logic Systems: Precision and High-Quality Prediction Intervals](https://mlsanity.com/paper/2404.12800/) (2024)
- [A Self-Constructing Multi-Expert Fuzzy System for High-dimensional Data Classification](https://mlsanity.com/paper/2410.13390/) (2024)
- [Gradient-Based Neuroplastic Adaptation for Concurrent Optimization of Neuro-Fuzzy Networks](https://mlsanity.com/paper/2506.21771/) (2025)
- [An Integrated Fusion Framework for Ensemble Learning Leveraging Gradient Boosting and Fuzzy Rule-Based Models](https://mlsanity.com/paper/2511.08077/) (2025)
- [Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems](https://mlsanity.com/paper/2602.19253/) (2026)
- [Explainable machine learning workflows for radio astronomical data processing](https://mlsanity.com/paper/2603.16350/) (2026)
- [SoftTri: Smooth Triangular Membership Functions for Adaptive Fuzzy Inference Systems](https://mlsanity.com/paper/2609.20194/) (2026)

## Abstract (arXiv)

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.