Fuzzy Tensor Algebra for Large-Scale AI Models

Abstract

Complex real-world judgments face hurdles due to data complexity and ambiguity, exposing the limitations of traditional fuzzy logic models. Intuitionistic fuzzy logic (IFL) improves reliability by analyzing entities and intermediary states but struggles with large-scale applications. Integrating IFL with mathematics and Artificial Intelligence (AI) is challenging, particularly without tensor algebra. Research suggests matrix algebra can aid IFL, but systematic validation is limited. The study advocates for a tensor algebra-based AI-augmented IFL framework to enhance scalability, involving tensor decompositions for dimensionality reduction and feature extraction, alongside AI for dynamic optimization. The theoretical analysis show tensor matrices can expand IFL, promoting scalability and improved decision-making accuracy without empirical validations yet. Current literature reflects a scarcity of tensor decomposition exploration in IFL. This research work presents a model using Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) with tensor decomposition techniques, resulting in a more robust and noise-resistant IFL model. Compatibility with AI techniques is highlighted, enhancing forecasting accuracy and resilience. The findings imply tensor decomposition is crucial for reinforcing IFL as a decision support system, pointing towards the need for adaptive deep learning frameworks in future research.

Introduction

It explains that traditional fuzzy set theory and even basic intuitionistic fuzzy sets struggle with high-dimensional, large-scale, and uncertain environments. To overcome this, researchers have extended these models using Type-2 fuzzy sets, intuitionistic fuzzy sets, and interval-valued intuitionistic fuzzy sets (IVIFS), which better represent uncertainty in real-world systems.

A major focus of the work is the integration of tensor algebra and tensor decomposition techniques. Tensors, which are multidimensional data structures, allow fuzzy and intuitionistic fuzzy models to represent complex relationships in higher dimensions. Techniques like CP decomposition help reduce complexity while preserving important information, making models more scalable and efficient.

The literature review highlights that many studies have applied:

• Fuzzy logic in decision-making systems
• Tensor-based models in machine learning, clustering, forecasting, and recommendation systems
• AI methods for improving prediction, classification, and optimization tasks

However, the text identifies a key research gap: most existing studies either use fuzzy logic or tensor methods separately, but there is limited work that fully integrates intelligence (AI) + tensor decomposition + intuitionistic fuzzy logic in a unified scalable framework, especially with real-world validation.

The motivation of the study is therefore to develop a unified AI-driven tensor-based intuitionistic fuzzy model that can:

• Handle complex, uncertain, and high-dimensional data
• Improve decision-making accuracy
• Scale efficiently for large problems

The proposed approach focuses on:

• Using Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) as the base model
• Representing data as higher-order fuzzy tensors
• Applying tensor decomposition (e.g., CP decomposition) for dimensionality reduction
• Incorporating AI for parameter optimization and improved performance

The study is mainly theoretical and conceptual, without experimental simulation, but aims to provide a framework for future research.

Conclusion

The current analysis as presented above has successfully achieved its proposed objectives by introducing a decomposition-selective and progressive conceptualization of higher order multi-criteria based tensor model. Base model of this study is an Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) augmented with a tensor matrix decomposition structure. The current conceptual framework introduces a generalized and scalable method to intuitionistic fuzzy model integrating CP and Singular Value Decomposition (SVD) methods as part of its consecutive improvements. As a step towards a further optimized and valid model for large scale ML decision-making applications, 4th-orderNeutrosophic Tensor (DxMxKxN) enhances the scale of machine learning. It also adds deep extraction of features by dimensionality reduction using HOSVD and PCA. The model is theoretically established to be improved in its robustness, interpretability, and statistical resilience, addressing noise and incompleteness than the conventional fuzzy models. The analysis successfully develops and validates a theoretical basis for fuzzy tensor matrix decomposition, which plays a role in generalizing a intuitionistic fuzzy logic model. Moreover, the study confirms the compatibility and applicability of the model within the limits of intelligent automation (AI), particularly using a machine learning (ML) approach to improve the accuracy of the forecast and real-time resilience. The results confirm that tensor decomposition and introduction of multiple criteria-based conditions enable a systematic generalization, transforming intuitionistic fuzzy logic into a data-resilient, explainable, and scalable decision support mechanism. Finally, some future works and improvement areas recommended are: • Development of an adaptive deep learning model for automating tensor decomposition choice and optimization. • Neural architecture search (NAS) and reinforcement learning for dynamic selection among CP, Tucker, and other decomposition models. • Optimization on information scaffolding and choice complexity to enhance model decisions. • AI-driven technique removes the necessity for manual parameter tuning. • AI model can achieve an intuitionistic fuzzy model that is self-optimizing and scalable in an objective, active resolution environment.

References

[1] Ahmad Al Zoubi, W., Mahmoud As`ad Alnaser, A., Moh`d Said\" Hatamleh, H., Al Wadi, Y., & Oqla Massa`deh, M. (2019). Introduction to Cartesian, Tensor and Lexicographic Product of Bipolar Interval Valued Fuzzy Graph. Journal of Engineering and Applied Sciences, 15(2), 581–585. https://doi.org/10.36478/jeasci.2020.581.585 [2] Atanassov, K.T. (1986). Intuitionistic Fuzzy sets. Fuzzy Sets and Systems 20, 87–96. [3] Bilal, M., & Lucian-Popa, I. (2025). A novel hesitant fuzzy tensor-based group decision-making approach with application to heterogeneous wireless network evaluation. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-15496-6 [4] Bilal, M., Chaoqian, L., & Popa, I. L. (2025). Fuzzy soft tensor-based group decision making approach with application to heterogeneous wireless network evaluation. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-20639-w [5] Chen, J. (1999). Anti-fuzzy sets and anti-fuzzy relations. Fuzzy Sets and Systems, 101(2), 159–162. https://doi.org/10.1016/S0165-0114(97)00292-4 [6] Chen, J. (2003). Anti-fuzzy subgroups and anti-fuzzy ideals. Fuzzy Sets and Systems, 135(1), 57–66. https://doi.org/10.1016/S0165-0114(02)00246-4 [7] Chen, L. (2020). Three-value cutting tensors of intuitionistic fuzzy tensors. Soft Computing, 24(24), 18953–18958. https://doi.org/10.1007/s00500-020-05125-x [8] De Braganca Pereira, C. A., Polpo, A., & Rodrigues, A. S. (2020). Data Science: Measuring Uncertainties. Entropy, 22(12), 1438. https://doi.org/10.3390/e22121438 [9] Deng, S., Liu, J., & Wang, X. (2018). The Properties of Fuzzy Tensor and Its Application in Multiple Attribute Group Decision Making. IEEE Transactions on Fuzzy Systems, 27(3), 589–597. https://doi.org/10.1109/tfuzz.2018.2865923 [10] Deng, S., Liu, J., Tan, J., & Zhou, L. (2019). A Novel Method Based on Fuzzy Tensor Technique for Interval-Valued Intuitionistic Fuzzy Decision-Making with High-Dimension Data. International Journal of Computational Intelligence Systems, 12(2), 580. https://doi.org/10.2991/ijcis.d.190424.001 [11] Dogra, S., & Pal, M. (2020). Picture fuzzy matrix and its application. Soft Computing, 24(13), 9413–9428. https://doi.org/10.1007/s00500-020-05021-4 [12] Drakopoulos, G., Kanavos, A., Mylonas, P., &Pintelas, P. (2020). Extending Fuzzy Cognitive Maps with Tensor-Based Distance Metrics. Mathematics, 8(11), 1898–1898. https://doi.org/10.3390/math8111898 [13] Fan, X., & Wang, Z. (2021). A Fuzzy Lyapunov Function Method to Stability Analysis of Fractional-Order T–S Fuzzy Systems. IEEE Transactions on Fuzzy Systems, 30(7), 2769–2776. https://doi.org/10.1109/tfuzz.2021.3078289 [14] Hitchcock, F. L. (1927). The expression of a tensor or a polyadic as a sum of products. Journal of Mathematics and Physics, 6, 164–189. [15] Huang, S., Zhao, G., Weng, Z., & Ma, S. (2021). Trapezoidal type-2 fuzzy inference system with tensor unfolding structure learning method. Neurocomputing, 473, 54–67. https://doi.org/10.1016/j.neucom.2021.12.011 [16] Kolda, T. G., & Bader, B. W. (2009). Tensor Decompositions and Applications. SIAM Review, 51(3), 455–500. https://doi.org/10.1137/07070111x [17] Kuppusamy, M., Singh, J., Raman, P., & Abdolrahimi, N. (2024). Advanced decision analytics with fuzzy logic: Integrating AI and computational thinking for personnel selection. Journal of Fuzzy Extension and Applications, 5(4), 679–699. https://doi.org/10.22105/jfea.2024.477833.1621 [18] Lu, X., & Zhang, W. (2014). Fuzzy tensors and their applications in decision making. Information Sciences,272, 152–168. https://doi.org/10.1016/j.ins.2014.02.059 [19] Lu, X., Zhang, W., & Deng, Y. (2018). Intuitionistic fuzzy tensors and their application to multi-attribute decision making. Applied Soft Computing, 68, 336–351. https://doi.org/10.1016/j.asoc.2018.04.040 [20] Lyu, H. L., Wang, W., & Liu, X. P. (2025). Semi-tensor product-based fuzzy relation matrix technique for gear system state forecasting. ISA Transactions. https://doi.org/10.1016/j.isatra.2025.07.035 [21] Minaev, Y., Filimonova, O., & Minaeva, J. (2021, June 5). Tensor models for data extraction and use of hidden knowledge in the environment of uncertainty modeled by fuzzy sets of 1 and 2 types. In Proceedings of the 3rd International Workshop on Modern Machine Learning Technologies and Data Science, Lviv–Shatsk, Ukraine. [22] Minaev, Yu. M., Filimonova, O. Yu., & Minaeva, Yu. I. (2023). Forecasting of Fuzzy Time Series Based on the Concept of the Nearest Fuzzy Sets and Tensor Models of Time Series. Cybernetics and Systems Analysis, 59(1), 165–176. https://doi.org/10.1007/s10559-023-00551-9 [23] Minaeva, J. (2021, December 1–3). The concept of nearest fuzzy sets (Kronecker proximity) and its application for data representation and decision making under uncertainty. In Information Technology and Implementation (IT&I-2021), Kyiv, Ukraine. [24] Misaghian, N., Motameni, H., & Rabbani, M. (2019). Prioritizing interdependent software requirements using tensor and fuzzy graphs. TURKISH JOURNAL of ELECTRICAL ENGINEERING & COMPUTER SCIENCES, 27(4), 2697–2717. https://doi.org/10.3906/elk-1806-179 [25] Mosleh, M., Otadi, M., &Khanmirzaie, A. (2009). Decomposition method for solving fully fuzzy linear systems. Iranian Journal of Optimization, 1(2009), 188–198. [26] Muthuraji, T., & Sriram, S. (2017). Representation and decomposition of an intuitionistic fuzzy matrix using some (?, ??) cuts. Applications and Applied Mathematics: An International Journal (AAM), 12(1), Article 16. [27] Rabanser, S., Falck, F., &Günnemann, S. (2017). Introduction to Tensor Decompositions and Their Applications in Machine Learning. arXiv preprint arXiv:1711.10781. [28] Sahoo, S., & Pal, M. (2015). Different types of products on intuitionistic fuzzy graphs. Pacific Science Review. A: Natural Science and Engineering, 17(3), 87–96. https://doi.org/10.1016/j.psra.2015.12.007 [29] Sarwar, M., & Akram, M. (2018). Certain Algorithms for Modeling Uncertain Data Using Fuzzy Tensor Product Bézier Surfaces. Mathematics, 6(3), 42–42. https://doi.org/10.3390/math6030042 [30] Sethia, K., Singh, J., & Gosain, A. (2025). An effective imputation approach for handling missing data using intuitionistic fuzzy clustering algorithms. Deleted Journal, 28(1). https://doi.org/10.1007/s10791-025-09639-6 [31] Shah, S., Tewary, R., Sahni, M., Sahni, R., Castro, E. L., & Lindahl, J. M. (2023). Study of Intuitionistic Fuzzy Super Matrices and Its Application in Decision Making. Mathematics and Statistics, 11(5), 753–766. https://doi.org/10.13189/ms.2023.110501 [32] Shi, Z., Ma, Y., & Fu, M. (2022). Fuzzy Support Tensor Product Adaptive Image Classification for the Internet of Things. Computational Intelligence and Neuroscience, 2022, 1–11. https://doi.org/10.1155/2022/3532605 [33] Singh, M., Pant, M., Kong, L., Alijani, Z., &Snášel, V. (2022). A PCA-based fuzzy tensor evaluation model for multiple-criteria group decision making. Applied Soft Computing, 132, 109753–109753. https://doi.org/10.1016/j.asoc.2022.109753 [34] Smarandache, F. (1998). A unifying field in logics: Neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability, and neutrosophic statistics. American Research Press. [35] Uddin, F., Ishtiaq, U., Javed, K., Aiadi, S. S., Arshad, M., Souayah, N., &Mlaiki, N. (2022). A New Extension to the Intuitionistic Fuzzy Metric-like Spaces. Symmetry, 14(7), 1400–1400. https://doi.org/10.3390/sym14071400 [36] Wang, S., Yang, J., Chen, Z., Yuan, H., Geng, J., & Hai, Z. (2020). Global and Local Tensor Factorization for Multi-criteria Recommender System. Patterns, 1(2), 100023–100023. https://doi.org/10.1016/j.patter.2020.100023 [37] Xu, Z., & Gou, X. (2016). An overview of interval-valued intuitionistic fuzzy information aggregations and applications. Granular Computing, 2(1), 13–39. https://doi.org/10.1007/s41066-016-0023-4 [38] Zhao, J., Zhang, J., Lei, Y., & Yi, B. (2025). Proportional grey picture fuzzy sets and their application in multi-criteria decision-making with high-dimensional data. AIMS Mathematics, 10(1), 208–233. https://doi.org/10.3934/math.2025011 [39] Zhu, Y., Xi, C., Wang, S., Xu, L., Chen, X., & Wang, Z. (2025). Knowledge?Driven and Low?Rank Tensor Regularized Multiview Fuzzy Clustering for Alzheimer’s Diagnosis. International Journal of Intelligent Systems, 2025(1). https://doi.org/10.1155/int/1458773

Copyright

Copyright © 2026 Md Raza Ansari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.