Mathematical foundations of interpretable and explainable artificial intelligence : A theoretical framework for trustworthy learning models
Thamarai Selvi Vaidhyanathanvaidhyanathan@su.edu.omDepartment of General Foundation ProgrammeSohar UniversitySohar, Sultanate of OmanView full profile → , *Vaidhyanathan PandianCorresponding authorPandian.vaidhyanathan@utas.edu.omDepartment of Supportive Requirements and MathematicsUniversity of Technology and Applied SciencesShinas, Sultanate of OmanView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jan 2026
- Published Online:
- 30 Sep 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2664
- Pages:
- 2721–2735
Abstract
The recent advancements in Artificial Intelligence (AI) models, particularly in finance, transportation, and autonomous systems, have resulted in a growing demand for the development of models that are both accurate and interpretable. Numerous contemporary state-of-the-art XAI methodologies, like LIME, SHAP, and gradient-based visualisations, offer intuitive elucidations of model behaviour; nonetheless, they are predominantly heuristic and lack robust mathematical foundations or possess tenuous underpinnings. This limitation raises questions about the interpretability and generalisability of explanations derived from deep learning models. To overcome these restrictions, we present a systematic theory of interpretability and explainability, incorporating axioms and computable properties that precisely delineate the optimal framework for transparent models. It regards interpretability as a structural characteristic of the functional mappings and information flow inside the model, examined from information-theoretic, topological, and optimisation viewpoints. It delineates four essential criteria: fidelity, stability, consistency, and completeness. Such assumptions ensure that an explanation accurately characterises the behaviour of a model within the permissible errors dictated by Lipschitz continuity. We present a Soft Unified Interpretability Score that integrates mutual information relevance, stability against input perturbations, and distortion criteria to assess explanation quality. Analytical parameterisation utilising linear models, decision trees, and a neural network indicates that interpretability diminishes as model complexity increases, although this trade-off may be quantified by the suggested approach. This initiative establishes a foundation for verifiably reliable AI by anchoring the explanation in mathematical principles, while also presenting a compelling open question regarding future standardised measurements of explainability.
Keywords
Subject Classifications
References
[1] F. Sovrano and F. Vitali, “An objective metric for Explainable AI: How and why to estimate the degree of explainability,” Knowl.-Based Syst., vol. 278, Art. no. 110866 (2023), doi: 10.1016/j.knosys.2023.110866.
[2] Y.-L. Chou, C. Moreira, P. Bruza, C. Ouyang, and J. Jorge, “Counterfactuals and causability in explainable artificial intelligence: Theory, algorithms, and applications,” Information Fusion, vol. 81, pp. 59–83 (May 2022), doi: 10.1016/j.inffus.2021.11.003.
[3] P. P. Angelov, E. A. Soares, R. Jiang, N. I. Arnold, and P. M. Atkinson, “Explainable artificial intelligence: an analytical review,” WIREs Data Mining and Knowledge Discovery, vol. 11, no. 5 (Sep. 2021), doi: 10.1002/widm.1424.
[4] M. F. Iqbal, A. Zafar, U. Khalil, and A. Ishaq, “Investigating Aptitude in Learning Programming Language Using Machine Learning and Natural Language Processing,” International Journal of Data Informatics and Intelligent Computing, vol. 3, no. 4, pp. 40–61 (Dec. 2024), doi: 10.59461/ijdiic.v3i4.145.
[5] U. Cali, M. Kuzlu, M. Pipattanasomporn, J. Kempf, and L. Bai, “Foundations of Big Data, Machine Learning, and Artificial Intelligence and Explainable Artificial Intelligence,” in Digitalization of Power Markets and Systems Using Energy Informatics, Cham: Springer International Publishing, pp. 115–137 (2021), doi: 10.1007/978-3-030-83301-5_6.
[6] K. Sokol and P. Flach, “Explainability Is in the Mind of the Beholder: Establishing the Foundations of Explainable Artificial Intelligence,” arXiv preprint arXiv:2112.14466, (2021), doi: 10.48550/arXiv.2112.14466.
[7] J. Borrego-Díaz and J. Galán-Páez, “Explainable Artificial Intelligence in Data Science,” Minds Mach (Dordr), vol. 32, no. 3, pp. 485–531 (Sep. 2022), doi: 10.1007/s11023-022-09603-z.
[8] W. Samek, T. Wiegand, and K.-R. Müller, “Explainable artificial intelligence: Understanding, visualizing and interpreting deep learning models,” in Explainable AI: Interpreting, Explaining and Visualizing Deep Learning, W. Samek, G. Montavon, A. Vedaldi, L. Hansen, and K.-R. Müller, Eds., Cham, Switzerland: Springer, LNCS 10590, pp. 1–28 (2017), doi: 10.1007/978-3-319-65854-6_1.
[9] L. Longo, R. Goebel, F. Lecue, P. Kieseberg, and A. Holzinger, “Explainable Artificial Intelligence: Concepts, Applications, Research Challenges and Visions,” pp. 1–16 (2020), doi: 10.1007/978-3-030-57321-8_1.
[10] J. J. Colombini, F. Bonchi, F. Giannini, F. Giannotti, R. Pellungrini, and P. Frosini, “Mathematical foundation of interpretable equivariant surrogate models,” in Explainable Artificial Intelligence (xAI 2025), R. Guidotti, U. Schmid, and L. Longo, Eds., Commun. Comput. Inf. Sci., vol. 2577. Cham, Switzerland: Springer, pp. 294–318 (2025), doi: 10.1007/978-3-032-08324-1_13.
[11] R. Marcinkevičs and J. E. Vogt, “Interpretable and explainable machine learning: A methods-centric overview with concrete examples,” WIREs Data Mining and Knowledge Discovery, vol. 13, no. 3 (May 2023), doi: 10.1002/widm.1493.
[12] Kofi Immanuel Jones and Swati Sah, “The Implementation of Machine Learning in The Insurance Industry with Big Data Analytics,” International Journal of Data Informatics and Intelligent Computing, vol. 2, no. 2, pp. 21–38 (Jun. 2023), doi: 10.59461/ijdiic.v2i2.47.
[13] B. Kovalerchuk, M. A. Ahmad, and A. Teredesai, “Survey of explainable machine learning with visual and granular methods beyond quasi-explanations,” in Interpretable Artificial Intelligence: A Perspective of Granular Computing, W. Pedrycz and S.-M. Chen, Eds., Stud. Comput. Intell., vol. 937. Cham, Switzerland: Springer, pp. 217–267 (2021), doi: 10.1007/978-3-030-64949-4_8.
[14] M. Ferrara, “Explainable artificial intelligence and mathematics: new frontiers (and challenges) of research not only as ‘AppliedMath,’’ International Journal of Mathematical Analysis, vol. 18, no. 1, pp. 11–19 (2024), doi: 10.12988/ijma.2024.912544.
[15] R. Kazmierczak, S. Azzolin, E. Berthier, A. Hedström, P. Delhomme, D. Filliat, N. Bousquet, G. Frehse, M. Mancini, B. Caramiaux, A. Passerini, and G. Franchi, “Benchmarking XAI Explanations with Human-Aligned Evaluations,” in Proc. AAAI Conf. Artif. Intell., vol. 40, no. 44, pp. 37491–37500 (2026), doi: 10.1609/aaai.v40i44.41082
[16] N. Thangarasu, R. Rajalakshmi, G. Manivasagam, and V. Vijayalakshmi, “Performance of re-ranking techniques used for recommendation method to the user CF- Model,” International Journal of Data Informatics and Intelligent Computing, vol. 1, no. 1, pp. 30–38 (Sep. 2022), doi: 10.59461/ijdiic.v1i1.9.
[17] D. E. Mathew, D. U. Ebem, A. C. Ikegwu, P. E. Ukeoma, and N. F. Dibiaezue, “Recent Emerging Techniques in Explainable Artificial Intelligence to Enhance the Interpretable and Understanding of AI Models for Human,” Neural Process Lett, vol. 57, no. 1, p. 16 (Feb. 2025), doi: 10.1007/s11063-025-11732-2.
[18] L. Longo, M. Brcic, F. Cabitza, J. Choi, R. Confalonieri, J. Del Ser, R. Guidotti, Y. Hayashi, F. Herrera, A. Holzinger, R. Jiang, H. Khosravi, F. Lecue, G. Malgieri, A. Páez, W. Samek, J. Schneider, T. Speith, and S. Stumpf, “Explainable Artificial Intelligence (XAI) 2.0: A manifesto of open challenges and interdisciplinary research directions,” Information Fusion, vol. 106, art. no. 102301 (Jun. 2024), doi: 10.1016/j.inffus.2024.102301.
[19] B. Antunes, “Statistical Quality and Reproducibility of Pseudorandom Number Generators in Machine Learning Technologies,” International Journal of Data Informatics and Intelligent Computing, vol. 4, no. 3, pp. 23–32 (Aug. 2025), doi: 10.59461/ijdiic.v4i3.214.
[20] Ashish Kumar Pandey, “Analysis Role of ML and Big Data Play in Driving Digital Marketing’s Paradigm Shift,” International Journal of Data Informatics and Intelligent Computing, vol. 2, no. 3, pp. 38–46 (Sep. 2023), doi: 10.59461/ijdiic.v2i3.75.
