Mathematical Foundations of Explainable AI: A Framework based on Topological Data Analysis

This paper presents a mathematically grounded framework for Explainable Artificial Intelligence (XAI) based on Topological Data Analysis (TDA). By leveraging persistent homology, we construct robust topological feature representations—including persistence images, landscapes, and Betti curves—that enrich traditional machine learning models with geometric and structural insights. We evaluate the framework across five benchmark datasets—Circles, Moons, Iris, MNIST, and Fashion-MNIST—spanning both synthetic and real-world domains with varying dimensionality. Experimental results demonstrate that TDA-derived features significantly enhance both predictive performance and interpretability. Combined models achieved up to +8.9% accuracy improvement, with the highest gains observed in non-linearly separable datasets. Explainability metrics such as Local Fidelity (0.86), Stability (0.92), and Faithfulness (0.91) improved substantially compared to raw-only models. Explanations were also more concise, with sparsity reduced from 5.2 to 3.1 features on average. Sensitivity analysis identified persistence threshold τ = 0.010 as optimal for filtering topological noise. The proposed TDA-XAI framework is model-agnostic, scalable, and compatible with standard interpretability tools like SHAP and LIME. It provides a principled way to bridge data geometry with explainable learning, offering substantial gains in accuracy, robustness, and transparency—particularly in high-stakes or complex decision-making domains.