A unified spectral-persistent homology framework for stable and generalizable topological deep learning

Topological Deep Learning (TDL) boosts the capabilities of neural networks by integrating topological features from complex data structures like graphs and simplicial complexes. However, there is a critical gap in the area of structural perturbations within a model and how these affect its stability and generalization. Overfitting and brittle predictions are concerns when dealing with topological noise. We propose a comprehensive theoretical and empirical evaluation framework of TDL robustness. We define two complementary metrics, Topological Drift which quantifies model sensitivity to structural noise using bottleneck distance in persistent homology, and Spectral Variance which measures shifts in the eigenvalues of the Hodge Laplacian. These define the degradation of learned topological representations. We conducted controlled experiments on synthetic Vietoris-Rips complexes, together with real-world data from the TDA Benchmark’s Enzyme Function Prediction dataset. These experiments support our conclusions. High levels of perturbation resulted in a drop in classification accuracy of 19.1%, and an increase in spectral variance of 4.3×, confirming theoretical expectations. Comparison with neglecting frameworks after 2022 on Simplicial Neural Networks and spectral graph models showcases the need for our rigorous dual-metric interpretable approach to stability analysis. Our persistent homology and spectral topology merger lay metric derived from persistent homology to quantify topological drift adaptable for creating TDL architectures with robustness. The results obtained in this study can be used to formulate model equipped with topological state descriptors derived from persistence diagrams and eigen-spectra methods of deep learning for advanced fields like bioinformatics, neuroscience, and structural biology, which involve data with inherent topological variations.