On the Use of Persistent Homology to Control the Generalization Capacity of a Neural Network

Analyzing neural network (NN) generalization is vital for ensuring effective performance on new, unseen data, beyond the training set. Traditional methods involve evaluating NN across multiple testing datasets, a resource-intensive process involving data acquisition, preprocessing, and labeling. The primary challenge is determining the optimal capacity for training observations, requiring adaptable adjustments based on the task and available data information. This paper leverages Algebraic Topology and relevance measures to investigate NN behavior during learning. We define NN on a topological space as a functional topology graph and compute topological summaries to estimate generalization gaps. Simultaneously, we assess the relevance of NN units, progressively pruning network units. The generalization gap estimation helps identify overfitting, enabling timely early-stopping decisions and identifying the architecture with optimal generalization. This approach offers a comprehensive insight into NN generalization and supports the exploration of NN extensibility and interpretability.