Topological Persistence of the Neural Embedding of the Archetypal Subspace

Suryaka Suresh, Vinayak Abrol, Kaushik Kalyanaraman, Sumantra Dutta Roy

Abstract: Persistent homology (PH) has recently emerged as a topological tool to analyse the learnability in the layer-by-layer feature embedding space of a well-trained deep neural network. However, the computation of the PH is computationally impractical in higher dimensions or for large-scale datasets. To address this, we analyse the homological features of the embedding space of the principal convex hull of the data. This is achieved using Archetypal Analysis (AA), a factor analysis method that expresses the data as a convex combination of extremal points on the convex hull. Through homological inference of point clouds, we show that, due to geometric properties, adding archetypes to the data does not change the homology of the underlying space approximated by the data. We use this analysis to prove that the homological complexity of archetypes of a class can be used as the minimal homological expressivity needed for a classifier to avoid misclassification. Since inferring the homological complexity of the underlying space for real-world data can be ambiguous, we attempt to quantify the transformation of the homological complexity of the embedding space of the class archetypes through the topological persistence of their embedding space. For this, we employ “topological average life,” which measures the average persistence of the homologies in the point cloud. We find a negative correlation between the fine tuned model’s performance and the topological average life of the archetypal subspace.

External IDs:doi:10.1109/jstsp.2025.3631403