On the Limitations of Fractal Dimension as a Measure of Generalization

Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. Neural network optimization trajectories have been proposed to possess fractal structure, leading to bounds and generalization measures based on notions of fractal dimension on these trajectories. Prominently, both the Hausdorff dimension and the persistent homology dimension have been proposed to correlate with generalization gap, thus serving as a measure of generalization. This work performs an extended evaluation of these topological generalization measures. We demonstrate that fractal dimension fails to predict generalization of models trained from poor initializations. We further identify that the $\ell^2$ norm of the final parameter iterate, one of the simplest complexity measures in learning theory, correlates more strongly with the generalization gap than these notions of fractal dimension. Finally, our study reveals the intriguing manifestation of model-wise double descent in persistent homology-based generalization measures. This work lays the ground for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.

Computable Stability for Persistence Rank Function Machine Learning

Persistent homology barcodes and diagrams are a cornerstone of topological data analysis. Widely used in many real data settings, they relate variation in topological information (as measured by cellular homology) with variation in data, however, they are challenging to use in statistical settings due to their complex geometric structure. In this paper, we revisit the persistent homology rank function -- an invariant measure of ``shape" that was introduced before barcodes and persistence diagrams and captures the same information in a form that is more amenable to data and computation. In particular, since they are functions, techniques from functional data analysis -- a domain of statistics adapted for functions -- apply directly to persistent homology when represented by rank functions. Rank functions, however, have been less popular than barcodes because they face the challenge that stability -- a property that is crucial to validate their use in data analysis -- is difficult to guarantee, mainly due to metric concerns on rank function space. However, rank functions extend more naturally to the increasingly popular and important case of multiparameter persistent homology. In this paper, we study the performance of rank functions in functional inferential statistics and machine learning on both simulated and real data, and in both single and multiparameter persistent homology. We find that the use of persistent homology captured by rank functions offers a clear improvement over existing approaches. We then provide theoretical justification for our numerical experiments and applications to data by deriving several stability results for single- and multiparameter persistence rank functions under various metrics with the underlying aim of computational feasibility and interpretability.