Shapley Homology: Topological Analysis of Sample Influence for Neural Networks

Area of Science:

• Machine Learning
• Topological Data Analysis
• Data Science

Background:

• The independent and identically distributed (i.i.d.) assumption simplifies structured data manifolds, impacting areas like data poisoning and model explainability.
• Understanding sample influence on data topology is crucial for robust machine learning model development.

Purpose of the Study:

• To develop a quantitative metric for assessing a data sample's influence on the topological features of its underlying manifold.
• To introduce the Shapley homology framework for decomposing topological properties of data complexes.

Main Methods:

• Proposing the Shapley homology framework, combining homology analysis (Betti numbers) and Shapley value decomposition.
• Defining an entropy measure based on sample influence to reflect data manifold complexity.
• Exploring connections between Shapley homology and Vapnik-Chervonenkis dimension.

Main Results:

• Empirical studies demonstrate that high zero-dimensional Shapley homology scores correlate with greater impact on neural network accuracy for graph connectivity tasks.
• Higher entropy values, derived from Shapley homology, indicate increased learning difficulty for regular grammars.

Conclusions:

• Shapley homology offers a novel approach to quantify sample influence on data topology, moving beyond i.i.d. assumptions.
• This framework provides insights into data complexity and its effect on machine learning model performance and learnability.