  Data samples collected for training machine learning models are typically
assumed to be independent and identically distributed (iid). Recent research
has demonstrated that this assumption can be problematic as it simplifies the
manifold of structured data. This has motivated different research areas such
as data poisoning, model improvement, and explanation of machine learning
models. In this work, we study the influence of a sample on determining the
intrinsic topological features of its underlying manifold. We propose the
Shapley Homology framework, which provides a quantitative metric for the
influence of a sample of the homology of a simplicial complex. By interpreting
the influence as a probability measure, we further define an entropy which
reflects the complexity of the data manifold. Our empirical studies show that
when using the 0-dimensional homology, on neighboring graphs, samples with
higher influence scores have more impact on the accuracy of neural networks for
determining the graph connectivity and on several regular grammars whose higher
entropy values imply more difficulty in being learned.
