  Data valuation -- quantifying the contribution of individual data sources to
certain predictive behaviors of a model -- is of great importance to enhancing
the transparency of machine learning and designing incentive systems for data
sharing. Existing work has focused on evaluating data sources with the shared
feature or sample space. How to valuate fragmented data sources of which each
only contains partial features and samples remains an open question. We start
by presenting a method to calculate the counterfactual of removing a fragment
from the aggregated data matrix. Based on the counterfactual calculation, we
further propose 2D-Shapley, a theoretical framework for fragmented data
valuation that uniquely satisfies some appealing axioms in the fragmented data
context. 2D-Shapley empowers a range of new use cases, such as selecting useful
data fragments, providing interpretation for sample-wise data values, and
fine-grained data issue diagnosis.
