  We propose a framework to analyze how multivariate representations
disentangle ground-truth generative factors. A quantitative analysis of
disentanglement has been based on metrics designed to compare how one variable
explains each generative factor. Current metrics, however, may fail to detect
entanglement that involves more than two variables, e.g., representations that
duplicate and rotate generative factors in high dimensional spaces. In this
work, we establish a framework to analyze information sharing in a multivariate
representation with Partial Information Decomposition and propose a new
disentanglement metric. This framework enables us to understand disentanglement
in terms of uniqueness, redundancy, and synergy. We develop an experimental
protocol to assess how increasingly entangled representations are evaluated
with each metric and confirm that the proposed metric correctly responds to
entanglement. Through experiments on variational autoencoders, we find that
models with similar disentanglement scores have a variety of characteristics in
entanglement, for each of which a distinct strategy may be required to obtain a
disentangled representation.
