  Many machine learning algorithms represent input data with vector embeddings
or discrete codes. When inputs exhibit compositional structure (e.g. objects
built from parts or procedures from subroutines), it is natural to ask whether
this compositional structure is reflected in the the inputs' learned
representations. While the assessment of compositionality in languages has
received significant attention in linguistics and adjacent fields, the machine
learning literature lacks general-purpose tools for producing graded
measurements of compositional structure in more general (e.g. vector-valued)
representation spaces. We describe a procedure for evaluating compositionality
by measuring how well the true representation-producing model can be
approximated by a model that explicitly composes a collection of inferred
representational primitives. We use the procedure to provide formal and
empirical characterizations of compositional structure in a variety of
settings, exploring the relationship between compositionality and learning
dynamics, human judgments, representational similarity, and generalization.
