  Modern generative models exhibit unprecedented capabilities to generate
extremely realistic data. However, given the inherent compositionality of the
real world, reliable use of these models in practical applications requires
that they exhibit the capability to compose a novel set of concepts to generate
outputs not seen in the training data set. Prior work demonstrates that recent
diffusion models do exhibit intriguing compositional generalization abilities,
but also fail unpredictably. Motivated by this, we perform a controlled study
for understanding compositional generalization in conditional diffusion models
in a synthetic setting, varying different attributes of the training data and
measuring the model's ability to generate samples out-of-distribution. Our
results show: (i) the order in which the ability to generate samples from a
concept and compose them emerges is governed by the structure of the underlying
data-generating process; (ii) performance on compositional tasks exhibits a
sudden "emergence" due to multiplicative reliance on the performance of
constituent tasks, partially explaining emergent phenomena seen in generative
models; and (iii) composing concepts with lower frequency in the training data
to generate out-of-distribution samples requires considerably more optimization
steps compared to generating in-distribution samples. Overall, our study lays a
foundation for understanding capabilities and compositionality in generative
models from a data-centric perspective.
