  Although the variational autoencoder (VAE) and its conditional extension
(CVAE) are capable of state-of-the-art results across multiple domains, their
precise behavior is still not fully understood, particularly in the context of
data (like images) that lie on or near a low-dimensional manifold. For example,
while prior work has suggested that the globally optimal VAE solution can learn
the correct manifold dimension, a necessary (but not sufficient) condition for
producing samples from the true data distribution, this has never been
rigorously proven. Moreover, it remains unclear how such considerations would
change when various types of conditioning variables are introduced, or when the
data support is extended to a union of manifolds (e.g., as is likely the case
for MNIST digits and related). In this work, we address these points by first
proving that VAE global minima are indeed capable of recovering the correct
manifold dimension. We then extend this result to more general CVAEs,
demonstrating practical scenarios whereby the conditioning variables allow the
model to adaptively learn manifolds of varying dimension across samples. Our
analyses, which have practical implications for various CVAE design choices,
are also supported by numerical results on both synthetic and real-world
datasets.
