  We present a new approach for understanding the periodicity structure and
semantics of motion datasets, independently of the morphology and skeletal
structure of characters. Unlike existing methods using an overly sparse
high-dimensional latent, we propose a phase manifold consisting of multiple
closed curves, each corresponding to a latent amplitude. With our proposed
vector quantized periodic autoencoder, we learn a shared phase manifold for
multiple characters, such as a human and a dog, without any supervision. This
is achieved by exploiting the discrete structure and a shallow network as
bottlenecks, such that semantically similar motions are clustered into the same
curve of the manifold, and the motions within the same component are aligned
temporally by the phase variable. In combination with an improved motion
matching framework, we demonstrate the manifold's capability of timing and
semantics alignment in several applications, including motion retrieval,
transfer and stylization. Code and pre-trained models for this paper are
available at https://peizhuoli.github.io/walkthedog.
