  In this paper, we propose $\tau$GAN a tensor-based method for modeling the
latent space of generative models. The objective is to identify semantic
directions in latent space. To this end, we propose to fit a multilinear tensor
model on a structured facial expression database, which is initially embedded
into latent space. We validate our approach on StyleGAN trained on FFHQ using
BU-3DFE as a structured facial expression database. We show how the parameters
of the multilinear tensor model can be approximated by Alternating Least
Squares. Further, we introduce a tacked style-separated tensor model, defined
as an ensemble of style-specific models to integrate our approach with the
extended latent space of StyleGAN. We show that taking the individual styles of
the extended latent space into account leads to higher model flexibility and
lower reconstruction error. Finally, we do several experiments comparing our
approach to former work on both GANs and multilinear models. Concretely, we
analyze the expression subspace and find that the expression trajectories meet
at an apathetic face that is consistent with earlier work. We also show that by
changing the pose of a person, the generated image from our approach is closer
to the ground truth than results from two competing approaches.
