  Non-adversarial generative models such as variational auto-encoder (VAE),
Wasserstein auto-encoders with maximum mean discrepancy (WAE-MMD),
sliced-Wasserstein auto-encoder (SWAE) are relatively easy to train and have
less mode collapse compared to Wasserstein auto-encoder with generative
adversarial network (WAE-GAN). However, they are not very accurate in
approximating the target distribution in the latent space because they don't
have a discriminator to detect the minor difference between real and fake. To
this end, we develop a novel non-adversarial framework called Tessellated
Wasserstein Auto-encoders (TWAE) to tessellate the support of the target
distribution into a given number of regions by the centroidal Voronoi
tessellation (CVT) technique and design batches of data according to the
tessellation instead of random shuffling for accurate computation of
discrepancy. Theoretically, we demonstrate that the error of estimate to the
discrepancy decreases when the numbers of samples $n$ and regions $m$ of the
tessellation become larger with rates of $\mathcal{O}(\frac{1}{\sqrt{n}})$ and
$\mathcal{O}(\frac{1}{\sqrt{m}})$, respectively. Given fixed $n$ and $m$, a
necessary condition for the upper bound of measurement error to be minimized is
that the tessellation is the one determined by CVT. TWAE is very flexible to
different non-adversarial metrics and can substantially enhance their
generative performance in terms of Fr\'{e}chet inception distance (FID)
compared to VAE, WAE-MMD, SWAE. Moreover, numerical results indeed demonstrate
that TWAE is competitive to the adversarial model WAE-GAN, demonstrating its
powerful generative ability.
