  Barrett's Esophagus (BE) is the only precursor known to Esophageal
Adenocarcinoma (EAC), a type of esophageal cancer with poor prognosis upon
diagnosis. Therefore, diagnosing BE is crucial in preventing and treating
esophageal cancer. While supervised machine learning supports BE diagnosis,
high interobserver variability in histopathological training data limits these
methods. Unsupervised representation learning via Variational Autoencoders
(VAEs) shows promise, as they map input data to a lower-dimensional manifold
with only useful features, characterizing BE progression for improved
downstream tasks and insights. However, the VAE's Euclidean latent space
distorts point relationships, hindering disease progression modeling. Geometric
VAEs provide additional geometric structure to the latent space, with RHVAE
assuming a Riemannian manifold and $\mathcal{S}$-VAE a hyperspherical manifold.
Our study shows that $\mathcal{S}$-VAE outperforms vanilla VAE with better
reconstruction losses, representation classification accuracies, and
higher-quality generated images and interpolations in lower-dimensional
settings. By disentangling rotation information from the latent space, we
improve results further using a group-based architecture. Additionally, we take
initial steps towards $\mathcal{S}$-AE, a novel autoencoder model generating
qualitative images without a variational framework, but retaining benefits of
autoencoders such as stability and reconstruction quality.
