  We consider the task of representation learning for unsupervised segmentation
of 3D voxel-grid biomedical images. We show that models that capture implicit
hierarchical relationships between subvolumes are better suited for this task.
To that end, we consider encoder-decoder architectures with a hyperbolic latent
space, to explicitly capture hierarchical relationships present in subvolumes
of the data. We propose utilizing a 3D hyperbolic variational autoencoder with
a novel gyroplane convolutional layer to map from the embedding space back to
3D images. To capture these relationships, we introduce an essential
self-supervised loss -- in addition to the standard VAE loss -- which infers
approximate hierarchies and encourages implicitly related subvolumes to be
mapped closer in the embedding space. We present experiments on both synthetic
data and biomedical data to validate our hypothesis.
