  In this paper, we propose a new method to perform data augmentation in a
reliable way in the High Dimensional Low Sample Size (HDLSS) setting using a
geometry-based variational autoencoder. Our approach combines a proper latent
space modeling of the VAE seen as a Riemannian manifold with a new generation
scheme which produces more meaningful samples especially in the context of
small data sets. The proposed method is tested through a wide experimental
study where its robustness to data sets, classifiers and training samples size
is stressed. It is also validated on a medical imaging classification task on
the challenging ADNI database where a small number of 3D brain MRIs are
considered and augmented using the proposed VAE framework. In each case, the
proposed method allows for a significant and reliable gain in the
classification metrics. For instance, balanced accuracy jumps from 66.3% to
74.3% for a state-of-the-art CNN classifier trained with 50 MRIs of cognitively
normal (CN) and 50 Alzheimer disease (AD) patients and from 77.7% to 86.3% when
trained with 243 CN and 210 AD while improving greatly sensitivity and
specificity metrics.
