  Archetypal analysis is a data decomposition method that describes each
observation in a dataset as a convex combination of "pure types" or archetypes.
These archetypes represent extrema of a data space in which there is a
trade-off between features, such as in biology where different combinations of
traits provide optimal fitness for different environments. Existing methods for
archetypal analysis work well when a linear relationship exists between the
feature space and the archetypal space. However, such methods are not
applicable to systems where the feature space is generated non-linearly from
the combination of archetypes, such as in biological systems or image
transformations. Here, we propose a reformulation of the problem such that the
goal is to learn a non-linear transformation of the data into a latent
archetypal space. To solve this problem, we introduce Archetypal Analysis
network (AAnet), which is a deep neural network framework for learning and
generating from a latent archetypal representation of data. We demonstrate
state-of-the-art recovery of ground-truth archetypes in non-linear data
domains, show AAnet can generate from data geometry rather than from data
density, and use AAnet to identify biologically meaningful archetypes in
single-cell gene expression data.
