  Unsupervised feature learning often finds low-dimensional embeddings that
capture the structure of complex data. For tasks for which prior expert
topological knowledge is available, incorporating this into the learned
representation may lead to higher quality embeddings. For example, this may
help one to embed the data into a given number of clusters, or to accommodate
for noise that prevents one from deriving the distribution of the data over the
model directly, which can then be learned more effectively. However, a general
tool for integrating different prior topological knowledge into embeddings is
lacking. Although differentiable topology layers have been recently developed
that can (re)shape embeddings into prespecified topological models, they have
two important limitations for representation learning, which we address in this
paper. First, the currently suggested topological losses fail to represent
simple models such as clusters and flares in a natural manner. Second, these
losses neglect all original structural (such as neighborhood) information in
the data that is useful for learning. We overcome these limitations by
introducing a new set of topological losses, and proposing their usage as a way
for topologically regularizing data embeddings to naturally represent a
prespecified model. We include thorough experiments on synthetic and real data
that highlight the usefulness and versatility of this approach, with
applications ranging from modeling high-dimensional single-cell data, to graph
embedding.
