  Unsupervised representation learning methods are widely used for gaining
insight into high-dimensional, unstructured, or structured data. In some cases,
users may have prior topological knowledge about the data, such as a known
cluster structure or the fact that the data is known to lie along a tree- or
graph-structured topology. However, generic methods to ensure such structure is
salient in the low-dimensional representations are lacking. This negatively
impacts the interpretability of low-dimensional embeddings, and plausibly
downstream learning tasks. To address this issue, we introduce topological
regularization: a generic approach based on algebraic topology to incorporate
topological prior knowledge into low-dimensional embeddings. We introduce a
class of topological loss functions, and show that jointly optimizing an
embedding loss with such a topological loss function as a regularizer yields
embeddings that reflect not only local proximities but also the desired
topological structure. We include a self-contained overview of the required
foundational concepts in algebraic topology, and provide intuitive guidance on
how to design topological loss functions for a variety of shapes, such as
clusters, cycles, and bifurcations. We empirically evaluate the proposed
approach on computational efficiency, robustness, and versatility in
combination with linear and non-linear dimensionality reduction and graph
embedding methods.
