Topologically Regularized Data Embeddings

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.