Incorporating Topological Priors Into Low-Dimensional Visualizations Through Topological Regularization

Unsupervised representation learning techniques are commonly employed to analyze high-dimensional or unstructured data. In some cases, users may have prior knowledge of the topology of the data, such as a known cluster structure or the fact that it follows a tree- or graph-based structure. However, generic methods for ensuring this inherent structure is evident in low-dimensional representations are lacking and it is unknown how imposing topological constraints affects downstream learning tasks. To fill this gap, we propose topological regularization - a generic approach based on algebraic topology to incorporate topological prior knowledge into low-dimensional representations. We introduce a class of topological loss functions and demonstrate that optimizing an embedding loss together with one of these loss functions as a regularizer results in embeddings that consider not only local proximities but also the desired topological structure. We provide a self-contained introduction to essential concepts in algebraic topology and offer intuitive guidance for designing topological loss functions suitable for a variety of data shapes, such as clusters, cycles, or bifurcations. We empirically assess the efficiency, robustness, and versatility of the proposed method when combined with linear and non-linear dimensionality reduction techniques, as well as graph embedding methods.