On the Importance of Looking at the Manifold
Authors: Nil Adell Mill, Jannis Born, Nathaniel Park, James Hedrick, María Rodríguez Martínez, Matteo Manica
Keywords: Topological Learning, GNN, VAE
Abstract: Data rarely lies on uniquely Euclidean spaces. Even data typically represented in regular domains, such as images, can have a higher level of relational information, either between data samples or even relations within samples, e.g., how the objects in an image are linked. With this perspective our data points can be enriched by explicitly accounting for this connectivity and analyzing them as a graph. Herein, we analyze various approaches for unsupervised representation learning and investigate the importance of considering topological information and its impact when learning representations. We explore a spectrum of models, ranging from uniquely learning representations based on the isolated features of the nodes (focusing on Variational Autoencoders), to uniquely learning representations based on the topology (using node2vec) passing through models that integrate both node features and topological information in a hybrid fashion. For the latter we use Graph Neural Networks, precisely Deep Graph Infomax (DGI), and an extension of the typical formulation of the VAE where the topological structure is accounted for via an explicit regularization of the loss (Graph-Regularized VAEs, introduced in this work). To extensively investigate these methodologies, we consider a wide variety of data types: synthetic data point clouds, MNIST, citation networks, and chemical reactions. We show that each of the representations learned by these models may have critical importance for further downstream tasks, and that accounting for the topological features can greatly improve the modeling capabilities for certain problems. We further provide a framework to analyze these, and future models under different scenarios and types of data.