Neural Eigenfunctions are Structured Representation Learners

This paper revisits the canonical concept of learning structured representations without label supervision by eigendecomposition. Yet, unlike prior spectral methods such as Laplacian Eigenmap which operate in a nonparametric manner, we aim to parametrically model the principal eigenfunctions of an integral operator defined by a kernel and a data distribution using a neural network for enhanced scalability and reasonable out-of-sample generalization. To achieve this goal, we first present a new series of objective functions that generalize the EigenGame Gemp et al. 2020 to function space for learning neural eigenfunctions. We then show that, when the similarity metric is derived from positive relations in a data augmentation setup, a representation learning objective function that resembles those of popular self-supervised learning methods emerges, with an additional symmetry-breaking property for producing structured representations where features are ordered by importance. We call such a structured, adaptive-length deep representation Neural Eigenmap. We demonstrate using Neural Eigenmap as adaptive-length codes in image retrieval systems. By truncation according to feature importance, our method requires up to $16\times$16× shorter representation length than leading self-supervised learning ones to achieve similar retrieval performance. We further apply our method to graph data and report strong results on a node representation learning benchmark with more than one million nodes.