Neural Eigenfunctions Are Structured Representation Learners

This paper introduces a structured, adaptive-length deep representation called Neural Eigenmap. Unlike prior spectral methods such as Laplacian Eigenmap that operate in a nonparametric manner, Neural Eigenmap leverages NeuralEF (Deng et al., 2022) to parametrically model eigenfunctions using a neural network. We show that, when the eigenfunction is derived from positive relations in a data augmentation setup, applying NeuralEF results in an objective function that resembles those of popular self-supervised learning methods, with an additional symmetry-breaking property that leads to structured representations where features are ordered by importance. We demonstrate using such representations as adaptive-length codes in image retrieval systems. By truncation according to feature importance, our method requires up to 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. Automatically learning representations from unlabelled data is a long-standing challenge in machine learning. Often, the motivation is to map data to a vector space where the geometric distance reflects semantic closeness. This enables, for example, retrieving semantically related information via finding nearest neighbors, or discovering concepts with clustering. One can also pass such representations as inputs to supervised learning procedures, which removes the need for feature engineering.