## Generalized Laplacian Eigenmaps

Published: 31 Oct 2022, 18:00, Last Modified: 11 Jan 2023, 11:48NeurIPS 2022 AcceptReaders: Everyone
Keywords: GCL, graph contrastive learning, node embedding, logdet, rank minimization
TL;DR: We propose GLEN, an NP-hard rank difference minimization problem for graph node embedding that enjoys the intra-class separation guarantee and can be solved with a logdet relaxation.
Abstract: Graph contrastive learning attracts/disperses node representations for similar/dissimilar node pairs under some notion of similarity. It may be combined with a low-dimensional embedding of nodes to preserve intrinsic and structural properties of a graph. COLES, a recent graph contrastive method combines traditional graph embedding and negative sampling into one framework. COLES in fact minimizes the trace difference between the within-class scatter matrix encapsulating the graph connectivity and the total scatter matrix encapsulating negative sampling. In this paper, we propose a more essential framework for graph embedding, called Generalized Laplacian EigeNmaps (GLEN), which learns a graph representation by maximizing the rank difference between the total scatter matrix and the within-class scatter matrix, resulting in the minimum class separation guarantee. However, the rank difference minimization is an NP-hard problem. Thus, we replace the trace difference that corresponds to the difference of nuclear norms by the difference of LogDet expressions, which we argue is a more accurate surrogate for the NP-hard rank difference than the trace difference. While enjoying a lesser computational cost, the difference of LogDet terms is lower-bounded by the Affine-invariant Riemannian metric (AIRM) and Jesen-Bregman the LogDet Divergence (JBLD), and upper-bounded by AIRM scaled by the factor of $\sqrt{m}$. We show that GLEN offers favourable accuracy/scalability compared to state-of-the-art baselines.
Supplementary Material: pdf
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