Efficient Dissimilarity Preservation in Graph Embeddings via Dimension Regularization

Core Concepts

Replacing costly node-wise repulsion in skip-gram graph embeddings with efficient dimension-wise regularization can preserve dissimilarity while improving scalability and representation quality.

Abstract

The content discusses an efficient approach to preserving dissimilarity in graph embeddings. The key insights are:

The negative function in skip-gram graph embedding objectives, which enforces dissimilarity between node embeddings, can be approximated via dimension regularization instead of costly node-wise repulsion.

As the need for node repulsion grows (i.e., the embeddings start collapsing due to only optimizing for similarity), the dimension regularization approach converges to optimizing the original skip-gram negative function.

The authors propose an algorithm augmentation framework that replaces skip-gram negative sampling (SGNS) with dimension regularization. This reduces the time complexity from linear in the number of nodes to linear in the number of embedding dimensions.

Empirical evaluations show that the augmented LINE and node2vec algorithms preserve downstream performance on link prediction tasks while dramatically reducing training runtime compared to the original algorithms using SGNS.

The authors also demonstrate that the Positive Only baselines, which remove the negative function entirely, are sensitive to graph connectivity and prone to collapsing embeddings, whereas the dimension regularization approach is more robust.

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Key Insights Distilled From

Re-visiting Skip-Gram Negative Sampling: Dimension Regularization for More Efficient Dissimilarity Preservation in Graph Embeddings

by David Liu,Ar... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00172.pdf

Deeper Inquiries

How generalizable is the dimension regularization approach to other graph embedding objectives beyond skip-gram, such as matrix factorization or graph neural networks

The dimension regularization approach proposed in the context of skip-gram negative sampling can be generalized to other graph embedding objectives beyond skip-gram. This approach is based on the idea of repelling dissimilar pairs by regularizing the embedding dimensions to be centered around the origin. In matrix factorization, this regularization can be applied by penalizing the covariance among dimensions, similar to how it was done for spectral embeddings. For graph neural networks (GNNs), the dimension regularization can be incorporated by enforcing orthogonality among the embedding dimensions, which would help in preserving dissimilarity between nodes. By reinterpreting the negative function as a dimension regularizer, the approach can be extended to various graph embedding techniques that involve balancing similarity and dissimilarity objectives.

Can the dimension regularization be further improved by incorporating node-specific weighting, similar to how SGNS allows tuning the negative sampling distribution

Incorporating node-specific weighting into the dimension regularization approach can potentially enhance its performance, similar to how SGNS allows tuning the negative sampling distribution based on node degrees. By introducing node-specific weights, the regularization can be tailored to prioritize certain nodes over others during the repulsion phase. This can be achieved by adjusting the regularization term based on node properties such as degree centrality, community membership, or other relevant features. By incorporating node-specific weighting, the dimension regularization approach can be further optimized to better capture the dissimilarity between nodes and improve the overall quality of the learned embeddings.

What are the theoretical guarantees on the convergence and optimality of the dimension regularization approach compared to the original skip-gram negative function

The dimension regularization approach offers theoretical guarantees on convergence and optimality compared to the original skip-gram negative function. By replacing the costly node-wise repulsion with dimension-wise operations, the approach reduces the time complexity of the optimization process. The regularization term, which centers the embeddings around the origin, ensures that the embeddings maintain a certain level of dissimilarity while preventing collapse into a degenerate solution. The theoretical guarantees stem from the connection between optimizing the dimension regularizer and achieving dissimilarity preservation in the embeddings. Under mild initialization conditions and with appropriate regularization parameters, the dimension regularization approach converges to optimizing the skip-gram loss function, providing a more efficient and scalable solution for learning graph embeddings.