Latent Space Model for Higher-order Networks and Generalized Tensor Decomposition

TL;DR

This paper introduces a unified latent space framework for higher-order networks, employing generalized tensor decomposition with theoretical guarantees and demonstrating effectiveness on synthetic and real data.

Contribution

It presents a novel generalized tensor decomposition algorithm with convergence guarantees for latent space models in complex higher-order networks.

Findings

Algorithm converges linearly under mild conditions.

Finite-sample error rates depend on signal strength and model complexity.

Method achieves accurate link prediction on real-world datasets.

Abstract

We introduce a unified framework, formulated as general latent space models, to study complex higher-order network interactions among multiple entities. Our framework covers several popular models in recent network analysis literature, including mixture multi-layer latent space model and hypergraph latent space model. We formulate the relationship between the latent positions and the observed data via a generalized multilinear kernel as the link function. While our model enjoys decent generality, its maximum likelihood parameter estimation is also convenient via a generalized tensor decomposition procedure.We propose a novel algorithm using projected gradient descent on Grassmannians. We also develop original theoretical guarantees for our algorithm. First, we show its linear convergence under mild conditions. Second, we establish finite-sample statistical error rates of latent position estimation, determined by the signal strength, degrees of freedom and the smoothness of link function, for both general and specific latent space models. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction.