Optimal link prediction with matrix logistic regression

TL;DR

This paper introduces a matrix logistic regression model for link prediction in large networks, analyzing its optimal performance and computational limitations in high-dimensional settings.

Contribution

It establishes the minimax rate for Frobenius-norm risk and proposes a penalised maximum likelihood estimator that achieves this rate, highlighting computational hardness.

Findings

Minimax rate for Frobenius-norm risk is derived.

A penalised maximum likelihood estimator attains the optimal rate.

Computational complexity limits the attainability of this rate by polynomial-time algorithms.

Abstract

We consider the problem of link prediction, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analysed in a high-dimensional regime under a structural assumption. The minimax rate for the Frobenius-norm risk is established and a combinatorial estimator based on the penalised maximum likelihood approach is shown to achieve it. Furthermore, it is shown that this rate cannot be attained by any (randomised) algorithm computable in polynomial time under a computational complexity assumption.