# Hyperlink Regression via Bregman Divergence

## TL;DR

This paper introduces Bregman hyperlink regression (BHLR), a flexible framework for hyper-relational learning that predicts hyperlink weights from data tuples using Bregman divergence, with proven statistical consistency and computational efficiency.

## Contribution

It proposes BHLR, a unified and general approach for hyper-relational learning that encompasses existing methods and provides theoretical guarantees for consistency and tractability.

## Key findings

- BHLR is statistically consistent and asymptotically recovers true hyperlink weights.
- The framework is computationally tractable with stochastic optimization and novel minibatch sampling.
- It unifies and extends various existing hyper-relational learning methods.

## Abstract

A collection of $U \: (\in \mathbb{N})$ data vectors is called a $U$-tuple, and the association strength among the vectors of a tuple is termed as the \emph{hyperlink weight}, that is assumed to be symmetric with respect to permutation of the entries in the index. We herein propose Bregman hyperlink regression (BHLR), which learns a user-specified symmetric similarity function such that it predicts the tuple's hyperlink weight from data vectors stored in the $U$-tuple. BHLR is a simple and general framework for hyper-relational learning, that minimizes Bregman-divergence (BD) between the hyperlink weights and estimated similarities defined for the corresponding tuples; BHLR encompasses various existing methods, such as logistic regression ($U=1$), Poisson regression ($U=1$), link prediction ($U=2$), and those for representation learning, such as graph embedding ($U=2$), matrix factorization ($U=2$), tensor factorization ($U \geq 2$), and their variants equipped with arbitrary BD. Nonlinear functions (e.g., neural networks), can be employed for the similarity functions. However, there are theoretical challenges such that some of different tuples of BHLR may share data vectors therein, unlike the i.i.d. setting of classical regression. We address these theoretical issues, and proved that BHLR equipped with arbitrary BD and $U \in \mathbb{N}$ is (P-1) statistically consistent, that is, it asymptotically recovers the underlying true conditional expectation of hyperlink weights given data vectors, and (P-2) computationally tractable, that is, it is efficiently computed by stochastic optimization algorithms using a novel generalized minibatch sampling procedure for hyper-relational data. Consequently, theoretical guarantees for BHLR including several existing methods, that have been examined experimentally, are provided in a unified manner.