  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.
