A collection of $U(\in \mathbb{N})$ data vectors $\boldsymbol{X}_{\boldsymbol{i}}:=(\boldsymbol{x}_{i_1},\boldsymbol{x}_{i_2},\ldots,\boldsymbol{x}_{i_U}) \in \mathcal{X}^{U} \quad (\mathcal{X} \subset \mathbb{R}^p)$ is called a $U$-tuple, and the association strength $w_{\boldsymbol{i}}$ among the vectors of a tuple $\boldsymbol{X}_{\boldsymbol{i}}$ is termed as the hyperlink weight, that is assumed to be symmetric with respect to permutation of the entries in the index $\boldsymbol{i}=(i_1,i_2,\ldots,i_U)$.

EX_tuples
Figure: $U=3$; for $\boldsymbol{i}=(2,3,5)$, $w_{\boldsymbol{i}}=w_{235}(=w_{253}=w_{325}=w_{352}=w_{523}=w_{532})$ represents the hyperlink weight defined for the $3$-tuple $\boldsymbol{X}_{\boldsymbol{i}}=(\boldsymbol{x}_2,\boldsymbol{x}_3,\boldsymbol{x}_5)$.

Okuno and Shimodaira (arXiv:1908.02573, submitted) proposes Bregman hyperlink regression (BHLR), which learns a user-specified symmetric similarity function $\mu_{\boldsymbol{\theta}}:\mathcal{X}^U \to \mathcal{S}(\subset \mathbb{R})$ such that it predicts the tuple's hyperlink weight from data vectors stored in the $U$-tuple. Nonlinear functions, such as neural networks, can be employed for the similarity function. BHLR is based on Bregman divergence (BD) and encompasses various existing methods such as logistic regression ($U=1$), Poisson regression ($U=1$), graph embedding ($U=2$), matrix factorization ($U=2$), tensor factorization ($U \ge 2$), and their variants equipped with arbitrary BD. We demonstrate that, regardless of the choice of BD and $U \in \mathbb{N}$, the proposed BHLR is generally (P-1) robust against the distributional misspecification, that is, it asymptotically recovers the underlying true conditional expectation of hyperlink weights given data vectors regardless of its conditional distribution, 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. Furthermore, a theoretical guarantee for the optimization is presented. Numerical experiments demonstrate the promising performance of the proposed BHLR.