Abstract
In this paper, we develop asymptotic theories for a class of latent variable models for large-scale multi-relational networks. In particular, we establish consistency results and asymptotic error bounds for the (penalized) maximum likelihood estimators when the size of the network tends to infinity. The basic technique is to develop a non-asymptotic error bound for the maximum likelihood estimators through large deviations analysis of random fields. We also show that these estimators are nearly optimal in terms of minimax risk.
Highlights
A multi-relational network (MRN) describes multiple relations among a set of entities simultaneously
Our objective function usually has many global maximizers, but, empirically, we found the algorithm works well on MRN recovery and the recovery performance is insensitive to the choice of the starting point of stochastic gradient descent (SGD)
For a set V of valid triples, the prediction performance can be evaluated by rank-based criteria, mean rank (MR), mean reciprocal rank (MRR), and hits at q (Hits@q), which are defined as
Summary
A multi-relational network (MRN) describes multiple relations among a set of entities simultaneously. The probability Mijk is modeled as a function f of the embeddings, θ i and θ j, and a relation-specific parameter vector wk This is a natural generalization of the latent space model for single-relational networks [6]. The results in this paper fill in the void by studying the error bounds and asymptotic behaviors of the estimators for Mijk’s for a general class of models. This is a challenging problem due to the following facts. We show that the distribution of MRN can be consistently estimated in terms of average Kullback-Leibler (KL) divergence even when the latent dimension increases slowly as the sample size tends to infinity.
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