### Abstract

Graph representation learning is a ubiquitous task in machine learning where the goal is to embed each vertex into a low-dimensional vector space. We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution. The bipartite graph is assumed to be generated by a semiparametric exponential family distribution, whose parametric component is given by the proximity of outputs of two one-layer neural networks, while nonparametric (nuisance) component is the base measure. Neural networks take high-dimensional features as inputs and output embedding vectors. In this setting, the representation learning problem is equivalent to recovering the weight matrices. The main challenges of estimation arise from the nonlinearity of activation functions and the nonparametric nuisance component of the distribution. To overcome these challenges, we propose a pseudo-likelihood objective based on the rank-order decomposition technique and focus on its local geometry. We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Moreover, we prove that the sample complexity of the problem is linear in dimensions (up to logarithmic factors), which is consistent with parametric Gaussian models. However, our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.

Publication

*arXiv:2003.01013*

###### PhD Student

Sen Na is a PhD student in the Department of Statistics at The University of Chicago. Prior to graduate school, he obtained BS in mathematics at Nanjing University, China. His research interests lie in nonlinear and nonconvex optimization, dynamic programming, high-dimensional statistics and their interface.

###### MS Student (2019-2020)

Yuwei Luo received his M.S. degree in Statistics at University of Chicago in March 2020. Prior to graduate school, he received B.S.degree in Mathematics at University of Science and Technology of China (USTC) in June 2018. His research interests include reinforcement learning, control, optimization and network analysis.

###### Associate Professor of Econometrics and Statistics

Mladen Kolar is an Associate Professor of Econometrics and Statistics at the University of Chicago Booth School of Business. His research is focused on high-dimensional statistical methods, graphical models, varying-coefficient models and data mining, driven by the need to uncover interesting and scientifically meaningful structures from observational data.