LINE [1], as an efficient network embedding method, has shown its effectiveness in dealing with large-scale undirected, directed, and/or weighted networks. Particularly, it proposes to preserve both the local structure (represented by First-order Proximity) and global structure (represented by Second-order Proximity) of the network. In this study, we prove that LINE with these two proximities (LINE(1st) and LINE(2nd)) are actually factoring two different matrices separately. Specifically, LINE(1st) is factoring a matrix M (1), whose entries are the doubled Pointwise Mutual Information (PMI) of vertex pairs in undirected networks, shifted by a constant. LINE(2nd) is factoring a matrix M (2), whose entries are the PMI of vertex and context pairs in directed networks, shifted by a constant. We hope this finding would provide a basis for further extensions and generalizations of LINE.
@article{arxiv.1707.05926,
title = {Equivalence between LINE and Matrix Factorization},
author = {Qiao Wang and Zheng Wang and Xiaojun Ye},
journal= {arXiv preprint arXiv:1707.05926},
year = {2017}
}