Highlighting data clusters by graph embedding

We propose a novel method, modularity embedding, to embed high-dimensional data or graphs in a low-dimensional space. Central to our work is a model that quantifies the relationship of two data points by their pairwise modular value. A larger value indicates a higher chance that they should be placed near to each other, and vice versa. The objective function of the model has a simple formulation of minimizing the sum of squared distances between data points weighted by pairwise modular values. It is naturally relaxed as a semi-definite program that learns a low-rank kernel matrix with only one linear constraint, which can be solved efficiently by modern mathematical optimization solvers. Compared with traditional graph embedding algorithms, the proposed method is shown to be able to highlight cluster structures inherent in high-dimensional data and graphs, which provides a promising tool in data analysis applications.