One-Hot Graph Encoder Embedding.

In this article we propose a lightning fast graph embedding method called one-hot graph encoder embedding. It has a linear computational complexity and the capacity to process billions of edges within minutes on standard PC - making it an ideal candidate for huge graph processing. It is applicable to either adjacency matrix or graph Laplacian, and can be viewed as a transformation of the spectral embedding. Under random graph models, the graph encoder embedding is approximately normally distributed per vertex, and asymptotically converges to its mean. We showcase three applications: vertex classification, vertex clustering, and graph bootstrap. In every case, the graph encoder embedding exhibits unrivalled computational advantages.

The Chi-Square Test of Distance Correlation.

Distance correlation has gained much recent attention in the data science community: the sample statistic is straightforward to compute and asymptotically equals zero if and only if independence, making it an ideal choice to discover any type of dependency structure given sufficient sample size. One major bottleneck is the testing process: because the null distribution of distance correlation depends on the underlying random variables and metric choice, it typically requires a permutation test to estimate the null and compute the p-value, which is very costly for large amount of data. To overcome the difficulty, in this paper we propose a chi-square test for distance correlation. Method-wise, the chi-square test is non-parametric, extremely fast, and applicable to bias-corrected distance correlation using any strong negative type metric or characteristic kernel. The test exhibits a similar testing power as the standard permutation test, and can be utilized for K-sample and partial testing. Theory-wise, we show that the underlying chi-square distribution well approximates and dominates the limiting null distribution in upper tail, prove the chi-square test can be valid and universally consistent for testing independence, and establish a testing power inequality with respect to the permutation test.

A Simple Spectral Failure Mode for Graph Convolutional Networks.

Neural networks have achieved remarkable successes in machine learning tasks. This has recently been extended to graph learning using neural networks. However, there is limited theoretical work in understanding how and when they perform well, especially relative to established statistical learning techniques such as spectral embedding. In this short paper, we present a simple generative model where unsupervised graph convolutional network fails, while the adjacency spectral embedding succeeds. Specifically, unsupervised graph convolutional network is unable to look beyond the first eigenvector in certain approximately regular graphs, thus missing inference signals in non-leading eigenvectors. The phenomenon is demonstrated by visual illustrations and comprehensive simulations.

Discovering and deciphering relationships across disparate data modalities.

Understanding the relationships between different properties of data, such as whether a genome or connectome has information about disease status, is increasingly important. While existing approaches can test whether two properties are related, they may require unfeasibly large sample sizes and often are not interpretable. Our approach, 'Multiscale Graph Correlation' (MGC), is a dependence test that juxtaposes disparate data science techniques, including k-nearest neighbors, kernel methods, and multiscale analysis. Other methods may require double or triple the number of samples to achieve the same statistical power as MGC in a benchmark suite including high-dimensional and nonlinear relationships, with dimensionality ranging from 1 to 1000. Moreover, MGC uniquely characterizes the latent geometry underlying the relationship, while maintaining computational efficiency. In real data, including brain imaging and cancer genetics, MGC detects the presence of a dependency and provides guidance for the next experiments to conduct.

Robust Vertex Classification.

For random graphs distributed according to stochastic blockmodels, a special case of latent position graphs, adjacency spectral embedding followed by appropriate vertex classification is asymptotically Bayes optimal; but this approach requires knowledge of and critically depends on the model dimension. In this paper, we propose a sparse representation vertex classifier which does not require information about the model dimension. This classifier represents a test vertex as a sparse combination of the vertices in the training set and uses the recovered coefficients to classify the test vertex. We prove consistency of our proposed classifier for stochastic blockmodels, and demonstrate that the sparse representation classifier can predict vertex labels with higher accuracy than adjacency spectral embedding approaches via both simulation studies and real data experiments. Our results demonstrate the robustness and effectiveness of our proposed vertex classifier when the model dimension is unknown.