Georgia Institute of Technology College of Sciences

We study the problem of estimating the left and right singular subspaces for a collection of heterogeneous random graphs with a shared common structure. We analyze an algorithm that first estimates the orthogonal projection matrices corresponding to these subspaces for each individual graph, then computes the average of the projection matrices, and finally finds the matrices whose columns are the eigenvectors corresponding to the d largest eigenvalues of the sample averages. We show that the algorithm yields an estimate of the left and right singular vectors whose row-wise fluctuations are normally distributed around the rows of the true singular vectors. We then consider a two-sample hypothesis test for the null hypothesis that two graphs have the same edge probabilities matrices against the alternative hypothesis that their edge probabilities matrices are different. Using the limiting distributions for the singular subspaces, we present a test statistic whose limiting distribution converges to a central chi-square (resp. non-central chi-square) under the null (resp. alternative) hypothesis. Finally, we adapt the theoretical analysis for multiple networks to the setting of distributed PCA; in particular, we derive normal approximations for the rows of the estimated eigenvectors using distributed PCA when the data exhibit a spiked covariance matrix structure.

We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell_{p}$ norm. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p\ge 2$, there exists a critical inverse temperature $\beta_{c}(p)$ such that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. On the other hand, for $\beta>\beta_{c}(p)$, the magnetization is not concentrated at zero similar to the classical case. We further generalize the model for general symmetric spin distributions and prove similar phase transition. In this talk, we discuss a brief overview of classical Curie-Weiss model, a generalized Hubbard-Stratonovich transforms, and how we apply the transform to Curie-Weiss model under $\ell^p$ constraint. This is based on joint work with Partha Dey.