Efficient inference in stochastic block models with vertex labels

We study the stochastic block model with two communities where vertices contain side information in the form of a vertex label. These vertex labels may have arbitrary label distributions, depending on the community memberships. We analyze a linearized version of the popular belief propagation algorithm. We show that this algorithm achieves the highest accuracy possible whenever a certain function of the network parameters has a unique fixed point. Whenever this function has multiple fixed points, the belief propagation algorithm may not perform optimally. We show that increasing the information in the vertex labels may reduce the number of fixed points and hence lead to optimality of belief propagation

Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap

A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model where the cluster means are sparse vectors. Here we provide an exact asymptotic characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity, i.e. when the fraction of non-zero components of the cluster means $\rho$, as well as the ratio $\alpha$ between the number of samples and the dimension are fixed, while the dimension diverges. We identify the information-theoretic threshold below which obtaining a positive correlation with the true cluster means is statistically impossible. Additionally, we investigate the performance of the approximate message passing (AMP) algorithm analyzed via its state evolution, which is conjectured to be optimal among polynomial algorithm for this task. We identify in particular the existence of a statistical-to-computational gap between the algorithm that require a signal-to-noise ratio $\lambda_{\text{alg}} \ge k / \sqrt{\alpha}$ to perform better than random, and the information theoretic threshold at $\lambda_{\text{it}} \approx \sqrt{-k \rho \log{\rho}} / \sqrt{\alpha}$. Finally, we discuss the case of sub-extensive sparsity $\rho$ by comparing the performance of the AMP with other sparsity-enhancing algorithms, such as sparse-PCA and diagonal thresholding.Comment: NeurIPS camera-ready versio

Statistical Mechanics of Generalization In Graph Convolution Networks

Graph neural networks (GNN) have become the default machine learning model for relational datasets, including protein interaction networks, biological neural networks, and scientific collaboration graphs. We use tools from statistical physics and random matrix theory to precisely characterize generalization in simple graph convolution networks on the contextual stochastic block model. The derived curves are phenomenologically rich: they explain the distinction between learning on homophilic and heterophilic graphs and they predict double descent whose existence in GNNs has been questioned by recent work. Our results are the first to accurately explain the behavior not only of a stylized graph learning model but also of complex GNNs on messy real-world datasets. To wit, we use our analytic insights about homophily and heterophily to improve performance of state-of-the-art graph neural networks on several heterophilic benchmarks by a simple addition of negative self-loop filters

Phase transitions in spiked matrix estimation: information-theoretic analysis

We study here the so-called spiked Wigner and Wishart models, where one observes a low-rank matrix perturbed by some Gaussian noise. These models encompass many classical statistical tasks such as sparse PCA, submatrix localization, community detection or Gaussian mixture clustering. The goal of these notes is to present in a unified manner recent results (as well as new developments) on the information-theoretic limits of these spiked matrix models. We compute the minimal mean squared error for the estimation of the low-rank signal and compare it to the performance of spectral estimators and message passing algorithms. Phase transition phenomena are observed: depending on the noise level it is either impossible, easy (i.e. using polynomial-time estimators) or hard (information-theoretically possible, but no efficient algorithm is known to succeed) to recover the signal.Comment: These notes present in a unified manner recent results (as well as new developments) on the information-theoretic limits in spiked matrix estimatio