Abstract: In sparse principal components analysis (PCA), the task is to infer a sparse, low-rank matrix from noisy observations. Johnstone and Lu proposed the popular “spiked covariance” model, wherein the population distribution is equivariant with the exception of a single direction, called the spike. Assuming that the spike direction is sparse in some basis, they also proposed a simple scheme to estimate its support based on the diagonal entries of the sample covariance. Indeed, later information-theoretic analysis demonstrated that the…

We will present some results and some open problems related to shotgun assembly of graphs for random generating models.Shotgun assembly of graphs is the problem of recovering a random graph or a randomly labelled graphs from small pieces. This problem generalizes the theoretically elegant and practically important problem of shotgun assembly of DNA sequences. The general problem of shotgun assembly presents novel problems in random graphs, percolation, and random constraint satisfaction problems. Based on joint works with Nathan Ross, with…

Prediction problems typically assume the training data are independent samples, but in many modern applications samples come from individuals connected by a network. For example, in adolescent health studies of risk-taking behaviors, information on the subjects’ social networks is often available and plays an important role through network cohesion, the empirically observed phenomenon of friends behaving similarly. Taking cohesion into account should allow us to improve prediction. Here we propose a regression-based framework with a network penalty on individual node…