Alessandro Rinaldo

On the asymptotic properties of the group lasso estimator for linear models

We establish estimation and model selection consistency, pre- diction and estimation boundsand persistencefor the group-lassoestimator and model selectorproposed by Yuan and Lin (2006) for least squares prob- lems when the covariates have a natural grouping structure. We consider the case of a fixed-dimensionalparameter space with increasing sample size and the double asymptotic scenario where the model complexity changes with the sample size.

Consistency of spectral clustering in stochastic block models

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as

On the geometry of discrete exponential families with application to exponential random graph models

\log n

Consistency under sampling of exponential random graph models

, with

Generalized density clustering

n

Autoregressive process modeling via the Lasso procedure

the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical

Properties and Refinements of The Fused Lasso

k

Maximum likelihood estimation in log-linear models

-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

Differential privacy and the risk-utility tradeoff for multi-dimensional contingency tables

There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with diculties in computing maximum likelihood estimates. The issues associated with these diculties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the clo- sure of k-dimensional exponential families of distribution with discrete base measure and polyhedral convex support P. We show that the normal fan of P is a geometric object that plays a fundamental role in deriving the statis- tical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. By means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy.

Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models

The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focusing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGMs expressive power. These results are actually special cases of more general results about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses.