Georgia Institute of Technology College of Sciences

We study four discrete time stochastic systems on $\bbN$ modelingprocesses of rumour spreading. The involved individuals can eitherhave an active ora passive role, speaking up or asking for the rumour. The appetite inspreading or hearing the rumour is represented by a set of randomvariables whose distributionsmay depend on the individuals. Our goal is to understand - based on those randomvariables distribution - whether the probability of having an infiniteset of individuals knowing the rumour is positive or not.

Semimartingales constitute the larges class of "good integrators" for which Ito integral could reasonably be defined and the stochastic analysis machinery applied. In this talk we identify semimartingales within certain infinitely divisible processes. Examples include stationary (but not independent) increment processes, such as fractional and moving average processes, as well as their mixtures. Such processes are non-Markovian, often possess long range memory, and are of interest as stochastic integrators. The talk is based on a joint work with Andreas Basse-O'Connor.

 Let (X,Y) be a random couple with unknown distribution P, X being an observation and Y - a binary label to be predicted. In practice, distribution P remains unknown but the learning algorithm has access to the training data - the sample from P. It often happens that the cost of obtaining the training data is associated with labeling the observations while the pool of observations itself is almost unlimited. This suggests to measure the performance of a learning algorithm in terms of its label complexity, the number of labels required to obtain a classifier with the desired accuracy. Active Learning theory explores the possible advantages of this modified framework.We will present a new active learning algorithm based on nonparametric estimators of the regression function and explain main improvements over the previous work.Our investigation provides upper and lower bounds for the performance of proposed method over a broad class of underlying distributions. 

Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. The resulting inverse problems are often ill-posed because there are fewer measurements available than the ambient dimension of the model to be estimated. In practice, however, many interesting signals or models contain few degrees of freedom relative to their ambient dimension: a small number of genes may constitute the signature of a disease, very few parameters may specify the correlation structure of a time series, or a sparse collection of geometric constraints may determine a molecular configuration. Discovering, leveraging, or recognizing such low-dimensional structure plays an important role in making inverse problems well-posed. In this talk, I will propose a unified approach to transform notions of simplicity and latent low-dimensionality into convex penalty functions. This approach builds on the success of generalizing compressed sensing to matrix completion, and greatly extends the catalog of objects and structures that can be recovered from partial information. I will focus on a suite of data analysis algorithms designed to decompose general signals into sums of atoms from a simple---but not necessarily discrete---set. These algorithms are derived in a convex optimization framework that encompasses previous methods based on l1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices. I will provide sharp estimates of the number of generic measurements required for exact and robust recovery of a variety of structured models. These estimates are based on computing certain Gaussian statistics related to the latent model geometry. I will detail several example applications and describe how to scale the corresponding inference algorithms to very large data sets. (Joint work with Venkat Chandrasekaran, Pablo Parrilo, and Alan Willsky)