Random phenomena are ubiquitous throughout science and engineering. Beyond the study of stochastic models in their own right, it is of importance in many applications to understand what information can be extracted on the basis of observed data. Mathematically, such ``data assimilation'' problems motivate the investigation of probabilistic phenomena that arise from conditioning. This topic has connections to several areas of probability, ergodic theory, measure theory, and statistical mechanics, as well as practical implications for the design and analysis of…

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree dependencies between neighboring nodes. In assortative networks the degree-degree dependencies are positive (nodes with similar degrees tend to connect to each other), while in disassortative networks these dependencies are negative. One of the problems with the commonly used Pearson correlation coefficient, also known as the assortativity coefficient, is that its magnitude decreases with the network size in…

Over the last decade we have witnessed the rapid proliferation of online networks and Internet activity. Such activity is considered as a blessing but it brings with it a large increase in risk of computer malware --- malignant software that actively spreads from one computer to another. To date, the majority of existing models of malware spread use stochastic behavior, where the set of neighbors infected from the current set of infected nodes is chosen obliviously. In this work, we…

Optimization problems are ubiquitous in contemporary engineering. A principal barrier to solving several real-world optimization problems is input uncertainty. In this talk, I will present new tools to study probabilistic instances of integer programs. As an application, I will show a phase-transition phenomenon in a simple distribution model for random integer programs. Our main tool is an elementary connection between integer programming and matrix discrepancy. I will describe this connection and derive matching upper and lower bounds on the discrepancy…

We consider the regression model with observation error in the design when the dimension can be much larger than the sample size and the true parameter is sparse. We propose two new estimators, based on linear and conic programming, and we prove that they satisfy oracle inequalities similar to those for the model with exactly known covariates. The only difference is that they contain additional scaling with the l1 or l2 norm of the true parameter. The scaling with the…

Abstract: We consider the problem of regression in three scenarios: (a) random design under the assumption that the model F is correctly specified, (b) distribution-free statistical learning with respect to a reference class F; and (c) online regression with no assumption on the generative process. The first problem is often studied in the literature on nonparametric estimation, the second falls within the purview of statistical learning theory, and the third is studied within the online learning community. It is recognized…

We are interested in the following question: suppose we generate a large graph according to the linear preferential attachment model---can we say anything about the initial (seed) graph? A precise answer to this question could lead to new insights for the diverse applications of the preferential attachment model. In this work we focus on the case of trees grown according to the preferential attachment model. We first show that the seed has no effect from a weak local limit point…

Given n vectors rj in n-space with all coefficients in one wants a vector x=(x1,...,xn) with all xi=+1 or −1 so that all dot products x⋅rj are at most Kn‾√ in absolute value, K an absolute constant. A random x would make x⋅rj roughly Gaussian but there would be outliers. The existence of such an x was first shown by the speaker, resolving a discrepancy question of Paul Erdős. However, the original argument did not give an effective algorithm. The…

Modern techniques in data accumulation and sensing have led to an explosion in both the volume and variety of data. Many of the resulting estimation problems are high-dimensional, meaning that the number of parameters to estimate can be far greater than the number of examples. A major focus of my work has been developing an understanding of how hidden low-complexity structure in large datasets can be used to develop computationally efficient estimation methods. I will discuss a framework for establishing…

In this talk we will introduce a new model of a growing sequence of random graphs where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied model of preferential attachment random graphs where higher degree vertices are preferred, we will call our model de-preferential attachment random graph model. We will consider two types of de-preferential attachment models,…