Computationally and Statistically Efficient Estimation in High-Dimensions

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 the error behavior of a broad class of estimators under high-dimensional scaling. I will then discuss how to compute these estimates and draw connections between the statistical and computational properties of our methods. Interestingly, the same tools used to establish good high-dimensional estimation performance have a direct impact for optimization: better conditioned statistical problems lead to more efficient computational methods.