High dimensional estimation and data analysis: Entropy and regularized regression

High-dimensional data are a frequent occurrence in many areas of application of statistics. For example, the analysis of data from neuroscience often involves fundamentally high-dimensional variables such as natural images or patterns of spiking in neural spike trains. These applications are often concerned with the relationship between these variables and another variate. What is the strength of the relationship? What is the nature of the relationship? This work is concerned with some of the statistical challenges in high-dimensional data analysis that arise when answering these questions; it is grounded in applications to data problems in neuroscience, and examines some challenges in entropy estimation and regularized regression that arise there.