Lectures on Ultra High Dimensional Regression
Tony Cai at Department of Biostatistics, Harvard University, April 16, 2010 |
The analysis of high-dimensional data now commonly arising in scientific investigations poses many statistical challenges not present in smaller scale studies. In these lectures I will discuss high-dimensional linear regression with large p and small n. This problem has attracted much recent interest in a number of fields including applied mathematics, electrical engineering, and statistics.
To provide a proper background and foundation for the main topics, we shall begin with discussions on the high-dimensional Gaussian sequence model. We then consider the linear model y = Xβ + z, where the dimension of the signal β is much larger than the number of observations. It is now well understood that l_{1} minimization methods provide effective ways for high dimensional sparse regression. I will present an elementary and unified analysis of l_{1} minimization methods including Lasso and the Dantzig Selector in three settings: noiseless, bounded error and Gaussian noise. Time permitting, I will also discuss l_{1} minimization approaches to sparse precision matrix estimation.
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