Abstract
With the abundance of high dimensional data in various disciplines, regularization
techniques are very popular these days. Despite the success of these techniques, some
challenges remain. One challenge is the development of efficient methods
incorporating structure information among predictors. Typically, the structure information
among predictors can be modeled by the connectivity of an undirected graph using
all predictors as nodes of the graph. In this talk, I will introduce an efficient
regularization technique incorporating graphical structure information among predictors.
Specifically, according to the undirected graph, we use a latent group lasso penalty to
utilize the graph node-by-node. The predictors connected in the graph are encouraged
to be selected jointly. This new regularization technique can be used for many
supervised learning problems. For sparse regression, our new method using the proposed
regularization technique includes adaptive Lasso, group Lasso, and ridge regression as
special cases. Theoretical studies show that it enjoys model selection consistency and
acquires tight finite sample bounds for estimation and prediction. For the multi-task
learning problem, our proposed graph-guided multi-task method includes the popular
l2,1-norm regularized multi-task learning method as a special case. Numerical studies
using simulated datasets and the Alzheimer's Disease Neuroimaging Initiative (ADNI)
dataset also demonstrate the effectiveness of the proposed methods. |