For many real-world applications, we often need to select correlated variables---such as genetic variations and imaging features associated with Alzheimer's disease---in a high dimensional space. The correlation between variables presents a challenge to classical variable selection methods. To address this challenge, the elastic net has been developed and successfully applied to many applications. Despite its great success, the elastic net does not exploit the correlation information embedded in the data to select correlated variables. To overcome this limitation, we present a novel hybrid model, EigenNet, that uses the eigenstructures of data to guide variable selection. Specifically, it integrates a sparse conditional classification model with a generative model capturing variable correlations in a principled Bayesian framework. We develop an efficient active-set algorithm to estimate the model via evidence maximization. Experiments on synthetic data and imaging genetics data demonstrated the superior predictive performance of the EigenNet over the lasso, the elastic net, and the automatic relevance determination.