Sparsity is often used to improve the interpretability of a statistical analysis and/or reduce the variance of a statistical estimator. This talk will explore another aspect—the utility of sparsity in model identifiability through two problems motivated by genetics applications. The first problem is about removing “batch effects” or latent confounders in multiple hypothesis testing. I will present a general framework called Confounder Adjusted Testing and Estimation (CATE) we proposed to unify several widely used but ad hoc proposals. If the latent confounders are strong enough and the signals are sparse enough, CATE can be as powerful as the oracle estimator which observes the latent confounders. The second problem is about tackling invalid instrumental variables in Mendelian randomization. I will describe a new statistical framework called Robust Adjusted Profile Score (MR-RAPS) which can provide efficient and robust inference in such problems, by exploiting weak genetic instruments and limiting the importance of invalid instruments. Finally, connections to related work and potential future work will be discussed.