We introduce a new methodology for post-silicon characterization of the gate-level variations in a manufactured Integrated Circuit (IC). The estimated characteristics are based on the power and the delay measurements that are affected by the process variations. The power (delay) variations are spatially correlated. Thus, there exists a basis in which variations are sparse. The sparse representation suggests using the L1-regularization (the compressive sensing theory). We show how to use the compressive sensing theory to improve post-silicon characterization. We also address the problem by adding spatial constraints directly to the traditional L2-minimization. The proposed methodology is fast, inexpensive, non-invasive, and applicable to legacy designs. Noninvasive IC characterization has a range of emerging applications, including post-silicon optimization, IC identification, and variations' modeling/simulations. The evaluation results on standard benchmark circuits show that, in average, the gate level characteristics estimation accuracy can be improved by more than two times using the proposed methods.