This paper analyzes the use of non-Gaussian input distributions over the Gaussian interference channel. It has been recently proved that the iid Gaussian code ensemble together with a decoder that treats interference as noise is sum-capacity achieving, if the interference is below a threshold. We show that, when the decoder treats interference as noise, and when the interference is above a threshold, the iid Gaussian ensemble can be strictly improved upon. In the block synchronous setting, the improvement is obtained by a Gaussian but non iid ensemble, whereas in the asynchronous setting, it is obtained by an iid but non Gaussian ensemble. The analysis of non-Gaussian ensembles is made possible by the use of the Hermite coordinate system.