this mixture happens in a linear fashion so if you optimize for and you not only optimize for the training set but you optimize for every possible mixture of the training set a linear mixture your minimization your minimizer function will actually become a linear function it's not surprising but they have a formal proof of this and they also have a proof that if certain assumptions are given then the minimizer's if you apply the minimizer's the hidden representations will actually fall on a low dimensional subspace which is also