The well--studied separable form g=Xw, where w has independent entries and X is a fixed matrix, was solved, essentially completely, by Knowles and Yin. However, the separable case just scratches the surface of what should be true. We ask: what assumptions on g are really needed? We show that concentration of quadratic forms suffices for an optimal averaged local law, while a structural condition on cumulant tensors—interpolating between independence and generic dependence—suffices for the full anisotropic local law.
I'll discuss key examples where our assumptions can be verified: sign-invariant vectors, the 'random features model’, and some examples of spin-glass type. We will also outline limitations of the method, and suggest some key open problems.