Minimax estimation in linear models with unknown finite alphabet design

We provide minimax theory for joint estimation of $F$ and $\omega$ in linear models $Y = F \omega + Z$ where the parameter matrix $\omega$ and the design matrix $F$ are unknown but the latter takes values in a known finite set. This allows to separate $F$ and $\omega$, a task which is not doable, in general. We obtain in the noiseless case, i.e., $Z = 0$, stable recovery of $F$ and $\omega$ from the linear model. Based on this, we show for Gaussian error matrix $Z$ that the LSE attains minimax rates for the prediction error for $F \omega$. Notably, these are exponential in the dimension of one component of $Y$. The finite alphabet allows estimation of $F$ and $\omega$ itself and it is shown that the LSE achieves the minimax rate. As computation of the LSE is not feasible, an efficient algorithm is proposed. Simulations suggest that this approximates the LSE well.