  We consider the problem of estimating the mean of a noisy vector. When the
mean lies in a convex constraint set, the least squares projection of the
random vector onto the set is a natural estimator. Properties of the risk of
this estimator, such as its asymptotic behavior as the noise tends to zero,
have been well studied. We instead study the behavior of this estimator under
misspecification, that is, without the assumption that the mean lies in the
constraint set. For appropriately defined notions of risk in the misspecified
setting, we prove a generalization of a low noise characterization of the risk
due to Oymak and Hassibi in the case of a polyhedral constraint set. An
interesting consequence of our results is that the risk can be much smaller in
the misspecified setting than in the well-specified setting. We also discuss
consequences of our result for isotonic regression.
