We consider multi-target linear regression in the presence of a deployment-time attacker. Specifically, we examine the effects of incorporating a maximum-rank constraint on the learned weight matrix (i.e., performing reduced-rank regression). For a broad class of practically relevant defender and attacker settings, we derive theoretical bounds on the change in adversarial robustness as a function of the rank constraint. In the classical setting—where the learner minimizes Mean Squared Error and the attacker is constrained by an \(\ell _2\) constraint—we show that adversarial robustness is unaffected by rank reduction. In contrast, under more general and practical settings, rank constraints can dramatically alter robustness. In general, these bounds depend on the eigenvalues of matrix \(\textbf{W}\) , which defines defender loss. These bounds and accompanying analysis provide both practical value and further develop a foundational understanding of the robustness of linear methods.