Robust point set registration is a challenging problem, especially in the cases of noise, outliers, and partial overlapping. Previous methods generally formulate their objective functions based on the mean-square error (MSE) loss and, hence, are only able to register point sets under predefined constraints (e.g., with Gaussian noise). This article proposes a novel objective function based on a bidirectional kernel mean p -power error (KMPE) loss, to jointly deal with the above nonideal situations. KMPE is a nonsecond-order similarity measure in kernel space and shows a strong robustness against various noise and outliers. Moreover, a bidirectional measure is applied to judge the registration, which can avoid the ill-posed problem when a lot of points converges to the same point. In particular, we develop two effective optimization methods to deal with the point set registrations with the similarity and the affine transformations, respectively. The experimental results demonstrate the effectiveness of our methods.