Title: Multivariate Regression with Gross Errors on Manifold-valued Data,
Abstract: We consider the topic of multivariate regression on manifold-valued output,
that is, for a multivariate observation, its output response lies on a
manifold. Moreover, we propose a new regression model to deal with the presence
of grossly corrupted manifold-valued responses, a bottleneck issue commonly
encountered in practical scenarios. Our model first takes a correction step on
the grossly corrupted responses via geodesic curves on the manifold, and then
performs multivariate linear regression on the corrected data. This results in
a nonconvex and nonsmooth optimization problem on manifolds. To this end, we
propose a dedicated approach named PALMR, by utilizing and extending the
proximal alternating linearized minimization techniques. Theoretically, we
investigate its convergence property, where it is shown to converge to a
critical point under mild conditions. Empirically, we test our model on both
synthetic and real diffusion tensor imaging data, and show that our model
outperforms other multivariate regression models when manifold-valued responses
contain gross errors, and is effective in identifying gross errors.