Multivariate Regression with Gross Errors on Manifold-valued Data
This addresses a bottleneck in practical scenarios like medical imaging where manifold-valued data is prone to gross errors, offering an incremental improvement over existing multivariate regression models.
The paper tackles multivariate regression with manifold-valued outputs corrupted by gross errors by proposing a model that corrects responses via geodesic curves and then performs linear regression, resulting in a nonconvex optimization problem addressed with the PALMR method. Empirically, it outperforms other models on synthetic and real diffusion tensor imaging data in handling gross errors and identifying them.
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.