# Multivariate Regression with Gross Errors on Manifold-valued Data

## TL;DR

This paper introduces a novel multivariate regression model for manifold-valued data that effectively handles gross errors by correcting responses via geodesic curves, and employs a specialized optimization algorithm with proven convergence.

## Contribution

The paper proposes PALMR, a new approach for robust multivariate regression on manifolds with gross errors, extending proximal alternating linearized minimization techniques.

## Key findings

- Outperforms existing models on synthetic data.
- Effective in identifying gross errors in diffusion tensor imaging.
- Converges to a critical point under mild conditions.

## 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.