Mahalanobis distance on extended Grassmann manifolds for variational pattern analysis

Solving Partial Least Squares Regression via Manifold Optimization Approaches

Partial least squares regression (PLSR) has been a popular technique to explore the linear relationship between two data sets. However, all existing approaches often optimize a PLSR model in Euclidean space and take a successive strategy to calculate all the factors one by one for keeping the mutually orthogonal PLSR factors. Thus, a suboptimal solution is often generated. To overcome the shortcoming, this paper takes statistically inspired modification of PLSR (SIMPLSR) as a representative of PLSR, proposes a novel approach to transform SIMPLSR into optimization problems on Riemannian manifolds, and develops corresponding optimization algorithms. These algorithms can calculate all the PLSR factors simultaneously to avoid any suboptimal solutions. Moreover, we propose sparse SIMPLSR on Riemannian manifolds, which is simple and intuitive. A number of experiments on classification problems have demonstrated that the proposed models and algorithms can get lower classification error rates compared with other linear regression methods in Euclidean space. We have made the experimental code public at https://github.com/Haoran2014.