A Novel Riemannian Conjugate Gradient Method on Quaternion Stiefel Manifold for Computing Truncated Quaternion Singular Value Decomposition

ABSTRACT A novel Riemannian conjugate gradient method on the quaternion Stiefel manifold is presented to solve the truncated quaternion singular value decomposition (QSVD) problem. It yields an efficient numerical implementation and can be feasibly embedded into modern machine learning methods for color image processing. The fundamental Riemannian geometry is built on a new structured Stiefel manifold, including tangent space, orthogonal projection, reaction, vector transport, and Riemannian gradient. Three new quaternion Riemannian conjugate gradient algorithms are presented, and their global convergence is proposed for the truncated QSVD computation. Numerical experiments on color image low‐rank approximation and color face recognition indicate the superiority of the newly proposed algorithms over the state‐of‐the‐art methods.