Riemannian optimization on unit sphere with $p$-norm and its applications

Abstract: This paper deals with Riemannian optimization on the unit sphere in terms of $p$-norm with general $p > 1$. As a Riemannian submanifold of the Euclidean space, the geometry of the sphere with $p$-norm is investigated, and several geometric tools used for Riemannian optimization, such as retractions and vector transports, are proposed and analyzed. Applications to Riemannian optimization on the sphere with nonnegative constraints and $L_p$-regularization-related optimization are also discussed. As practical examples, the former includes nonnegative principal component analysis and the latter is closely related to the Lasso regression and box-constrained problems. Numerical experiments verify that Riemannian optimization on the sphere with $p$-norm has substantial potential for such applications, and the proposed framework provides a theoretical basis for such optimization.