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

TL;DR

This paper explores Riemannian optimization on the unit sphere with p-norms, developing geometric tools and demonstrating applications to constrained problems like nonnegative PCA and Lasso regression, supported by numerical experiments.

Contribution

It introduces a geometric framework for Riemannian optimization on p-norm spheres, including retractions and vector transports, with applications to nonnegative constraints and Lp-regularization.

Findings

Numerical experiments confirm the effectiveness of p-norm Riemannian optimization.

The framework provides a theoretical basis for constrained optimization problems.

Applications include nonnegative PCA and Lasso regression.

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.