论文标题
$ p $ norm及其应用
Riemannian optimization on unit sphere with $p$-norm and its applications
论文作者
论文摘要
本文以$ p $ norm的形式介绍了对单位领域的Riemannian优化，一般$ p> 1 $。作为欧几里得空间的Riemannian submanifold，提出和分析了几种几何工具，并提出了几种几何工具，并提出和分析了几种几何工具，例如撤回和矢量运输。还讨论了在球体上的应用，并通过非负约束和$ l_p $ - 重新化相关的优化进行了应用。作为实际示例，前者包括非负主成分分析，后者与套索的回归和盒子约束问题密切相关。数值实验验证了$ p $ norm在球体上优化此类应用的潜力很大，并且所提出的框架为这种优化提供了理论基础。
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.
