Projecting onto rectangular hyperbolic paraboloids in Hilbert space

Heinz H. Bauschke, Manish Krishan Lal, Xianfu Wang
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Abstract

In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence.
投影到希尔伯特空间中的直角双曲抛物面上
在$\mathbb{R}^3$中,双曲抛物面是一个经典的鞍形二次曲面。最近,Elser在$\mathbb{R}^n$中使用矩形双曲抛物面对深度学习中出现的问题进行了建模。在他工作的激励下,我们对相关的投影进行了严格的分析。在某些情况下,找到这个投影等于找到一个五次或三次多项式的某个根。我们还观察了投影不是单态的情况,并指出了与图形和集合收敛的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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