论文标题
投射到希尔伯特空间中的矩形双曲线抛物面
Projecting onto rectangular hyperbolic paraboloids in Hilbert space
论文作者
论文摘要
在$ \ mathbb {r}^3 $中，双曲线抛物面是一个经典的鞍形四个表面。最近，埃尔瑟（Elser）模拟了在$ \ mathbb {r}^n $中使用矩形双曲线抛物线的深度学习中产生的问题。在他的工作的激励下，我们对相关预测进行了严格的分析。在某些情况下，发现此投影等于找到五五季度或立方多项式的某种根。我们还观察到投影不是单胎，并指出与图形和集合的连接。
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.
