Projecting onto rectangular hyperbolic paraboloids in Hilbert space

TL;DR

This paper rigorously analyzes the projection onto rectangular hyperbolic paraboloids in Hilbert space, revealing polynomial root-finding challenges and connections to convergence properties, motivated by deep learning applications.

Contribution

It provides a detailed mathematical analysis of projections onto hyperbolic paraboloids in high-dimensional spaces, including polynomial root solutions and convergence insights.

Findings

Projection involves solving cubic or quintic polynomials.

Conditions when the projection is not unique are identified.

Connections to graphical and set convergence are established.

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.