# Projecting onto rectangular hyperbolic paraboloids in Hilbert space

arXiv: https://arxiv.org/abs/2206.04878 · HTML: https://mlsanity.com/paper/2206.04878/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2022.

## Built on by (most cited first, 2 of 2)

- [Learning with Boolean threshold functions](https://mlsanity.com/paper/2602.17493/) (2026)
- [A projection-based framework for gradient-free and parallel learning](https://mlsanity.com/paper/2506.05878/) (2025)

## Abstract (arXiv)

In ℝ3, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in ℝn. 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.