A formula for the area of a triangle: Useless, but explicitly in Deep Sets form

Abstract

Any permutation-invariant function of data points ${\vec{r}}_i$ can be written in the form $\rho ({\sum }_i\varphi ({\vec{r}}_i))$ for suitable functions $\rho$ and $\varphi$. This form - known in the machine-learning literature as Deep Sets - also generates a map-reduce algorithm. The area of a triangle is a permutation-invariant function of the locations ${\vec{r}}_i$ of the three corners $1\le i\le 3$. We find the polynomial formula for the area of a triangle that is explicitly in Deep Sets form. This project was motivated by questions about the fundamental computational complexity of $n$-point statistics in cosmology; that said, no insights of any kind were gained from these results.

arXiv comment11 pages, 1 figure