A formula for the area of a triangle: Useless, but explicitly in Deep Sets form
arXiv:2503.22786v1 Announce Type: cross
Abstract: Any permutation-invariant function of data points $vec{r}_i$ can be written in the form $rho(sum_iphi(vec{r}_i))$ for suitable functions $rho$ and $phi$. 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 $1leq ileq 3$. 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.
Abstract: Any permutation-invariant function of data points $vec{r}_i$ can be written in the form $rho(sum_iphi(vec{r}_i))$ for suitable functions $rho$ and $phi$. 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 $1leq ileq 3$. 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.
Connor Hainje, David W. Hogg
Go to original source