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

2025/03/28 by Connor Hainje, David W. Hogg, Hainje, Connor +1 · 2 voices
Mathematics · Physics and Astronomy · #Cosmology and Nongalactic Astrophysics (astro-ph.CO) #FOS: Computer and information sciences #FOS: Physical sciences #Machine Learning (stat.ML) #astro-ph.CO #stat.ML

paper · pdf · doi:10.48550/arxiv.2503.22786

Abstract

Any permutation-invariant function of data points r_i can be written in the form ρ(∑_iφ(r_i)) for suitable functions ρ and φ. 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 r_i of the three corners 1≤ i≤ 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.

Discussions

• A formula for the area of a triangle: Useless, but explicitly in Deep Sets form [hn, 3 points, 0 comments]
• In a tight competition, my favourite new arXiv article today is this: www.arxiv.org/abs/2503.22786 "A formula for the area of a triangle: Useless, but explicitly in Deep Sets form." Can't beat the bru [bsky, 0 points, 0 comments]

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