Papers › A formula for the area of a triangle: Useless, but explicitly in Deep Sets form
A formula for the area of a triangle: Useless, but explicitly in Deep Sets form
Connor Hainje, David W. Hogg
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Any permutation-invariant function of data points r⃗ᵢ can be written in the form ρ(∑ᵢϕ(r⃗ᵢ)) 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 r⃗ᵢ 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.
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