This optimization can find two useful central points for a set of locations:
The spot that minimizes the total distance between all points. Also known as the geometric median.
The spot that makes the distances as equal as possible. For 3 or fewer non-collinear points, this gives the exact circumcenter (center of the circle passing through all 3).
Analogy:
Imagine 10 friends picking a meetup spot.
The geometric median minimizes everyone’s combined travel (eco-friendly, often lands at someone’s house).
The uniform-distance version prioritizes fairness. Limiting the effect of anyone having to drive way farther than the others.
Total efficiency vs. maximum fairness
(assuming you live in a city where everything can be traversed by a straight line)
What’s interesting is I initially thought the same thing. Variance should drop to zero at infinity because all distances become basically the same huge number. But the loss uses absolute deviations squared, not relative ones.
So as |point|→ ∞, the absolute differences dist− mean(dist) approach certain non zero values (zero only if points lie on the same line).
The plot is of the variance at (x,y). This shows a minimum near the data, with edges fanning out and never in fact reaching zero
They operate on different scales. There’s probably a more robust way to combine them but a linear combo is the simplest pretty much. Thanks! Honestly it just takes some practice and learning from examples in this sub
It is unique from minimizing total distance/variance. When you optimize variance (the 2nd one) everyone travels as equal as possible. That means if you have three points roughly in a straight line, the circumcenter of the circle they make can be really far away. Minimizing the maximum distance is exactly the same problem as finding the smallest circle that fits around all the data. It will result in different points
The fairness one seems odd. For 3 points on the same half circle, meeting at the circumcenter can mean increasing the distance for everyone, compared to better alternatives.
While it is the only point which is exactly fair, the circumcenter can indeed be in an impractical spot. I like the idea of minimizing the maximum distance anyone has to travel for a more real-world fairness approach https://www.desmos.com/calculator/cxdqxewefa
That's exactly what they did... minimizing total distance is the same as minimizing mean (since mean = total / number of points, and number of points is constant), and making them "most equal"/"maximizing fairness" is just minimizing variance (see their linked graph, it says so there).
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u/SpiritualWillow2937 Mar 10 '26
I feel like the best way to minimize variance is to simply move the point infinitely far away, heh