I was having a conversation on HN a few days ago about how the mean of a set can be seen as minimizer of an optimization problem -- roughly, it can be interpret

I was having a conversation on HN a few days ago about how the mean of a set can be seen as minimizer of an optimization problem -- roughly, it can be interpreted as that location that minimizes the 'distance' to all other points in a set. By choosing the 'distance' appropriately one can get all sorts of fun generalizations that includes all mentioned in the post. The familiar squared Euclidean distance yields the familiar arithmetic mean. One fruitful family of such 'distances' that lead to arithmetic mean, geometric mean, Holder's mean [1] and beyond etc is the Bregman divergence [0]. The sort of means that fall out from it have the form f⁻¹ ( 1/n ∑ᵢ₌₁ⁿ f(xᵢ) ) where f is a smooth monotonically increasing function. By f⁻¹ I mean the inverse function, not the reciprocal. [0] https://en.wikipedia.org/wiki/Bregman_divergence https://en.wikipedia.org/wiki/Bregman_divergence [1] https://en.wikipedia.org/wiki/Generalized_mean https://en.wikipedia.org/wiki/Generalized_mean [2] https://news.ycombinator.com/item?id=25221587 https://news.ycombinator.com/item?id=25221587