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Consider a sequence $(x_i)_{i \in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss

$$ \text{mode}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^0 $$

The median is the minimizing argument for the $L_1$ loss

$$ \text{median}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^1 $$

The arithmetic mean is the minimizing argument for the $L_2$ loss

$$ \text{arithmetic mean}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \in \mathbb{R}} \; \; \sum_{i \in I} \; | x_i - u|^2 $$

Can we find similar results for the harmonic mean or the geometric mean?

Thanks!

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A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean

$$\arg \min_{u \in \mathbb{R}^+} \sum_{i \in I}|\log \frac{x_i}{u}|^2$$

Interestingly, this might be generalised to complex values.

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