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Modes, Medians and Means: A Unifying Perspective defines the following centers based on the $L_p$ norms:

$$ \begin{aligned} \text{mode of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^0 \\ \text{median of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^1 \\ \text{mean of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^2 \\ \end{aligned} $$

Where:

$$ \begin{aligned} x = (x_1, x_2, \ldots, x_n) \ &:\ \text{a list of rational numbers} \\ \lvert x \rvert \ &:\ \text{number of elements in the list if $x$ is a list} \\ \end{aligned} $$

I wanted to extend this to the L4 center:

$$ \begin{aligned} \arg \min_s \sum_i \lvert x_i - s \rvert^4 = \text{L4 center} \\ \frac{d}{ds} \sum_i (x_i - s)^4 = 0 \\ \sum_i \left[ \frac{d}{ds}(x_i-s)^4 \right] = 0 \\ \sum_i \left[ -4(x_i-s)^3 \right] = 0 \\ \sum_i (x_i-s)^3 = 0 \\ \sum_i \left[ x_i^3 -3x_i^2s + 3x_is^2 - s^3 \right] = 0 \\ \sum_i x_i^3 + \sum_i -3x_i^2s + \sum_i 3x_is^2 + \sum_i -s^3 = 0 \\ \sum_i x_i^3 -3s \sum_i x_i^2 + 3s^2 \sum_i x_i - s^3 \lvert x \rvert = 0 \\ \end{aligned} $$

Which is a cubic with the following discriminant:

$$ \begin{aligned} &+ 162 \lvert x \rvert \sum_i x_i \sum_i x_i^2 \sum_i x_i^3 \\ &- 108 \left(\sum_i x_i\right)^3 \sum_i x_i^3 \\ &+ 18 \left(\sum_i x_i\right)^2 \left(\sum_i x_i^2\right)^2 \\ &- 108 \lvert x \rvert \left(\sum_i x_i^2\right)^3 \\ &- 27 \left(\lvert x \rvert\right)^2 \left(\sum_i x_i^3\right)^2 \end{aligned} $$

I was wondering if anything could be said about the roots of the L4 center given any list $x$ of rationals? For example, how many real roots, because intuitively I expect only one?

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  • $\begingroup$ Each term $(x_i-s)^4$ is a strictly convex function in $s$, so the sum $\sum_i$ is also strictly convex, so only one minimum...? $\endgroup$
    – Jukka Kohonen
    Commented Jan 6, 2022 at 13:34
  • $\begingroup$ @Jukka Kohonen I was thinking that if $x$ was a list with two clusters, then the midpoint of each cluster could each be a minimum and the midpoint between both clusters a maximum. It's too easy to imagine a sum of convex functions as $min(f_1,f_2)$ by forgetting there are no vertical asymptotes. Not something I was ever taught but easy to google or verify by plotting. $\endgroup$
    – user19087
    Commented Jan 7, 2022 at 2:26
  • $\begingroup$ Yes, it is a quite natural thought that this could happen, but as you can see from the strict convexity (or, from the fract that the derivative is strictly increasing as in Squala's answer), this cannot happen. There will be only one real minimum (of the sum of fourth powers) / only one real root (of the derivative). With small exponents things will be different. $\endgroup$
    – Jukka Kohonen
    Commented Jan 9, 2022 at 15:01

1 Answer 1

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The function $\sum_i(x_i-s)^3$ is strictly increasing in $s$, from $-\infty$ to $\infty$. It is also continuous, so there is a unique real root.

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  • $\begingroup$ I hadn't thought about it like this. So the sum is both strictly increasing and a cubic? Does this mean it is a real triple root (discriminant is zero and the inflection point has a zero tangent) or a unique real root (discriminant is greater than zero)? $\endgroup$
    – user19087
    Commented Jan 7, 2022 at 2:19
  • $\begingroup$ In general it is not a triple root, you would have a unique real root and two complex roots. $\endgroup$
    – Squala
    Commented Jan 7, 2022 at 11:33

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