Lp estimate via Newton’s method

We can obtain the value of the Lp estimates of central tendency described in Lp Estimators by using Newton’s method. Care must be taken since the function ||X–x||p is not differentiable at x = Lp(X).

Our goal is to find the value of x that minimizes ||X–x||p. It is sufficient to minimize the function

f(x)

This occurs when

f'(x) = 0

which is equivalent to

g(x) = 0

The derivative of g(x) is

g'(x)

for p ≠ 1.

Thus by Newton’s method, we iterate until convergence to find the value x = x(k) where

Newton's estimate

References

Zornoza, J. (2020) Distance metric for machine learning. Aigents
https://aigents.co/data-science-blog/publication/distance-metrics-for-machine-learning

Wikipedia (2020) Minkowski distance
https://en.wikipedia.org/wiki/Minkowski_distance

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