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We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$

For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of these have solutions which are computationally efficient to find. Is there a similar computationally efficient method for finding the general Lp estimator?

More specifically, we focus in $p>1$ as $0<p<1$ is non-convex. I imagine a closed form solution would be more efficient than naive search.

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  • $\begingroup$ what is the closed form for the median? $\endgroup$
    – mathworker21
    Commented Nov 8, 2023 at 2:54
  • $\begingroup$ @mathworker21 - I made an error in the post. Apologies. My intention was computationally efficient and translated that to closed-form. $\endgroup$
    – olivarb
    Commented Nov 8, 2023 at 7:58

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