The <coordinatewise median> suffers a contamination error of order $\epsilon\sqrt d$ with constant probability. In contrast, the <Tukey median> under an isotropic Gaussian model satisfies a high-probability Euclidean error bound of order
$$
\epsilon+\sqrt{\frac dn}+\sqrt{\frac{\log(1/\delta)}n}
$$
with probability at least $1-\delta$, up to universal constants. Its contamination term is dimension-free. Thus coordinatewise estimation loses a factor $\sqrt d$ in its dependence on adversarial contamination.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 4 / c / Solution

The coordinatewise median of vectors applies the univariate <sample median> separately to each coordinate. It is easy to compute but depends on the chosen coordinate axes.

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