The geometric median $\boldsymbol{z}$ of a set of vectors $\boldsymbol{x}^{(1)}, \ldots, \boldsymbol{x}^{(n)}$ minimizes the total Euclidean distance:

\[\boldsymbol{z} \in \arg\min_{\boldsymbol{y} \in \mathbb{R}^d} \sum_{i=1}^{n} \left\| \boldsymbol{y} - \boldsymbol{x}^{(i)} \right\|_2.\]

This makes it a robust alternative to the mean when data includes outliers.

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📚 This explanation is part of the Aalto Dictionary of Machine Learning — an open-access multi-lingual glossary developed at Aalto University to support accessible and precise communication in ML.