Nearest centroid classifier

Given labeled training samples ${\displaystyle \textstyle \{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}}$ with class labels ${\displaystyle y_{i}\in \mathbf {Y} }$, compute the per-class centroids ${\displaystyle \textstyle {\vec {\mu }}_{\ell }={\frac {1}{|C_{\ell }|}}{\underset {i\in C_{\ell }}{\sum }}{\vec {x}}_{i}}$ where ${\displaystyle C_{\ell }}$ is the set of indices of samples belonging to class ${\displaystyle \ell \in \mathbf {Y} }$.

The class assigned to an observation ${\displaystyle {\vec {x}}}$ is ${\displaystyle {\hat {y}}={\arg \min }_{\ell \in \mathbf {Y} }\|{\vec {\mu }}_{\ell }-{\vec {x}}\|}$.