I am trying to find a method with a low computational cost to compute the distance of a point $P$ and a space $S$ that is defined by the origin $O$ and $m$ vectors $v_1, v_2, ..., v_m$ in an $n$-dimensional space ($m<n$). The vectors are not restricted by any means other than that they are not 0. Furthermore, I would like to identify the point in $S$ that is closest to $P$.
This calculation is part of a 'fitting function' for a machine learning problem and thus has to be executed rather often and should be fast. The input to the function is as defined above, $P$ and $v_1, v_2, ..., v_m$. This is just for context and I am happy for a mathematical solution and can of course do the implementation myself.
Thanks in advance and please let me know if I need to specify anything in more detail.