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Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.
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Approximating a subspace by sampling a base without replacement
Starting with the case where $n = m + 1$, if I understand correctly you can take a point sampled from the uniform distribution on the surface of the unit sphere in $p$-space and obtain $d(x,S)^2$ by i …