Projections On Affine Sets
Why
What is the projection of a vector onto an affine set?
Result
Suppose $a \in \R ^n$ and $U \in \R ^{n
\times k}$ with $U^\top U = I$.
Define the affine set $W(a, U) = a +
\range(U)$.
Then
\[ \textstyle \proj_{W(a, U)}(x) = UU^\top x + (I - UU^\top )a. \]
The minimizer of $J: \R ^n \to \R $ defined by
\[ J(z) = \norm{a + Uz - x} = \norm{Uz - (x -a)}, \]
is $z^\star = U^\top (x-a)$. So the projection of $x$ onto $W(a, U)$ is\[ a + Uz^\star = a + UU^\top (x - a) = UU^\top x + (I - UU^\top )a. \]