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Real Affine Set Representations
Real Vector Projections
Projections On Subspaces
Real Matrix Range
Needed by:
Minimum Residual Affine Sets
Minimum Residual Affine Sets
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Projections On Affine Sets


What is the projection of a vector onto an affine set?


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

\[ a + Uz^\star = a + UU^\top (x - a) = UU^\top x + (I - UU^\top )a. \]

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