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I was wondering if there is a closed formular for the projection onto the intersection of the subspaces $Ax = b$ and $Zx = 0$. I know there is a closed formula for either one of those, but can you also project onto the interesection by use of the pseudoinverse?

I am aware of the alternating projection method, but this takes too long for my purposes.

Thanks!

Royi
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InspectorPing
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  • Since the intersection of affine subspaces is affine, is there any reason that this does not answer your question? https://math.stackexchange.com/questions/1320363/projection-of-z-onto-the-affine-set-x-mid-ax-b – aleph_two Dec 17 '18 at 04:23
  • Yes, what takes the place of $A$ in this case. Since my subspace is now the intersection of the spaces above, what is the matrix $A$ of which I can compute the generalized inverse now? It is neither $Ax = b$ nor $Zx = 0$ anymore. – InspectorPing Dec 17 '18 at 14:33

1 Answers1

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The problem is given by:

$$\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| x - z \right\|}_{2}^{2} \\ \text{subject to} \quad & A x = b \\ & Z x = 0 \end{aligned}$$

Which is equivalent of:

$$\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| x - z \right\|}_{2}^{2} \\ \text{subject to} \quad & C x = d \\ \end{aligned}$$

Where $ C = \begin{bmatrix} A \\ Z \end{bmatrix} $ and $ d = \begin{bmatrix} b \\ 0 \end{bmatrix} $.

Then the answer is given by $ x = z - {C}^{T} {\left( C {C}^{T} \right)}^{-1} \left( C z - d \right) $.

It can be easily generalized to more than 2 sub spaces.

Royi
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