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Suppose there is an $n \times n$ real symmetric matrix $X$. It is easy to project it onto the positive semidefinite cone $\mathcal{S}_n^+$. We can just apply the the eigenvalue decomposition to obtain $X = U S U^T$, where $S$ is the eigenvalue matrix, then we can project $S$ onto $S_+$ by changing all the negative eigenvalues in $S$ to be zeros. Then the projected matrix $\bar X $ equals to $US_+U^T$. $\bar X$ is the "closest" matrix to $X$ such that $\bar X \succeq 0$. ($\bar X \in \mathcal{S}_n^+$)

Now I want to project it to a more complicated cone, i.e., a face of cone. Let's denote the cone by $\mathcal{K}$, suppose $V$ is given, then $$\mathcal{K} = \left\{Z \in \mathcal{S}_n, Z = V P V^T, P \in \mathcal{S}_d^+, n > d \right\}$$ where $\mathcal{S}_n$ means the cone of $n \times n$ real symmetric matrices, $\mathcal{S}_d^+$ denotes the cone of $d \times d$ real symmetric positive semidefinite matrices, and $V$ is an $n \times d$ matrix. Therefore $Z$ will not be full rank, i.e., $Z$ is on the boundary (face) of the positive semidefinite cone. I would like to project $X$ to the cone $\mathcal{K}$ so the projected $\bar X$ is the "closest" one to $X$.

I am not sure how to do this projection similar to the first paragraph. Does there exist a formula or algorithm for doing this? I know I can use the matlab toolbox but that is an iterative algorithm which is not exact and also costly. Any ideas or suggestions would be greatly appreciated.

Rodrigo de Azevedo
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Pew
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  • Is the situation exactly as stated? So $V$ is given and $\mathcal{K}$ depends on $V$? Or is $\mathcal{K}$ the set of positive semidefinite matrices with rank $\le d$? – gerw Dec 18 '15 at 21:48
  • Yes, $\mathcal{K}$ depends on $V$, $\mathcal{K}$ is the set of positive semidefinite matrices such that the matrices of has null space $B$ since we require $V^T B = 0$. If $\mathcal{K}$ is the set of psd matrices with rank $ k \leq d$ then we can just select the $d$ greatest eigenvalues to do the projection – Pew Dec 19 '15 at 00:47

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Without loss of generality, $V^TV=I_d$ (if not do $QR$ decomposition or similar to get an orthonormal basis for the span of $V$). Note that by changing basis $\mathcal{K}$ is equivalent to the intersection of the positive semidefinite cone with block diagonal matrices. $$ Z \in \mathcal{K} \Leftrightarrow QZQ^T = \begin{pmatrix} A &0\\0 &0 \end{pmatrix}, A \in S_d^+ $$

So to project onto $\mathcal{K}$, first we calculate $V^TZV$ (equivalent to extracting out the $d \times d$ submatrix), project it onto $S_d^+$, then expand back onto $S_n$. That is, denoting projecting onto $C$ by $\Pi_C$:

$$ \Pi_\mathcal{K}(Z) = V[\Pi_\mathcal{S_d^+}(V^TZV)]V^T\\ $$

p.s.
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