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I'd like to solve the following minimization problem:

$$ \min_{\Phi^T D \Phi = I_k} \frac{1}{2}\|\Phi - Y\|_F^2, $$

where $\Phi \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{n \times n}$ is a (diagonal) positive definite matrix, $I_k \in \mathbb{R}^{k \times k}$ is the identity matrix, and $Y \in \mathbb{R}^{n \times k}$ with $n>k$.

I think I should be able to reduce this to an orthogonal Procrustes problem, but I'm getting tripped up when I try to use the substitution $\Phi = D^{-1/2}\,\Psi$.

Is there a closed form solution?

I'm trying to prove to myself the progression Equations (14-16) in "Compressed Manifold Modes for Mesh Processing" [Neumann et al. 2012]. They claim the solution is:

$$ \Phi = D^{1/2}(YVW^{-1/2}V^T), $$ based on the singular value decomposition $(D^{1/2} Y)^T (D^{1/2} Y) = V W V^T$. I'm not seeing how to prove this.

Edit: At the very least, I think [Neumann et al. 2012] has a typo. It should probably be $$ \Phi = YVW^{-1/2}V^T, $$ then we at least satisfy the constraint: $$ \require{cancel} \Phi^T D \Phi = VW^{-1/2}V^T Y^T D YVW^{-1/2}V^T = VW^{-1/2} V^T V W V^T VW^{-1/2}V^T = I_k. $$ I'm still not seeing why this choice is optimal, though.

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  • $\begingroup$ Ack. That was a typo. Phi/Psi/Y are thin (n>k) and yes, Y and D are given. $\endgroup$ Commented Mar 26, 2018 at 13:04
  • $\begingroup$ Isn't this a special case of the question you asked in November? $\endgroup$ Commented Mar 26, 2018 at 13:46
  • $\begingroup$ yes, I think it is a special case (I'm interested in both cases). This case is specifically already in the literature with a claimed solution I don't understand. $\endgroup$ Commented Mar 26, 2018 at 16:48
  • $\begingroup$ I think the same counterexample for the previous problem (set $k=1$, reduce to closest point on ellipsoid) works in this case too. Neumann et al. state that a proof of their claim appears in Lai and Osher 2014, but I could only find there a proof for the case when the objective is $\|\Psi-Y\|_F^2$, not $\|D^{-1/2}\Psi-Y\|_F^2$. $\endgroup$
    – user856
    Commented Jun 13, 2018 at 16:16

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