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1 vote

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

Here I'll present a few remarks and one reasonably efficient (IMHO) algorithm. Unlike my previous answer the conclusions here are experimental more than theoretical. You are absolutely right that the ...
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Find an optimizer for $g(x,y)$ if it exists

By conditions 2 and 3, $f(t,t)=t$ for all $t>0$. By continuity, $f(0,0)=\lim_{t\to 0^+} f(t,t)=\lim_{t\to 0^+}t=0$. From this it follows that $g(0,0)=0$. As you said, if $g$ has a maximizer, then ...
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How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

I'm still not sure whether a closed form solution exists. But I recognized that most implementations of SVD will be iterative (the simplest being power iterations), so the block-coordinate descent in ...
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How to find the maximum of a sum of squares of sums?

You can solve the problem via binary quadratic programming as follows. Let binary decision variable $x_{id}$ indicate whether row $i$ is rotated $d$ places. The problem is to maximize $$\sum_{j=0}^{...
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3 votes

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

OK, sorry for the delay: yesterday was hectic and then I got distracted by another question. Here is a small piece of theory promised. First, let us notice that $\|L-SR\|_F=\|LR^*-S\|_F$ so, as you ...
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2 votes

Eigenvectors that are tensor products?

Your $f(x) := \langle x^{\otimes r}, A x^{\otimes r}\rangle$ is a homogeneous polynomial of degree $2r$ of $d$ variables, $q(x) := ||x||^2$ is a homogeneous quadratic polynomial, and you look for the ...
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3 votes

Eigenvectors that are tensor products?

This is NP-hard already when $r = 2$. To see this, I will consider the problem of minimizing your function $f$ instead of maximizing it, but it's not too much work to flip things around to see that ...
3 votes

Eigenvectors that are tensor products?

I strongly doubt any efficient algorithm exists for even approximately finding the maximizer in general. With slightly different notation, your problem is the same as the problem of "tensor ...

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