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suppose I have a tensor $x \in \mathbb{R}^{n \times 2 \times 3}$. I take the seminorm of $x$ given by taking the Schatten-1 Norm in every $2 \times 3$ slice and then the $\ell_1$-Norm of the resulting vector of sums of singular values. I want to use this as a regularizer in a convex optimization problem. For my purposes, I need the dual norm of the proposed method, which is given by taking the Schatten-$\infty$-Norm (or spectral norm) in every $2 \times 3$ slice and then the $\ell_\infty$-Norm of the resulting vector.

Now imagine I vectorize $x$ to the form $u \in \mathbb{R}^{6n}$. Is there a way to write the condition "dual norm of u less than one" as a matrix constraint, i.e. $Au \leq 1$? I am trying to use this in $CVX$ in matlab. Of course I can just pass this on in a custom function and operate on the tensor $x$, but it would be much faster if I could avoid that.

Thanks!

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  • $\begingroup$ So you want the absolute value of the largest element of $u$ to be at most 1? $\endgroup$ – LinAlg Oct 31 '18 at 17:45

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