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YCor
YCor
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conjugate Conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~otherwise\end{cases}$$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~\text{otherwise}\end{cases}$$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

Conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~\text{otherwise}\end{cases}$$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

added 1 characters in body; edited body
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Bernard
Bernard
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I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1\\\infty ~~~otherwise\end{cases}$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1\\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

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Bernard
Bernard
  • 111
  • 2
  • 7

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1\\\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.