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Michael Hardy
Michael Hardy
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Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|.\|_F$$\|\cdot\|_F$ the Froebenius norm. Consider the following set: \begin{equation} \mathcal{M}=\{\Theta \in S_n(\mathbb{R})| \ \|\Theta\|_{0}\leq K, \|\Theta\|_F \leq r\} \end{equation}

I was wondering what was a good estimation of the covering number $\mathcal{N}(\mathcal{M},\|.\|_{F},\varepsilon)$$\mathcal{N}(\mathcal{M},\|\cdot\|_{F},\varepsilon)$ of $\mathcal{M}$ with respect to $\|.\|_F$$\|\cdot\|_F$ ?

Do we also have an estimation of it when $\Theta \in S^{++}_n(\mathbb{R})$ the space of positive definite matrices instead of $\Theta \in S_n(\mathbb{R})$ ?

Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|.\|_F$ the Froebenius norm. Consider the following set: \begin{equation} \mathcal{M}=\{\Theta \in S_n(\mathbb{R})| \ \|\Theta\|_{0}\leq K, \|\Theta\|_F \leq r\} \end{equation}

I was wondering what was a good estimation of the covering number $\mathcal{N}(\mathcal{M},\|.\|_{F},\varepsilon)$ of $\mathcal{M}$ with respect to $\|.\|_F$ ?

Do we also have an estimation of it when $\Theta \in S^{++}_n(\mathbb{R})$ the space of positive definite matrices instead of $\Theta \in S_n(\mathbb{R})$ ?

Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|\cdot\|_F$ the Froebenius norm. Consider the following set: \begin{equation} \mathcal{M}=\{\Theta \in S_n(\mathbb{R})| \ \|\Theta\|_{0}\leq K, \|\Theta\|_F \leq r\} \end{equation}

I was wondering what was a good estimation of the covering number $\mathcal{N}(\mathcal{M},\|\cdot\|_{F},\varepsilon)$ of $\mathcal{M}$ with respect to $\|\cdot\|_F$ ?

Do we also have an estimation of it when $\Theta \in S^{++}_n(\mathbb{R})$ the space of positive definite matrices instead of $\Theta \in S_n(\mathbb{R})$ ?

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YCor
YCor
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Covering number in the space of symmetric matrices

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Titouan Vayer
Titouan Vayer
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Covering number

Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|.\|_F$ the Froebenius norm. Consider the following set: \begin{equation} \mathcal{M}=\{\Theta \in S_n(\mathbb{R})| \ \|\Theta\|_{0}\leq K, \|\Theta\|_F \leq r\} \end{equation}

I was wondering what was a good estimation of the covering number $\mathcal{N}(\mathcal{M},\|.\|_{F},\varepsilon)$ of $\mathcal{M}$ with respect to $\|.\|_F$ ?

Do we also have an estimation of it when $\Theta \in S^{++}_n(\mathbb{R})$ the space of positive definite matrices instead of $\Theta \in S_n(\mathbb{R})$ ?