# Questions tagged [nuclear-norm]

The nuclear norm of a matrix is the sum of its singular values.

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### Power Method for SVD and nuclear norm.

As an exercise from a data analysis course from my school, I have proved the convergence of the following formula. It is as an exercise related to the Power Method for leading singular vector. The ...
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### inverse limits of epimorphisms in (nuclear) TVS

Definition (Bidirected set) A directed set $(I,\leq)$ is said to be bidirected, if $(I,\geq)$ is also a directed set. Definition(Preserved property) Let $\mathcal C$ be a category and $P$ a property ...
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### Is the $k$-th elementary symmetric polynomial of the singular values the nuclear norm of the $k$-th alternating power of the matrix?

If $A$ is a complex $m$ by $n$ matrix, thus representing a $\mathbb{C}$-linear map from $\mathbb{C}^n$ to $\mathbb{C}^m$, we denote by $s_1, \ldots , s_r$ its singular values. Let $e_k$ denote the $k$-...
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### Proof of the variational formulation of the nuclear norm [closed]

How to show that the nuclear norm can be written in the following way? $$\|X\|_* = \min\limits_{A,B: AB=X}\frac{\|A\|^2_2}{2} + \frac{\|B\|^2_2}{2}$$ Related: Variational characterization of nuclear ...
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### What is the $1$-norm of a matrix?

Let $A \in \mathbb C^{n \times n}$ be some matrix. I'm trying to understand what $\lvert\!\lvert A \rvert\!\rvert_1$ means. I found two definitions: $\lvert\!\lvert A \rvert\!\rvert_1$ is the ...
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### If $A\approx_\varepsilon D$, where $D$ is diagonal, is $A$ already almost diagonal?

Let $D$ and $A$ be symmetric positive semidefinite Hermitian matrices such that $\|D-A\|_1 \leq \varepsilon$, where $\|X\|_1 = \operatorname{tr}(\sqrt{X^*X})$. Moreover, let $D$ be diagonal. Does ...
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### How calculate the trace norm via convex optimization?

I am taking the convex optimization course by CMU (though I am not a CMU student), and got stuck on this problem. Formally, show that computing $\left \| X \right \|_{tr}$ can be expressed as the ...
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### How to show that $\mathcal{C}^{\infty} (\mathcal{M}, \mathbb{R})$ is a nuclear space ?.

Let $\mathcal{M}$ be a compact differential manifold. Let $A = \mathcal{C}^{\infty} (\mathcal{M}, \mathbb{R})$ be the space of smooth functions over $\mathcal{M}$. How to show that $A$ is a ...
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### Show that the nuclear norm is a norm

Let $A \in \mathbb{R}^{n\times m}$. Define $$\|A\| = \sum_{i=1}^{\min(n,m)} \sigma_i$$ where $\sigma_i$ are the singular values of $A$. How to show that $\|A\|$ is norm? We want to verify the ...
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### Proximal operator of the nuclear norm with non-negativity constraints

Let $\bf L$ and $\bf R$ be $n \times n$ matrices. Consider the following regularized least-squares problem \mathbf{L} = \min_{\mathbf{L} \geq \mathbf{0}} \mu \|\mathbf{L}\|_* + \...
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Let $L$ and $R$ be $n \times n$ matrices. Consider the following minimization problem \mathbf{L} = \min_{L \geq \mathbf{0}} \mu\|\mathbf{L}\|_* + \dfrac{1}{2\lambda}\|\mathbf{L-R}\|...