# Questions tagged [nuclear-norm]

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

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### Nuclear norm of matrix comprised of points sampled uniformly at random from the hypersphere?

In a subfield of machine learning called "self-supervised learning", many methods constrain network representations to lie on the hypersphere. I want to understand how such representations ...
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### Relationship between trace norm (a.k.a. Schatten-1 norm) of a matrix and the vector norm of the matrix's row average?

I'm trying to understand whether a connection exists between two seemingly different optimization problems in machine learning. Setup: Suppose I have $N$ points $x_1, ..., x_N \in \mathbb{R}^D$, where ...
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### Uniform continuity of $A\log A$

Let $A,B$ be finite dimensional full-rank positive semidefinite Hermitian matrices with unit trace. Let $|A-B|_1 \leq \varepsilon$ where $|X|_1 = \text{Tr}(\sqrt{X^*X})$ and $X^*$ is the transpose ...
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### How do I minimize $||L||_* + \lambda ||S||_1$

Assume that we got a $n * m$ picture called $X$ and $X$ contains the noise picture $S$. $X - S = L$ where $L$ is the clean filtered ideal picture and $X = L + S$ is the real picture taken by camera. I ...
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1 vote
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### Nuclear norm distance under Kronecker product

Let $A\otimes A$ denote the Kronecker product. Suppose $\|A - B\|_1 = \varepsilon$, where $\|\cdot \|_1$ is the nuclear norm a defined by $\|X\|_1 = \text{Tr}(\sqrt{X^\dagger X})$ and $X^\dagger$ is ...
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### Positive semidefinite ordering of matrices after small perturbations

We use the notation $A\leq B$ to denote that $B-A$ is positive semidefinite. The notation $|\cdot |_1$ denotes the nuclear norm i.e. $|X|_1 = \operatorname{Tr}\sqrt{X^*X}$, where $X^*$ is the ...
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1 vote
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### Singular values and trace norm of a submatrix

Let $A$ be an $m \times n$ matrix where $m \leq n$, and let $B$ the matrix obtained from $A$ by removing both its first row and its first column. Let us denote the singular values of $A$ by: \...
1 vote
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### Decompose nuclear norm

Let $A$ and $B$ be two square matrices such that $A^\top B = 0$ and $B^\top A = 0$. How can we show that $$\|A+B\|_{nuc} = \|A\|_{nuc} + \|B\|_{nuc}$$
162 views

### Nuclear norm bounded by Frobenius norm

We have matrices $A,B\in\mathbb{R}^{n,m}$. Let $P\in\mathbb{R}^{n,n}$ and $Q\in\mathbb{R}^{m,m}$ be the orthogonal projection matrices in the column space of $A$ and the row space of $A$ (or column ...
1 vote
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### Prove that $\|{X}\|_{*}=\min _{{A B}={X}}\|{A}\|_{F}\|{B}\|_{F}=\min _{{A B}={X}} \frac{1}{2}\left(\|{A}\|_{F}^{2}+\|{B}\|_{F}^{2}\right)$.

For any matrix $X\in\mathbb{R}^{m\times n}$, I am confused with $$\|{X}\|_{*}=\min _{{A B}={X}}\|{A}\|_{F}\|{B}\|_{F}=\min _{{A B}={X}} \frac{1}{2}\left(\|{A}\|_{F}^{2}+\|{B}\|_{F}^{2}\right),$$ ...
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### How to solve the truncated nuclear norm minimization problem to global optima?

Definition: Given a matrix $X\in\mathbb{R}^{m\times n}$ and a positive integer $r<\min\{m,n\}$, the truncated nuclear norm $\|X\|_{r,*}$ is defined as the sum of $\min\{m,n\}-r$ minimum singular ...
1 vote
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### Nuclear norm of a block matrix

Suppose that $$X = \begin{bmatrix} A & C^T \\ C & B \end{bmatrix}$$ where $X, A, B \succeq 0$ are (real) positive semidefinite matrices. Is it true that the Cauchy-Schwarz like inequality:...
1 vote
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### Subdifferential of the sum of two matrix norms

I've been reading the paper "Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion" by Koltchinskii et. al. Let $\partial F(x)$ be the subdifferential of the convex ...
Question: Given a symmetric matrix, $S$, what is the solution to the optimization problem $$\arg\min_{P \in \mathcal{S}_{\ge 0}} \| S - P \|_N$$ where $\| \cdot \|_N$ denotes the nuclear norm, i.e. ...