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

# Questions tagged [nuclear-norm]

115 questions

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### Why is minimizing the nuclear norm of a matrix a good surrogate for minimizing the rank?

A method called "Robust PCA" solves the matrix decomposition problem
$$L^*, S^* = \arg \min_{L, S} \|L\|_* + \|S\|_1 \quad \text{s.t. } L + S = X$$
as a surrogate for the actual problem
$$L^*, S^* = \arg \min_{L, S} rank(L) + \|S\|_0 \quad…

blubb

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### Derivative of the nuclear norm

The nuclear norm is defined in the following way
$$\|X\|_*=\mathrm{tr} \left(\sqrt{X^T X} \right)$$
I'm trying to take the derivative of the nuclear norm with respect to its argument
$$\frac{\partial \|X\|_*}{\partial X}$$
Note that $\|X\|_*$ is a…

Alt

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### Fast computation/estimation of the nuclear norm of a matrix

The nuclear norm of a matrix is defined as the sum of its singular values, as given by the singular value decomposition (SVD) of the matrix itself. It is of central importance in Signal Processing and Statistics, where it is used for matrix…

gappy

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### Trace Norm properties

Let $\|A\|_1=\operatorname{trace}(\sqrt{A^* A})$. I already proved that for arbitrary unitary matrices $U$ and $V$, $\|UAV^*\|_1=\|A\|_1$ and $\|A\|_1=\sigma_1+\dots+\sigma_k$. Now I would like to prove that $\|A\|_1$ defines a matrix norm, $A\in…

Montaigne

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### Deriving the sub-differential of the nuclear norm

Let $$f(K) = \| K \|_*$$ be the nuclear norm (sum of the singular values) of $K=U\Sigma V^T$. How can one compute the subdifferential $\partial f$?
This may be a basic question, I'm trying to work my way through a paper in which minimizing $f$ over…

Lepidopterist

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### Show that the dual norm of the spectral norm is the nuclear norm

Could someone help me understand why the dual norm of the spectral norm is the nuclear norm?
We can focus on the real field. Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by
$$\left \| X\right\| = \max\limits_{i…

guanglei

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### If matrix $A$ has entries $A_{ij}=\sin(\theta_i - \theta_j)$, why does $\|A\|_* = n$ always hold?

If we let $\theta\in\mathbb{R}^n$ be a vector that contains $n$ arbitrary phases $\theta_i\in[0,2\pi)$ for $i\in[n]$, then we can define a matrix $X\in\mathbb{R}^{n\times n}$, where
\begin{align*}
X_{ij} = \theta_i - \theta_j.
\end{align*}
Then the…

ChristophorusX

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### Rank, nuclear norm and Frobenius norm of a matrix.

The nuclear norm, denoted $\|\cdot\|_*$ is a good surrogate for the $rank$ when minimizing problems like
$$\label{pb1}\tag{1}
\min_X rank(X) : AX = B
$$
Here, we're trying to find an matrix X with low rank such that $AX=B$.
If I recall correctly,…

davcha

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### Sum of singular values of a matrix

Is there a "trick" to calculate the sum of singular values of a matrix $A$, without actually finding them?
For example, the sum of the squared singular values is $\operatorname{trace}(A^TA)$.

catch22

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### Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show
that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear norm? Thanks in advance.

hi9879

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### Prove that the nuclear norm is convex

For an $m \times n$ matrix, $A$, the nuclear norm of $A$ is defined as $\sum_{i}\sigma_{i}(A)$
where $\sigma_{i}(A)$ is the $i^{th}$ singular value of $A$. I've read that the nuclear norm is convex on the set of $m \times n$ matrices. I don't…

Mykie

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### Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$:
$$\| X\|_\Sigma=\min_{X=UV'}\|U\|_\mathrm{Fro}\|V\|_\mathrm{Fro}=\min_{X=UV'}\frac{1}{2}(\|U\|_\mathrm{Fro}^{2}+\|V\|_\mathrm{Fro}^2)$$
where…

new2you

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### Why is the nuclear norm called so?

A simple question. Why is the sum of the singular values of a matrix called its nuclear norm? What is the origin of, and motivation for, this term?
Apparently the term nucleus is sometimes used to refer to the kernel of a linear transformation, but…

user856

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votes

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### Inequality between Frobenius and nuclear norm

Let $M$ be a square matrix, $\|\cdot\|_*$ be the nuclear (trace) norm, and $\|\cdot\|_F$ be the Frobenius norm. The following inequality holds between the norms:
$$\|M\|^2_* \leq \text{rank}(M) \|M\|^2_F.$$
This is pretty easy to show by using the…

egrr

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### What is the definition of the nuclear norm (aka trace norm, Ky-Fan n-norm) of a tensor?

What is the direct definition for the trace norm of a tensor?
By direct I mean without matricization.
Edit 1: By tensor, I mean a multi-way array, a generalization of vectors and matrices. (The nuclear norm of a matrix, which is a special case of a…

user25004

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