Questions tagged [nuclear-norm]

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

65 questions
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Logarithms of matrices that are close in trace distance are close?

Let us define the 1-norm or nuclear norm by the following expressions where $\dagger$ represents the conjugate transpose of a matrix. $$||A||_{1} = \text{Tr}(\sqrt{A^\dagger A})$$ The one norm ...
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Derivative of matrix nuclear norm

I'm trying to find the derivative of $$|(L^TL - \sigma)|_1 = \mbox{Tr} \left( \sqrt{(L^TL - \sigma)^\dagger(L^TL - \sigma)} \right)$$ with respect to $L$, where $\dagger$ is the transpose conjugate ...
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Revisit “Inequality between Frobenius and nuclear norm”

I am reading the following question: Inequality between Frobenius and nuclear norm I know $$\|X\|_* = \sum_{i=1}^r \sigma_i(X)$$and $$\|X\|_F = \bigg(\sum_{i=1}^r \sigma^2_i(X)\bigg)^{1/2}$$ I try ...
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the acceleration of nuclear norm of similar sub-problems, perhaps SVT?

i have come up with a model with two sub-problems. It looks like: $$||A||_* + \lambda_1||A-G-H_1||_F^2$$ $$||B||_* + \lambda_2||B+G-H_2||_F^2$$ So it involves two independent problems of solving ...
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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_{...
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“Shadow prices” interpretation of the dual certificate of nuclear norm optimization

In nuclear norm based matrix completion, there exists a low rank matrix $M$ of which only the indicies in $\Omega$ are sampled. Candes and Recht (2008) recover the original matrix using following ...
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When is the Frobenius norm bounded by the nuclear norm?

I am reading the Recht (2011) paper titled, "A Simpler Approach to Matrix Completion", and I cannot figure out the last inequality of the last line on page 3422 (page 10 of the document). The ...
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Derivative of the nuclear norm ${\left\| {XA} \right\|_*}$ with respect to $X$

The nuclear norm (also known as trace norm) is defined as \begin{equation} {\left\| M \right\|_*} = \mbox{tr} \left( {\sqrt {{M^T}M} } \right) = \sum\limits_{i = 1}^{\min \left\{ {m,n} \right\}} {{\...
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Nuclear norm and Schatten norm in practice

I have a problem where the regularizer is the nuclear norm and the matrix being regularized is $n \times d$ with $d < n$. I was initially not getting low rank for the desired performance, the ...
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Convex relaxation of rank constraint

I would like to approximate a symmetric matrix $X\in \mathbb{R}^{n\times n}$ by a matrix of rank no more than $r$, i.e., a matrix $Z=EE'$ where $E\in\mathbb{R}^{n\times r}$, for some given $r\leq n$. ...
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How to convert the matrix completion problem to the standard SDP form?

Given the matrix completion problem defined below. \begin{equation} \begin{array}{ll} \text{minimize }{X \in \mathbb{R}^{m \times n}} & \sum_{(i,j) \in \Omega} ( X_{ij} - Z_{ij} )^2 + \lambda \...
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How to solve the constrained convex optimization problem with nuclear norm?

Let $f(x,y)=\sum_{i=1}^n\sum_{j=1}^{n}(x_{ij}+\sqrt{-1}y_{ij})B_{ij}(x,y)$ Now I want to solve the following problem: \begin{equation} \begin{split} &\arg\min_{C} \iint_{[0,1]^2}|(f_x+\...
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Revisit “How can I visualize the nuclear norm ball”

Revisit "How can I visualize the nuclear norm ball" Two eigenvalues are reproduced as following: $$s_{1,2}=\frac{1}{\sqrt{2}}\sqrt{x^2+2y^2+z^2\pm|x+z|\sqrt{(x-z)^2+4y^2}}.$$ According ...
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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 ...
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Problem in defining weighted nuclear norm in CVX

I want to define a weighted nuclear norm, i.e., $$\|X\|_{w,*}=\sum_{i=1}^{m}{w_i\sigma_i(X)}$$ in CVX, but I can't. CVX has a function for nuclear norm called ...