We have three ambiguous tags average, expectation and means. They are often used for probability and statistics problems. The tag expected-value has been synonymized with probability. The current tag excerpts and info for these three tags are not clear enough. The average tag includes arithme...

In Weighted Nuclear Norm Minimization with Application to Image Denoising, it is stated that nuclear norm of a matrix $\mathbf{X}$, given by $$\|\mathbf{X}\|_{*}=\sum_{i} \sigma_{i}(\mathbf{X})$$ where $\sigma_{i}(\mathbf{X})$ are the singular values, is convex. In the same paper, the weighted ...

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{X}} \frac{1}{2} \| \boldsymbol{X - Y} \|_F^2 + \lambda \| \boldsymbol{X} \|_{*} \end{equation} where $F$ denotes the Frobenius norm and $*$ denotes the nuclear norm. $\boldsymbol{Y}$ a...

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 see how this true and can't find a proof online.

The nuclear norm is defined in the following way $$\| X \|_* := \mbox{tr} \left( \sqrt{X^T X} \right)$$ and, from Derivative of the nuclear norm with respect to its argument, $$\frac{d}{dX} \| X \|_* = U\Sigma^{-1}\mid\Sigma \mid V^T$$ What is the second derivative of the nuclear norm? $$\...

The function is $$f(x) = \| x x^T - V \|_*$$ where $\| \cdot \|_*$ denotes the nuclear norm and $V$ is a given matrix. $x$ is a vector. Please tell me how to differentiate $f(x)$. And, if it is possible, please show me how to compute the 2nd derivative of $f(x)$.