# Tagged Questions

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### Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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### Frobenius norm and Gaussian noise

Why Frobenius norm is considered to a good tool for dealing with Gaussian noise?
I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ... 0answers 41 views ### An Orthogonal Projection with Weighted Norm In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written ... 1answer 35 views ### Why is$L_0$norm not convex? [closed] I have this confusion in understanding the convexity of the$L_0$norm. Why is$L_0$norm not convex? 1answer 53 views ### Why is L21 norm not smooth I have this confusion. I was reading this paper http://www.cis.temple.edu/~yuhong/research/papers/ijcai13b.pdf. I didn't understand why is L21 norm not smooth? 2answers 55 views ### Minimize Function over Convex Subset Suppose that C is a closed convex subset of$\mathbb R^n$and$x \in \mathbb R^n$. The projection of$\mathbf x$onto C is the closest point$\mathbf y \in C : \mathbf z = \mathbf y$minimizes ... 2answers 304 views ### Derivative of nuclear norm I'm trying to take the derivative of nuclear norm with respect to its argument. nuclear norm is defined in the following way: $$\|x\|_*=\mathrm{tr}(\sqrt{x^Tx})$$ I'm trying to calculate: ... 1answer 37 views ### An equation related to covariance matrix, square root of the matrix, and Euclidean norm. How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which$\Sigma $is a covariance matrix. I tried some numerical examples in ... 2answers 38 views ### Show that the set of points that are nearer$a$than$b$with respect to$\lVert \cdot \rVert_2$is convex I am trying to show the above statement: Show that the set of points that are nearer$a$than$b$in the sense of Euclidean$\lVert\cdot\rVert_2$are convex. My attempt This follows from the ... 0answers 172 views ### Minimum L1 norm may not obtain the sparsest solution? Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ... 1answer 85 views ### minimum trace norm on the set of matrices with fixed diagonal entries What is the min nuclear norm (sum of singular values) on all$n \times n$matrices$A$whose diagonal is fixed. i.e.$diag(A) = v$Is it true that the diagonal matrix is a minimizer? 1answer 146 views ### Gradient of a norm with a linear operator In mathematical image processing many algorithms are stated as an optimization problem, where we have an observation$f$and want recover an image$u$that minimizes a objective function. Further, to ... 1answer 48 views ### Minimization of norms How do I minimize the following?$ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $Also,$X_k^TX_k = 1 \ \ \forall k $I am given that the answer should be :$ \sqrt{Y^T - 2t} + Y^TX$... 1answer 33 views ### Square of 2-norm This might be silly but I am stuck with the following problem:$ || Y - Z_i/x||^2_2 $= 2t How would I solve to get$x $from this equation? 1answer 301 views ### Dual to the dual norm is the original norm (?) I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have$\|y\|_* $as the norm dual of ... 1answer 242 views ### Convexity of the squared Frobenius norm of a matrix I was reading this paper where the define an optimization problem as where K and L are kernel matrices and$\pi$is the permutation matrix. They have explained that the function is convex because ... 2answers 82 views ### About the convexity of Ky Fan's norm As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ... 0answers 82 views ### Examples of functions that are Lipschitz w.r.t. Schatten p-norm? A convex function$f$is$R$-Lipschitz w.r.t. to a norm$\|\cdot\|$if for all points$a, b$we have$|f(a)-f(b)| \leq R\|a-b\|$. For a real symmetric$n\times n$matrix$A$with eigenvalues denoted ... 2answers 314 views ### On the convexity of element-wise norm 1 of the inverse Let us define$\|A\|_1$the element wise norm 1 of a matrix$A \in \mathbb{R}^{n \times m}$as $$\|A\|_1= \sum_{i,j} |A_{i,j}|.$$ Obviously, this function is convex over$\mathbb{R}^{n \times m}$. ... 1answer 79 views ### On the convexity of the element-wise norm 1 of a pseudoinverse Let us define$\|A\|_1$the element wise norm 1 of a matrix$A \in \mathbb{R}^{n \times m}$as $$\|A\|_1= \sum_{i,j} |A_{i,j}|.$$ Obviously, this function is convex over$\mathbb{R}^{n \times m}. ... 0answers 131 views ### Upper bound for L1-L2 optimization problem I am interested in the following convex optimization problem: \begin{align*} \max & ||x||_1 \\ \text{s.t.} & ||x-a||_1 \le K \\ & ||b\circ x||_2 \le 1\\ & x \in R^n \end{align*} where ... 1answer 220 views ###L_1$projection of sum of convex functions onto polytopes Suppose I have a function$f(x) : \mathbb R^n \to \mathbb R$that is the sum of a given strictly convex function$g : \mathbb R \to \mathbb R$in a single variable, i.e.$f(x) = g(x_1) + g(x_2) + ...
Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...