0
votes
1answer
36 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
votes
1answer
16 views

derivative of 2 norm wrt matrix

I have a matrix A which is of size m,n, a vector B which of size ...
0
votes
1answer
16 views

gradient norm of a simple function

In this answer Derivation of soft thresholding operator how can I derive that $\nabla(||x-b||_2^2)=b-x$?
0
votes
0answers
30 views

Proving that derivative of a bounded linear map (at a point) is the map itself

I have been struggling to prove a claim. How can I show that If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$? Attempt: $f:X \rightarrow Y$ is a bounded linear map ...
1
vote
1answer
49 views

Check my answer - Finding the jacobi matrix of a function

We are given $f: \mathbb R^n \to \mathbb R^n$ such that: $0 \neq x \in \mathbb R^n$, $f(x)=\frac{x}{|x|}$, where $|x| = \sqrt {x_1^2 +x_2^2+...+x_n^2}$ Find the jacobi matrix (the differential ...
2
votes
1answer
43 views

Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
0
votes
1answer
145 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 ...
1
vote
2answers
110 views

Calculate the derivative of a complex norm

I'm stuck with a rather trivial looking question. How do you calculate the derivative of the norm of a complex number to it self? Like in $$ \frac{d|a|^2}{da} = ? $$ I think it would give rise to a ...
0
votes
1answer
30 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
0
votes
0answers
21 views

Matrix $L_1$ norm derivative [duplicate]

Say that $A\in \mathcal{M}_{n,n}$ what is the result of the following derivative: $\frac{\partial \|A- diag(A)\|_1}{\partial A}$, where $diag(A)$ is the matrix that contains the diagonal entries of ...
1
vote
0answers
210 views

Why is the subdifferential of norm of a matrix ||A|| defined like this?

I read in a paper called "Characterization of the subdifferential of some matrix norms" that it defines the subdifferential of the matrix norm like this: $$\partial ||A||=\{G \in R^{m\times n} : ...
1
vote
1answer
1k views

Gradient of squared Frobenius norm

I'd like to find the gradient of $\frac{1}{2} ||X A^T||_F^2$ with respect to $X_{ij}$. Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like $\partial \left[\frac{1}{2} ||X ...