0
$\begingroup$

Given an invertible matrix $G \in \mathbb{R}^{d \times d}$, what is the gradient of the following function w.r.t. $x$: $$ \left\| Gx\right\|_2^2$$

Also, is this function smooth!?

Please advise.

Thanks in advance.

$\endgroup$
0
$\begingroup$

Write the function in terms of the inner/Frobenius product (which I'll denote with a colon) $$f=\|Gx\|^2_F = Gx:Gx$$

Then finding the differential and gradient is pretty straightforward $$\eqalign{ df &= 2\,Gx:G\,dx \cr &= 2\,G^TGx:dx \cr \cr \frac{\partial f}{\partial x} &= 2\,G^TGx \cr\cr }$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.