What is the derivative of this?

Question

I have a function of the following form:

$J = \|W^TW-I\|_F^2$

Where, $W$ is a matrix and $F$ is the Frobenius Norm.

How can I find the derivative of $\frac{\partial J}{\partial W}$ ?

What does F mean? And is W a vector or a matrix? – KittyL Jan 29 '15 at 18:09 — KittyL, Jan 29 '15 at 18:09
I have edited. Please have a look. – user570593 Jan 29 '15 at 18:27 — user570593, Jan 29 '15 at 18:27

score 0 · Answer 1 · answered Jan 30 '15 at 00:11

This is what we will use: $||A||_F^2=tr(A^TA)$.

$||W^TW-I||_F^2=tr(W^TWW^TW-2W^TW+I)=||W^TW||_F^2-2tr(W^TW)+tr(I)$

Derivative of the $tr(W^TW)$ is $2W$. Derivative of the third term is $0$. My guess is derivative of the first term is $4WW^TW$ or $4W^TWW^T$. Now through some tedious calculation, you can verify that it is $4WW^TW$.

So $\frac{\partial J}{\partial W}=4WW^TW-4W$.

What is the derivative of this?

1 Answers1