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 $\frac{\partial J}{\partial W}$ ?
2026-04-06 23:04:12.1775516652
What is the derivative of this?
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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 $\frac{\partial J}{\partial W}=4WW^TW-4W$.