4
$\begingroup$

I am trying to take the matrix derivative of the following function with respect to $\bf W$:

\begin{equation} \| \left| \mathbf{X}\mathbf{W}\right|-\mathbf{1}_{n \times K} \| ^2_F \\ \end{equation}

Where $\mathbf{X}$ is $n \times d$, $\mathbf{W}$ is $d \times K$ and $\mathbf{1}_{n \times K}$ is a marix with all elements one. $\| \cdot \|_F$ is the Frobenius norm and $\left| \mathbf{X}\mathbf{W}\right|$ is the element wise absolute value of $\mathbf{X}\mathbf{W}$.

Any helps is highly appreciated.

$\endgroup$
3
$\begingroup$

For typing convenience, define the matrices $$\eqalign{ Y &= XW \\ J &= 1_{n\times K} \qquad&({\rm all\,ones\,matrix}) \\ S &= {\rm sign}(Y) \\ A &= S\odot Y \qquad&({\rm absolute\,value\,of\,}Y) \\ B &= A-J \\ Y &= S\odot A \qquad&({\rm sign\,property}) \\ }$$ where $\odot$ denotes the elementwise/Hadamard product and the sign function is applied element-wise. Use these new variables to rewrite the function, then calculate its gradient. $$\eqalign{ \phi &= \|B\|_F^2 \\&= B:B \\ d\phi &= 2B:dB \\ &= 2(A-J):dA \\ &= 2(A-J):S\odot dY \\ &= 2S\odot(A-J):dY \\ &= 2(Y-S):dY \\ &= 2(Y-S):X\,dW \\ &= 2X^T(Y-S):dW \\ \frac{\partial\phi}{\partial W} &= 2X^T(Y-S) \\ }$$ where a colon denotes the trace/Frobenius product, i.e. $$\eqalign{ A:B = {\rm Tr}(A^TB) = {\rm Tr}(AB^T) = B:A }$$ The cyclic property of the trace allows such products to be rearranged in various ways $$\eqalign{ A:BC &= B^TA:C \\ &= AC^T:B \\ }$$ Finally, when $(A,B,C)$ are all the same size, their Hadamard and Frobenius products commute with each other $$\eqalign{ A:B\odot C &= A\odot B:C \\\\ }$$ NB: When an element of $\,Y$ equals zero, the gradient is undefined. This behavior is similar to the derivative of $\,|x|\,$ in the scalar case.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ I just checked the derivative using TensorFlow and this is correct! Thank you so much! Could you please introduce me some references on this topic? $\endgroup$ – Arman Aug 18 at 16:31
  • 1
    $\begingroup$ I would recommend Matrix Differential Calculus by Magnus and Neudecker. $\endgroup$ – greg Aug 18 at 18:33
  • 1
    $\begingroup$ The answer from greg is great. However, in the last part of the explanation, the correct notation for a colon is as follows: $$\eqalign{ A:B = {\rm Tr}(A^TB) = {\rm Tr}(B^TA) = B:A }$$ $\endgroup$ – Hadi Zavareh Aug 19 at 22:05

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.