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I came across some problems that are related to partial derivative but I haven't learnt this yet. And I looked up many online resources but couldn't find answers to my doubts. Really hope someone can help me.
Here is my problem. $y=A^TB$, where A, B are two matrices. Now I want to know what $\frac{\partial y}{\partial A}$,$\frac{\partial y}{\partial B}$ are.

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    $\begingroup$ No, your problem isn't this. Your problem is that partial derivatives with respect to a single (real) variable are well defined (and you may or may be not aware of that definition), but there is no commonly accepted notion/notation for a vector/matrix variable. Please, "research" does not mean "ask at MSE"! Would you be so kind to do a minimum of own effort? Thank you ever so much! $\endgroup$ – user436658 Jan 13 at 14:11
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$\def\p#1#2{\frac{\partial #1}{\partial #2}}$Let $\,(\alpha,\beta)\,$ be fourth-order tensors with components $$\eqalign{ \alpha_{ijk\ell} &= \delta_{ik}\,\delta_{j\ell} \\ \beta_{ijk\ell} &= \delta_{i\ell}\,\delta_{jk} \\ }$$ and properties with respect to the matrices $(F,G,H)$ $$\eqalign{ \alpha:H &= H:\alpha = H \\ \beta:F &= F:\beta = F^T \\ HFG &= H\alpha G^T:F \\ }$$ where a colon denotes a double-contraction product, i.e. $$\eqalign{ \left(\alpha:H\right)_{ij} &= \sum_k\sum_\ell\alpha_{ijk\ell}\,H_{k\ell} \\ \left(F:\beta\right)_{k\ell} &= \sum_i\sum_jF_{ij}\,\beta_{ijk\ell} \\ }$$ and juxtaposition implies a single-contraction product $$\eqalign{ \left(H\alpha\right)_{mjk\ell} &= \sum_i H_{mi}\,\alpha_{ijk\ell} \\ \left(\alpha G^T\right)_{ijkm} &= \sum_\ell\alpha_{ijk\ell}\,G^T_{\ell m} \\ }$$

With these tensors, the posted question can be answered as follows $$\eqalign{ Y &= A^TB \\&= A^T\alpha:B \quad&\implies\quad\p{Y}{B} &= A^T\alpha \\ Y &= \alpha B^T:A^T \\ &= \alpha B^T:\beta:A \quad&\implies\quad\p{Y}{A} &= \alpha B^T:\beta \\ }$$ So the gradients in question are seen to be fourth-order tensors.

An approach which avoids higher-order tensors, is to transform the relationship into a vector equation using Kronecker products. $$\eqalign{ {\rm vec}(Y) &= (I\otimes A^T)\;{\rm vec}(B) \quad&\implies\quad \p{{\,\rm vec}\,Y}{{\,\rm vec}\,B} = (I\otimes A^T) \\ &= (B^T\otimes I)K\;{\rm vec}(A) \quad&\implies\quad \p{{\,\rm vec}\,Y}{{\,\rm vec}\,A} = (B^T\otimes I)K \\ }$$ where $K$ is the commutation matrix associated with vectorization.

Another approach is to use component-wise derivatives $$\eqalign{ \p{Y}{A_{ij}} &= E_{ij}^TB \qquad\quad \p{Y}{B_{ij}} &= A^TE_{ij} \\ }$$ where $E_{ij}$ is a matrix with all components equal to zero, except the $(i,j)$ component which equals one. And any matrix with independent components satisfies the identity $$\eqalign{ \p{G}{G_{k\ell}} &= E_{k\ell} \qquad\iff\qquad \p{G^T}{G_{k\ell}} &= E_{k\ell}^T \\ }$$ Finally, to bring things full circle $$\eqalign{ \p{G_{ij}}{G_{k\ell}} &= \alpha_{ijk\ell} \qquad\iff\qquad \p{G_{ij}^T}{G_{k\ell}} &= \beta_{ijk\ell} \\ }$$

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  • $\begingroup$ LOL — Tensors for someone who hasn't learned partial derivatives :) $\endgroup$ – Ted Shifrin Jan 13 at 21:24
  • $\begingroup$ @TedShifrin I believe the question is from someone who is familiar with partial derivatives but not necessarily matrix calculus. Since there are relatively few good references on the topic (aside from this forum), they deserve an answer to the question that was asked without any hint of condescension. $\endgroup$ – greg Jan 14 at 2:29
  • $\begingroup$ I'm not suggesting condescension, but read the first sentence. Of course, I don’t expect people to read before answering. Yes, condescension. $\endgroup$ – Ted Shifrin Jan 14 at 2:35

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