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I have a question about differentiating an expression which has multiple kronecker products.

I have the following objective function I would like to differentiate with respect to $\mathbf{Q}$: \begin{equation*} \lVert\mathbf{y}-\mathbf{A}(\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q})\mathbf{x}\rVert^2_2 \end{equation*} where $\mathbf{y}\in\mathbb{R}^m$, $\mathbf{A}\in\mathbb{R}^{m\times K^4}$, $\mathbf{Q}\in\mathbb{R}^{K\times K}$ and $\mathbf{x}\in\mathbb{R}^{K^4}$. I am confused with how the chain rule works with respect to matrix differentiation. This is how I proceeded:

Let $ f=\lVert\mathbf{y}-\mathbf{A}(\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q})\mathbf{x}\rVert^2_2$ and $\mathbf{B}=\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q}\otimes\mathbf{Q}$. Therefore $\frac{df}{d\mathbf{Q}}=\frac{df}{d\mathbf{B}}\frac{d\mathbf{B}}{d\mathbf{Q}}$

When I calculate $\frac{df}{d\mathbf{B}}=\mathbf{A}^T(\mathbf{y}-\mathbf{ABx})\mathbf{x}^T$ I gain a $\mathbb{R}^{K^4\times K^4}$ matrix not a $\mathbb{R}^{K\times K}$ matrix that I am hoping for. Therefore I am using the chain rule wrong because of the change in dimensions i.e scalar to matrix.

Thank you for your help in advance.

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  • $\begingroup$ Of course $df/dB$ is $K^4\times K^4$. $B$ itself is $K^4\times K^4$. There is no mystery here $\endgroup$ – Federico Dec 13 '18 at 16:50
  • $\begingroup$ Thanks for your comments! In reply to your first comment I agree there is no mystery but I was lead to believe a scalar-matrix differential has the same dimensions as the matrix i.e $dim(\frac{df}{d\mathbf{Q}})=dim(\mathbf{Q})$ therefore it seems like the chain rule in this case does not preserve dimensionality. Secondly thanks for your longer answer, that definitely clears up part of my question :) $\endgroup$ – shex95 Dec 13 '18 at 17:04
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    $\begingroup$ Yes, $\frac{df}{dQ}$ has the same dimension as $Q$, that is correct. The problem is how you interpret $\frac{df}{dB}\frac{dB}{dQ}$. The former is a $k^4\times k^4$ matrix, the second a $(k^4\times k^4)\times(k\times k)$ tensor, and you have to contract the correct indices to obtain the chain rule: $$ \frac{df}{dQ_{ij}} = \sum_{k,l=1}^{k^4} \frac{df}{dB_{kl}} \frac{dB_{kl}}{dQ_{ij}} = \sum_{k,l=1}^{k^4} \left(\frac{df}{dB}\right)_{kl} \frac{dB_{kl}}{dQ_{ij}} $$ $\endgroup$ – Federico Dec 13 '18 at 17:15
  • $\begingroup$ Thanks Federico, you have cleared things up there. I appreciate your help. $\endgroup$ – shex95 Dec 13 '18 at 17:17
  • $\begingroup$ So the contraction between $\frac{df}{dB}$ and $\frac{dB}{dQ}$ is what some people call double dot product. But my advice is to write out the indices in order to not commit mistakes $\endgroup$ – Federico Dec 13 '18 at 17:18
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Short answer: the derivative of $Q\otimes Q\otimes Q\otimes Q$ with respect to $Q$ is a mess, at first sight...

Let's start simple. Let $Q$ be a $K\times K$ matrix with entries $Q_{ij}$ and let $E^{ab}$ be the $K\times K$ matrix with all $0$ entries, except the entry $(a,b)$ which is $1$; in other words, $(E^{ab})_{ij} = \delta_a^i\delta_b^j$.

Then I claim that $$ \frac{\partial(Q\otimes Q)}{\partial Q_{ij}} = E^{ij}\otimes Q+Q\otimes E^{ij} . $$ I leave it to you to see why, because trying to write out the involved matrices will probably crash the entire Stack Exchange network...

Jokes aside, this is really immediate to see: just write $Q\times Q$ as in the first formula of the definition and think which elements are affected by $Q_{ij}$. There is the entire $(i,j)$th block, so you get $E^{ij}\otimes Q$, but there is also the $(i,j)$th entry in each block, which gives you $Q\otimes E^{ij}$.

Now, if $A$ and $B$ are matrices which are functions of $Q$, by the same reasoning you get $$ \frac{\partial(A\otimes B)}{\partial Q_{ij}} = \frac{\partial A}{\partial Q_{ij}}\otimes B + A\otimes \frac{\partial B}{\partial Q_{ij}} . $$

So you can iterate for instance $$ \begin{split} \frac{\partial (Q\otimes Q\otimes Q)}{\partial Q_{ij}} &= \frac{\partial(Q\otimes Q)}{\partial Q_{ij}}\otimes Q + (Q\otimes Q)\otimes \frac{\partial Q}{\partial Q_{ij}} \\ &= (E^{ij}\otimes Q+Q\otimes E^{ij})\otimes Q + (Q\otimes Q)\otimes E^{ij} \\ &= E^{ij}\otimes Q\otimes Q + Q\otimes E^{ij}\otimes Q + Q\otimes Q\otimes E^{ij}. \end{split} $$

Now you can prove by induction that $$ \frac{\partial \bigl(\bigotimes_{n=1}^N Q\bigr)}{\partial Q_{ij}} = \sum_{n=1}^N \left(\bigotimes_{h=1}^{n-1} Q\right) \otimes E^{ij} \otimes \left(\bigotimes_{h=n+1}^{N} Q\right). $$

Written more concisely, $$ \frac{\partial Q^{\otimes N}}{\partial Q_{ij}} = \sum_{n=1}^N Q^{\otimes (n-1)}\otimes E^{ij} \otimes Q^{\otimes (N-n)} . $$

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