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Let $A\in\mathcal{M}(m\times m)$ and $B\in \mathcal{M}(n\times n)$ and define the Kronecker product $\Sigma=A\otimes B.$ Denote $a\in\Bbb{R.}$

I would like to compute the derivative of $\log(a+(y-\mu)^T\Sigma^{-1}(y-\mu))$ respect to $A.$

There is the ${\rm tr}$ trick, so we have $(y-\mu)^T\Sigma^{-1}(y-\mu)={\rm tr}((y-\mu)^T(y-\mu)\Sigma^{-1}).$

Now we have $\Sigma^{-1}=A^{-1}\otimes B^{-1}$ so that

$\log(a+(y-\mu)^T\Sigma^{-1}(y-\mu))={\rm tr}((y-\mu)^T(y-\mu)A^{-1}\otimes B^{-1}).$

How can get $$\frac{\partial {\rm tr}((y-\mu)^T(y-\mu)A^{-1}\otimes B^{-1})}{\partial A}$$ ?

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For consistency, I'll use the convention were uppercase Latin letters are matrices, lowercase Latin are vectors, and Greek letters are scalars.

I'll also make frequent use of the trace/Frobenius product, i.e. $$A:B = {\rm tr}(A^TB)$$

Finally, define new variables which are easier to type $$\eqalign{ w &= y-u \cr S &= A\otimes B \cr \beta &= \alpha + ww^T:S^{-1} \cr \lambda &= \log\beta\cr }$$ Now find the differential of $\lambda$, which is the function that you asked about

$$\eqalign{ d\lambda &= \beta^{-1}d\beta \cr \beta\,d\lambda &= d\beta \cr &= ww^T:dS^{-1} \cr &= -ww^T:S^{-1}\,dS\,S^{-1} \cr &= -S^{-T}ww^TS^{-T}:dS \cr &= -S^{-T}ww^TS^{-T}:dA\otimes B \cr }$$ At this point, assume we can find a Kronecker factorization of the LHS, i.e. $$\eqalign{ F\otimes G &= S^{-T}ww^TS^{-T} \cr }$$ where the sizes of the matrices $(F,G)$ are the same as $(A,B)$, respectively.

Then we can proceed to find the gradient $$\eqalign{ \beta\,d\lambda &= -(F\otimes G):(dA\otimes B) \cr &= -(G:B)(F:dA) \cr \frac{\partial\lambda}{\partial A} &= -\beta^{-1}(B:G)F \cr }$$ Our assumed Kronecker factorization may not exist, but we can always find a decomposition by summing over multiple factors
$$\eqalign{ S^{-T}ww^TS^{-T} &= \sum_{k=1}^r F_k\otimes G_k }$$ Search for papers by vanLoan & Pitsianis on "Kronecker approximation".
Pitsianis's 1997 dissertation contains Matlab code to calculate this decomposition.
It's basically (yet another) a clever use of the SVD decomposition.

In this case, since the LHS is a rank-1 matrix and the Kronecker factors are square matrices of sizes $(m\times m)$ and $(n\times n)$, we know exactly how many terms are needed for the decomposition: $r=\min(m^2,n^2)$

This more complicated Kronecker decomposition changes our gradient expression to $$\eqalign{ \frac{\partial\lambda}{\partial A} &= -\beta^{-1}\sum_{k=1}^r \big(B:G_k\big)F_k \cr\cr }$$ NB: There are lots of ways to rearrange the terms in a Frobenius product, all of which follow from the cyclic properties of the trace function.

For example, all of the following are equivalent $$\eqalign{ A:BC &= BC:A \cr &= A^T:(BC)^T \cr &= B^TA:C \cr &= AC^T:B \cr }$$

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