Derivative of matrix-valued function with respect to matrix input

Question

I have the expression

$$\bf \phi = \bf X W$$

where $\bf X$ is a $20 \times 10$ matrix, $\bf W$ is a $10 \times 5$ matrix.

How can I calculate $\frac{d\phi}{d\bf W}$? What is the dimension of the result?

Answer 1

Let function $\mathrm F : \mathbb R^{n \times p} \to \mathbb R^{m \times p}$ be defined as follows

$$\rm F (X) := A X$$

where $\mathrm A \in \mathbb R^{m \times n}$ is given. The $(i,j)$-th entry of the output is

$$f_{ij} (\mathrm X) = \mathrm e_i^\top \mathrm A \, \mathrm X \, \mathrm e_j = \mbox{tr} \left( \mathrm e_j \mathrm e_i^\top \mathrm A \, \mathrm X \right) = \langle \mathrm A^\top \mathrm e_i \mathrm e_j^\top, \mathrm X \rangle$$

Hence,

$$\partial_{\mathrm X} \, f_{ij} (\mathrm X) = \color{blue}{\mathrm A^\top \mathrm e_i \mathrm e_j^\top}$$

Answer 2

There is a similar question.

Also, you could define it

$$C = \frac{\partial \phi}{\partial W} $$

where C is a 4D matrix (or tensor) with

$$ C_{a,b,c,d} = \frac{\partial \phi_{a,b}}{\partial W_{c,d}} $$

Actually, when derivatives are expressed as matrices, for example, $f=x^TAx$ where $x\in R^{n\times1}, A\in R^{n\times n}$, you could think of $\frac{\partial f}{\partial A}$ as

$$ \left[\frac{\partial f}{\partial A}\right]_{ij} = \frac{\partial f}{\partial A_{ij}} $$