# Numerator layout for derivatives and the chain rule

We have three matrices $$\mathbf{W_2}$$, $$\mathbf{W_1}$$ and $$\mathbf{h}$$ (technically a column vector):

$$\mathbf{W_1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \;\;\;\;\;\;\;\;\; \mathbf{W_2} = \begin{bmatrix} e & f \\ \end{bmatrix} \;\;\;\;\;\;\;\;\; \mathbf{h} = \begin{bmatrix} h_1 \\ h_2 \\ \end{bmatrix}$$

And a scalar $$y$$, where:

$$y = \mathbf{W_2} \mathbf{W_1} \mathbf{h}$$

I'd like to compute the derivative of $$y$$ with respect to $$\mathbf{W_1}$$, assuming numerator layout.

Using the chain rule:

$$y = \mathbf{W_2} \mathbf{u} \;\;\;\;\;\;\;\;\; \mathbf{u} = \mathbf{W_1} \mathbf{h} \\$$

\begin{align} \frac{\partial y}{\partial \mathbf{W_1}} &= \frac{\partial y}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial \mathbf{W_1}} \\ &= \mathbf{W_2} \frac{\partial \mathbf{u}}{\partial \mathbf{W_1}} \\ &= \mathbf{W_2} \mathbf{h}^{\top} \\ \end{align}

All well and good. Except - this isn't a $$2x2$$ matrix!! In fact, the dimensions don't match up for matrix multiplication, so something must be incorrect.

If we take the Wikipedia definition of the derivative of a scalar by a matrix, using numerator layout, we know that actually:

$$\frac{\partial y}{\partial \mathbf{W_1}} = \begin{bmatrix} \frac{\partial y}{\partial a} & \frac{\partial y}{\partial c} \\ \frac{\partial y}{\partial b} & \frac{\partial y}{\partial d} \\ \end{bmatrix}$$

Each element is just a scalar derivative, which we can calculate without any matric calculus. If we do that by hand and then factorise, we end up with:

$$\frac{\partial y}{\partial \mathbf{W_1}} = \mathbf{h} \mathbf{W_2}$$

Clearly, $$\mathbf{h} \mathbf{W_2} \neq \mathbf{W_2} \mathbf{h}^\top$$.

Can anybody suggest where I went wrong?

• From that same Wikipedia article: “The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives.” – amd Jan 16 at 18:49
• @amd yes true! But the result looks elegant enough that there should be a way to derive it easily, no? – chris838 Jan 17 at 14:11
• A common way to compute any of these types of derivatives is first compute the differential of the composite function, and then work out the derivative from that. The chain rule does apply to the differential and those often-surprising transposes appear organically. – amd Jan 17 at 17:51

For the ∂u/∂W1, u is 2x1 vector and W1 is 2x2 matrix. So ∂u/∂W1 is a 2x2x2 tensor and not h⊤.

Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a tensor of rank higher than 2, so that they do not fit neatly into a matrix