How do I calculate the delta term of a Convolutional Layer, given the delta terms and weights of the previous Convolutional Layer?

I am first deriving the error for a convolutional layer below for simplicity for a one dimensional array (input) which can easily be transferred to a multidimensional then:

We assume here that the $y^{l-1}$ of length $N$ are the inputs of the $l-1$-th conv. layer, $m$ is the kernel-size of weights $w$ denoting each weight by $w_i$ and the output is $x^l$.
Hence we can write (note the summation from zero): $$x_i^l = sumlimits_{a=0}^{m-1} w_a y_{a+i}^{l-1}$$ where $y_i^l = f(x_i^l)$ and $f$ the activation function (e.g. sigmoidal). With this at hand we can now consider some error function $E$ and the error function at the convolutional layer (the one of your previous layer) given by $partial E / partial y_i^l $. We now want to find out the dependency of the error in one the weights in the previous layer(s): begin{equation} frac{partial E}{partial w_a} = sumlimits_{a=0}^{N-m} frac{partial E}{partial x_i^l} frac{partial x_i^l}{partial w_a} = sumlimits_{a=0}^{N-m}frac{partial E}{partial w_a} y_{i+a}^{l-1} end{equation}
where we have the sum over all expression in which $w_a$ occurs, which are $N-m$. Note also that we know the last term arises from the fact that $frac{partial x_i^l}{partial w_a}= y_{i+a}^{l-1}$ which you can see from the first equation.
To compute the gradient we need to know the first term, which can be calculated by: $$ frac{partial E}{partial x_i^l} = frac{partial E}{partial y_i^l} frac{partial y_i^l}{partial x_i^l} = frac{partial E}{partial y_i^l} frac{partial}{partial x_i^l} f(x_i^{l})$$ where again the first term is the error in the previous layer and $f$ the nonlinear activation function.

Having all necessary entities we are now able to calculate the error and propagate it back efficiently to the precious layer: $$ delta^{l-1}_a = frac{partial E}{partial y_i^{l-1} } = sumlimits_{a=0}^{m-1} frac{partial E}{partial x_{i-a}^l} frac{partial x_{i-a}^l}{partial y_i^{l-1}} = sumlimits_{a=0}^{m-1} frac{partial E}{partial x^l_{i-a}} w_a^{flipped}$$ Note that the last step can be understood easy when writing down the $x_i^l$-s w.r.t. the $y_i^{l-1}$-s. The $flipped$ refers to a transposed weight maxtrix ($T$).

Therefore you can just calculate the error in the next layer by (now in vector notation):

$$delta^{l} = (w^{l})^{T} delta^{l+1} f'(x^{l})$$

which becomes for a convolutional and subsampling layer: $$delta^{l} = upsample((w^{l})^{T} delta^{l+1}) f'(x^{l})$$ where the $upsample$ operation propagates the error through the max pooling layer.

Please feel free to add or correct me!

For references see:

• Convolutional Neural Network by Stanford
• Convolutional Neural Network by Andrew Gibiansky

and for a C++ implementation this (without requirement to install)