# Backpropagation in capsule networks

Trying to create a capsule network implementation, I've browsed through several tutorials and code sources, but was unable to find how back-propagation for capsule networks is implemented. It is not hard to find out how the backpropagation is done when no routing is applied, but it's a bit tricky to figure out how to perform it when routing is on.

Anyone has a decent reference with clear formulas on this subject?

(I did read the original paper by Hinton et al.)

By examining the dynamic routing algorithm, one can clearly see that the logits $$b_{ij}$$ must always be set to zero (whether training the model or not), and then they are updated on each iteration according to the prediction $$\hat{u}_{j|i}$$, and in turn, update $$c_{ij}$$. What this means is that all $$c_{ij}$$ are not parameters for the model but rather values determined by the dynamic routing algorithm, hence the name "dynamic". In other words, $$c_{ij}$$ are not trainable parameters. Similar to weighted sums and activations of ordinary deep neural networks, you need to store these values during the feed forward pass to substitute them later on in the calculations of the backpropagation algorithm.
That said, the transformation matrices $$W_{ij}$$ are the trainable parameters and what you need to do is calculate their derivatives. To do that you need to substitute the equations $$\hat{u}_{j|i} = W_{ij} u_{i}$$ and $$s_j = \sum\limits_{i} c_{ij} \hat{u}_{j|i}$$ in the activation function $$v_j =\frac{\lVert s_j \rVert^{2}}{1+ \lVert s_j \rVert^{2}} \frac{s_j}{ \lVert s_j \rVert}$$, i.e., calculate $$s_j = \sum\limits_{i} c_{ij} W_{ij} u_{i}$$ and use that to calculate $$\lVert s_j \rVert$$ using the euclidean norm, then plug these in $$v_j$$. The final formula will be daunting but at least it will be in terms of the weights, the values $$c_{ij}$$ which should be treated as constants, and $$u_i$$.
To make things a bit easier you can use this trick: set $$z_{ij} = W_{ij} u_{i}$$ and then the derivations should be nicer in terms of $$z_{ij}$$.