# How do these matrices form an order-$4$-tensor?

I'm reading this paper on a convolutional neural network for modelling sentences, and I'm having some trouble understanding section $3.5$. Please consider the following text:

We denote a feature map of the $i$-th order by $F^i$. As in convolutional networks for object recognition, to increase the number of learnt feature detectors of a certain order, multiple feature maps $F_1^i,\ldots, F_n^i$ may be computed in parellel at the same layer. Each feature map $F_j^i$ is computed by convolving a distinct set of filters arranged in a matrix $m_{j,k}^i$ with each feature map $F_k^{i-1}$ of the lower order $i-1$ and summing the results: $$F_j^i = \sum_{k=1}^n m_{j,k}^i*F_k^{i-1}$$ where $*$ indicates the wide convolution. The weights $m_{j,k}^i$ form an order-$4$-tensor. After the wide convolution, first dynamic $k$-max pooling and then the non-linear function are applied individually to each map.

I've tried to figure out why the matrices of weights $m_{j,k}^i$ form an order-$4$-tensor (even though there are only three indices), by figuring out the feature maps:

For the first layer we have:

$$F_1^1 = \sum_{k=1}^1m_{1,k}^1 * F_k^0 = m_{1,1}^1*F_1^0\\ F_2^1 =\sum_{k=1}^1m_{2,k}^1 * F_k^0 = m_{2,1}^1*F_1^0$$ I think we just sum to one here since the input for the first layer is just the sentence matrix $s$ (so $F_1^0 = s$). For the second layer we have: $$F_1^2 = \sum_{k=1}^2m_{1,k}^2*F_k^1 = m_{1,1}^2*F_1^1 + m_{1,2}^2*F_2^1\\ F_2^2 = \sum_{k=1}^2m_{2,k}^2*F_k^1 = m_{2,1}^2*F_1^1 + m_{2,2}^2*F_2^1$$ If this is correct that means that we need six different matrices for the two layers combined: $$m_{1,1}^1, m_{2,1}^1, m_{1,1}^2, m_{1,2}^2, m_{2,1}^2, m_{2,2}^2$$ Question: Why/how do these six matrices form a order-$4$-tensor?

The only thing I can think of is that it has something to do with the fact that the word embeddings in this paper have dimension $d = 4$ (there are four different filters), and that there is a different matrix $m_{j,k}^i$ for each of those four dimensions. This doesn't make sense though, since the matrices are used on a two-dimensional input, where one of those two dimensions is the dimension of the filter. In order words: the fact that the $m_{j,k}^i$ are matrices means that every row already corresponds to a filter dimension.

Under any reasonable definition, you are right three indices would mean an order 3 tensor. In fact I would call the $m^i_{j,k}$ a set of two dimensional tensors (i.e. matrices) indexed by i. The point is that the i index is an index that takes finitely many integer values. The 'j' and 'k' components are different as they index a set of real numbers. However, you will have to check the paper to see how he is doing things. As Gertrude Stein once said, a definition is a definition is a definition.