while reading the Deep Parametric Continuous Convolutional Neural Networks paper, I struggled to understand the formula for their parametric continuous convolution layer (page 2) : $$ \large{ h_{k,i} = \sum_d^F \sum_j^N g_{d,k} (\textbf{y}_i - \textbf{x}_j) f_{d,j} } $$
My understanding:
As far as I understood there are $N$ $F$-dimensional input feature vectors, i.e., $\large{f_{d,j} \in \mathbb{R}^{F \times N}}$ is matrix where each column corresponds to a one input feature vector $\large{\textbf{f}_{\mathrm{in,j}}}$.
Each of these vectors has an associated location in the support domain $\large{\mathcal{S} = \{\textbf{y}_j \}_{j=1}^N}$. Furthermore, the output domain locations $\large{\mathcal{O} = \{\textbf{x}_i\}_{i=1}^{O}}$ need to be provided as an input, where $O$ defines the number of output features.
However that would only make sense if they actually meant: $$ \large{ h_{k,i} = \sum_d^F \sum_j^N g_{d,k} (\textbf{x}_i - \textbf{y}_j) f_{d,j} } $$ which would also make sense in relation to their previous approximation, where they use the following notation $$ \large{ h(\textbf{x}) \approx \sum_{i=1}^N \frac {1}{N} f(\textbf{y}_i) g(\textbf{x} - \textbf{y}_i) } $$
Anyway, I have the feeling that I misunderstood something and I would appreciate any help.