The paper ([arxiv](https://arxiv.org/abs/1610.09716)) introduces DCNNs (Doubly Convolutional Neural Networks). Those are CNNs which contain a new layer type which generalized convolutional layers.
CNNs seem to learn many filters which are similar to other learned filters in the same layer. The weights are only slightly shifted.
The idea of double convolution is to learn groups filters where filters within each group are translated versions of each other. To achieve this, a doubly convolutional layer allocates a set of meta filters which has filter sizes that are larger than the effective filter size. Effective filters can be then extracted from each meta filter, which corresponds to convolving the meta filters with an identity kernel. All the extracted filters are then concatenated, and convolved with the input.
> We have also confirmed that replacing a convolutional layer with a doubly convolutional layer consistently improves the performance, regardless of the depth of the layer.
* CIFAR-10+: 7.24% error
* CIFAR-100+: 26.53% error
* ImageNet: 8.23% Top-5 error
The k-translation correlation is effectively a [cosine similarity](https://en.wikipedia.org/wiki/Cosine_similarity). I think the authors should have mentioned that.