 I give an overview of the paper which proposes an exponential schedule of dilated convolutional layers as a way to combine local and global knowledge  I point out the connection between 2D dilated convolutions and Kronecker products  cascades of exponentially dilated convolutions  as proposed in the paper  can be thought of as parametrising a large convolution kernel as a Kronecker product of small kernels  the relationship to Kronecker factorisation only holds under particular assumptions, in this sense cascades of exponenetially diluted convolutions are a generalisation of the Kronecker layer (Zhou et al. 2015)  I note that dilated convolutions are equivariant under image translation, a property that other multiscale architectures often violate. #### Background The key application the dilated convolution authors have in mind is dense prediction: vision applications where the predicted object that has similar size and structure to the input image. For example, semantic segmentation with one label per pixel; image superresolution, denoising, demosaicing, bottomup saliency, keypoint detection, etc. In many such applications one wants to integrate information from different spatial scales and balance two properties: 1. local, pixellevel accuracy, such as precise detection of edges, and 2. integrating knowledge of the wider, global context To address this problem, people often use some kind of multiscale convolutional neural networks, which often relies on spatial pooling. Instead the authors here propose using layers dilated convolutions, which allow us to address the multiscale problem efficiently without increasing the number of parameters too much. #### Dilated Convolutions It's perhaps useful to first note why vanilla convolutions struggle to integrate global context. Consider a purely convolutional network composed of layers of $k\times k$ convolutions, without pooling. It is easy to see that size of the receptive field of each unit  the block of pixels which can influence its activation  is $l*(k1)+k$, where $l$ is the layer index. So the effective receptive field of units can only grow linearly with layers. This is very limiting, especially for highresolution input images. Dilated convolutions to the rescue! The dilated convolution between signal $f$ and kernel $k$ and dilution factor $l$ is defined as: $$ \left(k \ast_{l} f\right)_t = \sum_{\tau=\infty}^{\infty} k_\tau \cdot f_{t  l\tau} $$ Note that I'm using slightly different notation than the authors. The above formula differs from vanilla convolution in last subscript $f_{t  l\tau}$. For plain old convolution this would be $f_{t  \tau}$. In the dilated convolution, the kernel only touches the signal at every $l^{th}$ entry. This formula applies to a 1D signal, but it can be straightforwardly extended to 2D convolutions. The authors then build a network out of multiple layers of diluted convolutions, where the dilation factor $l$ increases exponentially at each layer. When you do that, even though the number of parameters grows only linearly with layers, the effective receptive field of units grows exponentially with layer depth. This is illustrated in the figure below: ![](http://www.inference.vc/content/images/2016/05/ScreenShot20160512at094712.png) What this figure doesn't really show is the parameter sharing and parameter dependencies across the receptive field (frankly, it's pretty hard to visualise exactly with more than 2 layers). The receptive field grows at a faster rate than the number of parameters, and it is obvious that this can only be achieved by introducing additional constraints on the parameters across the receptive field. The network won't be able to learn arbitrary receptive field behaviours, so one question is, how severe is that restriction? #### Relationship to Kronecker Products To me this whole dilated convolution paper cries Kronecker product, although this connection is never made in the paper itself. It's easy to see that a 2D dilated convolution with matrix/filter $K$ is the same as vanilla convolution with a diluted filter $\hat{K}_{l}$ which can be represented as the following Kronecker product: $$ \hat{K}_l = K \otimes \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & \ddots & & 0 \\ 0 & \ddots & \ddots & \ddots & \\ 0 & & \ddots & \ddots & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} $$ Using this, and properties of convolutions and Kronecker products (I suggest beginners to make extensive use of the matrix cookbook) we can even understand something about exponentially iterated dilated convolutions. Let's assume we apply several layers of dilated convolutions, without nonlinearity, as in Equation 3 of the paper. For simplicity, I assume that that all convolution kernels $K_l, L=1\ldots L$ are $a\times a$ in size, the dilation factor at layer $l$ is $a^{l}$, and we only have a single channel throughout ($C=1$). In this case we can show that: $$ F_{L+1} = K_L \ast_{a^L} \left( K_{L1} \ast_{a^{(L1)}} \left( \cdots K_1 \ast_{a} \left( K_0 \ast F_0 \right) \cdots \right) \right) = \left( K_L \otimes K_{L1} \otimes \cdots \otimes K_{0} \right) \ast F_0 $$ The lefthand side of this equation is the same construction as in Equation 3 in the paper, but expanded. The right hand side is a single vanilla convolution, but with a convolution kernel that is constructed as the Kronecker product of all the $a\times a$ kernels $K_l$. It turns out Kroneckerfactored parametrisations of convolution tensors are already used in CNNs, a quick googling revealed this paper: Shuchang Zhou, JiaNan Wu, Yuxin Wu, Xinyu Zhou (2015) Exploiting Local Structures with the Kronecker Layer in Convolutional Networks What can Kroneckerfactored filters represent? Let's look at what kind of kernels can we represent with Kronecker products, and hence what behaviour should we expect from dilated convolutions. Here are a few examples of $27\times 27$ kernels that result from taking the Kronecker product of three random $3\times 3$ kernels: ![](http://www.inference.vc/content/images/2016/05/VzORx0FEfAAAAAElFTkSuQmCC.png) These look somehow natural, at least to me. They look like pretty plausible texture patches taken from some pixellated video game. You will notice the repeated patterns and the hierarchical structure. Indeed, we can draw cool selfsimilar fractallike filters if we keep taking the Kronecker product of the same kernel with itself, some examples of such random fractals: ![](http://www.inference.vc/content/images/2016/05/YSJIkSZIkLYw3bCRJkiRJkhbGGzaSJEmSJEkL8zeSmRmMrhHPQgAAAABJRU5ErkJggg.png) I would say these kernels are not entirely unreasonable for a ConvNet, and if you allow for multiple channels ($C>1$) they can represent pretty nice structured patterns and shapes with reasonable number of parameters. Compare these filters to another common technique for reducing parameters of convolution tensors: lowrank decompositions (see e.g. Lebedev et al, 2014). Spatially, a lowrank approximation to a square 2D convolution filter can be understood as subsequently applying two smaller rectangular filters: one with a limited horizontal extent and one with limited vertical extent. Here are a few random samples of $27\times 27$ filters with a rank of 1. These can be represented using the same number of parameters (27) as the Kronecker samples above. To me, these don't look so natural. Notice also that for lowrank representations the number of parameters has to scale linearly with the spatial extent of the filter, whereas this scaling can be logarithmic if we use a Kronecker parametrisation. This is the real deal when using Kronecker products or dilated convolutions. Here is another cool illustration of the naturalness of the Kronecker approximation, taken out of the Kronecker layer paper: ![](http://www.inference.vc/content/images/2016/05/ScreenShot20160512at145833.png) So in general, parametrising convolution kernels as Kroneckerproducts seems like a pretty good idea. The dilated convolutions paper presents a more flexible approach than just Kroneckerfactors. Firstly, you can add nonlinearities after each layer of dilated convolution, which would now be different from Kronecker products. Secondly, the Kronecker analogy only holds if the dilation factor and the kernel size are the same. In the paper the authors used a kernel size of $3$ and dilation factor of $2$. #### Final note on translational equivariance One desirable property of convolutions is that they are translationally equivariant: if you shift the input image by any amount, the output remains the same, shifted by the same amount. This is a very useful inductive bias/prior assumtion to use in a dense prediction task. One way to introduce multiscale thinking to ConvNets is to use architectures that look like the figure below: we first decrease the spatial extent of featuremaps via pooling, then grow them back again via unpooling/deconvolution. Additional shortcut connections ensure that pixellevel local accuracy can be retained. The example below is from the SegNet paper, but there are multiple other papers such as this one on recombinator networks. ![](http://www.inference.vc/content/images/2016/05/convdeconv.png) However, as soon as you include spatial pooling, the translational equivariance property of the whole network might break. For example the SegNet above is not translationally equivariant anymore: the network's predictions are sensitive to small, singlepixel shifts to the input image, which is undesirable. Thankfully, layers of dilated convolutions are still translationally equivariant, which is a good thing. #### Summary This dilated convolutions idea is pretty cool, and I think these papers are just scratching the surface of this topic. The dilated convolution architecture generalises Kroneckerfactored convolutional filters, it allows for very large receptive fields while only growing the number o
Your comment:
