I'm trying to understand the structure of the temporal convolutional neural network as described in here. Figure 1 of the paper illustrates a residual block. Are the dilated convolutions on the left the same as the layers on the right? i.e., at the bottom layer, dilated convolutions are applied then -> weight normalisation -> ReLU -> dropout -> next dilated convolution layer? If so, why is the 1x1 convolution needed?
Yes they're the same. The 1x1 convolution is in both of them. For residual mappings, you're adding the old layer's input value to the input of the later layer down the line (aka up the image). If the convolutional layers in between decrease the dimensionality of the image (for example by using zero padding), then the input dimension will be smaller for the later layer than the earlier one. Using a 1x1 convolution is a way of resizing the earlier input to match the later input size.