Paper:
Successful training of convolutional neural networks is often associated with sufficiently deep architectures composed of high amounts of features. These networks typically rely on a variety of regularization and pruning techniques to converge to less redundant states. We introduce a novel bottom-up approach to expand representations in fixed-depth architectures. These architectures start from just a single feature per layer and greedily increase width of individual layers to attain effective representational capacities needed for a specific task. While network growth can rely on a family of metrics, we propose a computationally efficient version based on feature time evolution and demonstrate its potency in determining feature importance and a networks' effective capacity. We demonstrate how automatically expanded architectures converge to similar topologies that benefit from lesser amount of parameters or improved accuracy and exhibit systematic correspondence in representational complexity with the specified task. In contrast to conventional design patterns with a typical monotonic increase in the amount of features with increased depth, we observe that CNNs perform better when there is more learnable parameters in intermediate, with falloffs to earlier and later layers.

Review:
This paper introduces a simple correlation-based metric to measure whether filters in neural networks are being used effectively, as a proxy for effective capacity. The authors then introduce a greedy algorithm that expands the different layers in a neural network until the metric indicates that additional features will end up not being used effectively.
The application of this algorithm is shown to lead to architectures that differ substantially from hand-designed models with the same number of layers: most of the parameters end up in intermediate layers, with fewer parameters in earlier and later layers. This indicates that common heuristics to divide capacity over the layers of a network are suboptimal, as they tend to put most parameters in later layers. It's also nice that simpler tasks yield smaller models (e.g. MNIST vs. CIFAR in figure 3).
The experimental section is comprehensive and the results are convincing. I especially appreciate the detailed analysis of the results (figure 3 is great). Although most experiments were conducted on the classic benchmark datasets of MNIST, CIFAR-10 and CIFAR-100, the paper also includes some promising preliminary results on ImageNet, which nicely demonstrates that the technique scales to more practical problems as well. That said, it would be nice to demonstrate that the algorithm also works for other tasks than image classification.
I also like the alternative perspective compared to pruning approaches, which most research seems to have been focused on in the past. The observation that the cross-correlation of a weight vector with its initial values is a good measure for effective filter use seems obvious in retrospect, but hindsight is 20/20 and the fact is that apparently this hasn't been tried before. It is definitely surprising that a simple method like this ends up working this well.
The fact that all parameters are reinitialised whenever any layer width changes seems odd at first, but I think it is sufficiently justified. It would be nice to see some comparison experiments as well though, as the intuitive thing to do would be to just keep the existing weights as they are.
Other remarks:
Formula (2) seems needlessly complicated because of all the additional indices. Maybe removing some of those would make things easier to parse. It would also help to mention that it is basically just a normalised cross-correlation. This is mentioned two paragraphs down, but should probably be mentioned right before the formula is given instead.
page 6, section 3.1: "it requires convergent training of a huge architecture with lots of regularization before complexity can be introduced", I guess this should be "reduced" instead of "introduced".