Trajectory growth through random deep ReLU networks
This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected trajectory growth through random networks, for a more general class of weights distributions, including sparsely connected networks. We illustrate this approach by deriving bounds for sparse-Gaussian, sparse-uniform, and sparse-discrete-valued random nets. We prove that trajectory growth can remain exponential in depth with these new distributions, including their sparse variants, with the sparsity parameter appearing in the base of the exponent.
Summary: The authors examine trajectory growth of deep ReLU neural networks whose weights come from a (random) “sparse-Gaussian”, “sparse-uniform”, and “sparse-discrete” distribution. They give definitions of these distributions in the paper. They do this by extending the proof of Raghu (2017) so that it can handle more general distributions than the standard Gaussian. They also provide some numerical experiments verifying their theories. Strengths: The paper is well-written and the proofs are clearly explained. I’m grateful that the authors specifically mentioned where their proof deviates from the original and they clearly delineate how their proof method extends Raghu (2017) Weaknesses: This is an interesting direction, but I do not believe there are enough results to constitute an accept. If the authors are following Raghu (2017), then I would have also liked to see analysis on trained networks as done in Raghu (2017) for example. I also think the title of the paper is too general for the specific results contained in the paper, namely sparsity should at least be mentioned in the title.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Trajectory growth through random deep ReLU networks
This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected trajectory growth through random networks, for a more general class of weights distributions, including sparsely connected networks. We illustrate this approach by deriving bounds for sparse-Gaussian, sparse-uniform, and sparse-discrete-valued random nets. We prove that trajectory growth can remain exponential in depth with these new distributions, including their sparse variants, with the sparsity parameter appearing in the base of the exponent.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



Summary: The authors examine trajectory growth of deep ReLU neural networks whose weights come from a (random) “sparse-Gaussian”, “sparse-uniform”, and “sparse-discrete” distribution. They give definitions of these distributions in the paper. They do this by extending the proof of Raghu (2017) so that it can handle more general distributions than the standard Gaussian. They also provide some numerical experiments verifying their theories. Strengths: The paper is well-written and the proofs are clearly explained. I’m grateful that the authors specifically mentioned where their proof deviates from the original and they clearly delineate how their proof method extends Raghu (2017) Weaknesses: This is an interesting direction, but I do not believe there are enough results to constitute an accept. If the authors are following Raghu (2017), then I would have also liked to see analysis on trained networks as done in Raghu (2017) for example. I also think the title of the paper is too general for the specific results contained in the paper, namely sparsity should at least be mentioned in the title.
assistant
Summary: The authors examine trajectory growth of deep ReLU neural networks whose weights come from a (random) “sparse-Gaussian”, “sparse-uniform”, and “sparse-discrete” distribution. They give definitions of these distributions in the paper. They do this by extending the proof of Raghu (2017) so that it can handle more general distributions than the standard Gaussian. They also provide some numerical experiments verifying their theories. Strengths: The paper is well-written and the proofs are clearly explained. I’m grateful that the authors specifically mentioned where their proof deviates from the original and they clearly delineate how their proof method extends Raghu (2017) Weaknesses: This is an interesting direction, but I do not believe there are enough results to constitute an accept. If the authors are following Raghu (2017), then I would have also liked to see analysis on trained networks as done in Raghu (2017) for example. I also think the title of the paper is too general for the specific results contained in the paper, namely sparsity should at least be mentioned in the title.