Learning Two-layer Neural Networks with Symmetric Inputs
We give a new algorithm for learning a two-layer neural network under a very general class of input distributions. Assuming there is a ground-truth two-layer network y = A sigma(Wx) + xi, where A, W are weight matrices, xi represents noise, and the number of neurons in the hidden layer is no larger than the input or output, our algorithm is guaranteed to recover the parameters A, W of the ground-truth network. The only requirement on the input x is that it is symmetric, which still allows highly complicated and structured input. Our algorithm is based on the method-of-moments framework and extends several results in tensor decompositions. We use spectral algorithms to avoid the complicated non-convex optimization in learning neural networks. Experiments show that our algorithm can robustly learn the ground-truth neural network with a small number of samples for many symmetric input distributions.
 This paper pushes forward our understanding of learning neural networks. The authors show that they can learn a two-layer (one hidden layer) NN, under the assumption that the input distribution is symmetric. The authors convincingly argue that this is not an excessive limitation, particularly in view of the fact that this is intended to be a theoretical contribution. Specifically, the main result of the paper relies on the concept of smoothed analysis. It states that give data generated from a network, the input distribution can be perturbed so that their algorithm then returns an epsilon solution. The main machinery of this paper is using a tensor approach (method of moments) that allows them to obtain a system of equations that give them their “neuron detector.” The resulting quadratic equations are linearized through the standard lifting approach (making a single variable in the place of products of variables). This is an interesting paper. As with other papers in this area, it is somewhat difficult to imagine that the results would extend to tell us about guarantees on learning a general depth neural network. Nevertheless, the tools and ideas used are of interest, and while already quite difficult and sophisticated, perhaps do not yet seem stretched to their limits.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Learning Two-layer Neural Networks with Symmetric Inputs
We give a new algorithm for learning a two-layer neural network under a very general class of input distributions. Assuming there is a ground-truth two-layer network y = A sigma(Wx) + xi, where A, W are weight matrices, xi represents noise, and the number of neurons in the hidden layer is no larger than the input or output, our algorithm is guaranteed to recover the parameters A, W of the ground-truth network. The only requirement on the input x is that it is symmetric, which still allows highly complicated and structured input. Our algorithm is based on the method-of-moments framework and extends several results in tensor decompositions. We use spectral algorithms to avoid the complicated non-convex optimization in learning neural networks. Experiments show that our algorithm can robustly learn the ground-truth neural network with a small number of samples for many symmetric input distributions.
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>



 This paper pushes forward our understanding of learning neural networks. The authors show that they can learn a two-layer (one hidden layer) NN, under the assumption that the input distribution is symmetric. The authors convincingly argue that this is not an excessive limitation, particularly in view of the fact that this is intended to be a theoretical contribution. Specifically, the main result of the paper relies on the concept of smoothed analysis. It states that give data generated from a network, the input distribution can be perturbed so that their algorithm then returns an epsilon solution. The main machinery of this paper is using a tensor approach (method of moments) that allows them to obtain a system of equations that give them their “neuron detector.” The resulting quadratic equations are linearized through the standard lifting approach (making a single variable in the place of products of variables). This is an interesting paper. As with other papers in this area, it is somewhat difficult to imagine that the results would extend to tell us about guarantees on learning a general depth neural network. Nevertheless, the tools and ideas used are of interest, and while already quite difficult and sophisticated, perhaps do not yet seem stretched to their limits.
assistant
 This paper pushes forward our understanding of learning neural networks. The authors show that they can learn a two-layer (one hidden layer) NN, under the assumption that the input distribution is symmetric. The authors convincingly argue that this is not an excessive limitation, particularly in view of the fact that this is intended to be a theoretical contribution. Specifically, the main result of the paper relies on the concept of smoothed analysis. It states that give data generated from a network, the input distribution can be perturbed so that their algorithm then returns an epsilon solution. The main machinery of this paper is using a tensor approach (method of moments) that allows them to obtain a system of equations that give them their “neuron detector.” The resulting quadratic equations are linearized through the standard lifting approach (making a single variable in the place of products of variables). This is an interesting paper. As with other papers in this area, it is somewhat difficult to imagine that the results would extend to tell us about guarantees on learning a general depth neural network. Nevertheless, the tools and ideas used are of interest, and while already quite difficult and sophisticated, perhaps do not yet seem stretched to their limits.