Reviews: On the Expressive Power of Deep Polynomial Neural Networks

Post-rebuttal: After reading the authors' response and further consideration, I am downgrading my score to 7 from 9. While I am still very excited about the new perspective this work brings, I now realize that there is still a lot of work remaining in order to tie the theoretical results to real-world phenomena. Regardless of whether the paper gets accepted, I'd ask the authors to make the gap clearer and to lay out more clearly an agenda for future work that address the various issues discussed in the rebuttal, e.g.: approximation, empirical notions of filling, etc. ORIGINALITY The paper considers the functional space of polynomial networks as an algebraic object. They use tools from algebraic geometry to analyze the dimension of the Zariski closure of this space. The paper is highly original in relating recent results from algebra to basic issues about neural networks. QUALITY & CLARITY This work tackles head-on the problem of analyzing the functional space of polynomial varieties.