Paper:
The resemblance between the methods used in studying quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and a request of high bond dimension. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multipartite entanglement renormalization ansatz (MERA). This approach overcomes scalability issues and implies novel mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states can be defined, which optimally encodes each class of the images into a quantum many-body state. We study the quantum features of the TN states, including quantum entanglement and fidelity. We suggest these quantities could be novel properties that characterize the image classes, as well as the machine learning tasks. Our work could be further applied to identifying possible quantum properties of certain artificial intelligence methods.

Review:
Full disclosure: the authors' submission is not anonymous. They included a github link at the bottom of page 6 and I am aware of the name of the author and coauthors (and have previously read their work and am a fan of it). Thus, this review is not double blind. I notified the area chair last week and we agreed that I submit this review. 
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This is an interesting application of tensor networks to machine learning. The work proposes using a tree tensor network for image classification. Each image is first mapped into a higher-dimensional space. Then the input features are contracted with the tensors of the tensor network. The maximum value of the final layer of the network gives the predicted class. The training algorithm is inspired by the multipartite entanglement renormalization ansatz: it corresponds to updating each tensor in the network by performing a singular value decomposition of the environment tensor (everything in the cost function after removing the current tensor to be updated).
Overall, I think this is an interesting, novel contribution, but it is not accessible to non-physicists right now. The paper could be rewritten to be accessible to non-physicists and would be a highly-valuable interdisciplinary contribution.
* Consider redoing the experiments with a different cost function: least squares is an unnatural cost function to use for classification. Cross entropy would be better.
* discuss the scalability: why did you downsample MNIST from 28x28 pixels to 16x16 pixels? Why is training accuracy not reported on the 10-class model in Table 1? If it is because of a slow implementation, that's fine. But if it is because of the scalability of the method, it would be good to report that. In either case it wouldn't hurt the paper, it is just important to know. 
* In section 5, you say "input vectors are still initially arranged ... according to their spatial locations in the image". But don't you change the spatial locations of the image to follow equation (10)? It would be good to add a sentence clarifying this. 
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In its current form, reading the paper requires a physics background. 
There are a few things that would make it easier to read for a general machine learning audience:
* connect your method to matrix factorization and tensor decomposition approaches
* include an algorithm box for Strategy-I and Strategy-II
* include an appendix, with a brief review of upward and downward indices which is crucial for understanding your method (few people in machine learning are familiar with Einstein notation)
* relate your interesting ideas about quantum states to existing work in information theory. I am skeptical of the label 'quantum': how do quantum mechanical tools apply to images? What is a 'quantum' many-body state here? There is no intrinsic uncertainty principle at play in image classification. I would guess that the ideas you propose are equivalent to existing work in information theory. That would make it less confusing.
* in general, maybe mention the inspiration of your work from MERA, but avoid using physics language when there are no clear physical systems. This will make your work more understandable and easier to follow. A high-level motivation for MERA from a physics perspective suffices; the rest can be phrased in terms of tensor decompositions.  
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Minor nits:
* replace \citet with \citep everywhere - all citations are malformed
* figure 1 could be clarified - say that see-through gray dots are dimensions, blue squares are tensors, edges are contractions
* all figure x and y labels and legends are too small
* some typos: "which classify an input image by choosing"; "we apply different feature map to each"; small grammar issues in many places
* Figure 4: "up-down" and "left-right" not defined anywhere