Venue: ICLR 2017 conference

Reviews:

final_score: rating: 4: Ok but not good enough - rejection
review_content: I have not read the revised version in detail yet. 

SUMMARY 
This paper studies the preimages of outputs of a feedforward neural network with ReLUs. 

PROS 
The paper presents a neat idea for changes of coordinates at the individual layers. 

CONS 
Quite unpolished / not enough contributions for a finished paper. 

COMMENTS 
- In the first version the paper contains many typos and appears to be still quite unpolished. 

- The paper contains nice ideas but in my opinion it does not contribute sufficiently many results for a Conference paper. 
I would be happy to recommend for the Workshop track. 

- Irreversibly mixed and several other notions from the present paper are closely related to the concepts discussed in [Montufar, Pascanu, Cho, Bengio, NIPS 2014]. I feel that that paper should be cited here and the connections should be discussed. In particular, that paper also contains a discussion on the local linear maps of ReLU networks. 

- I am curious about the practical considerations when computing the pre-images. The definition should be rather straight forward really, but the implementation / computation could be troublesome. 


DETAILED COMMENTS 
- On page 1 ``can easily be shown to be many to one'' in general. 

- On page 2 ``For each point x^{l+1}'' The parentheses in the superscript are missing. 

- After eq. 6 ``the mapping is unique''  is missing `when w1 and w2 are linearly independent' 

- Eq. 1 should be a vector. 

- Above eq. 3. ``collected the weights a_i into the vector w'' and bias b. Period is missing. 

- On page 2 ``... illustrate the preimage for the case of points on the lines ... respectively'' 
Please indicate which is which.  

- In Figure 1. Is this a sketch, or the actual illustration of a network. In the latter case, please state the specific value of x and the weights that are depicted. Also define and explain the arrows precisely. 
What are the arrows in the gray part? 

- On page 3 `` This means that the preimage is just the point x^{(l)}''  the points that W maps to x^{(l+1)}. 

- On page 3 the first display equation. There is an index i on the left but not on the right hand side. 
The quantifier in the right hand side is not clear. 

- ``generated by the mapping ... w^i '' subscript

- ``get mapped to this hyperplane'' to zero 

- ``remaining'' remaining from what? 

- ``using e.g. Grassmann-Cayley algebra'' 
How about using elementary linear algebra?!

- ``gives rise to a linear manifold with dimension one lower at each intersection'' 
This holds if the hyperplanes are in general position. 

- ``is complete in the input space'' forms a basis 

- ``remaining kernel'' remaining from what? 

- ``kernel'' Here kernel is referring to nullspace or to a matrix of orthonormal basis vectors of the nullspace, or to what specifically? 

- Figure 3. Nullspaces of linear maps should pass through the origin. 

- `` from pairwise intersections'' \cap 

- ``indicated as arrows or the shaded area'' this description is far from clear. 

- typos: peieces, diminsions, netork, me, 
 


review_title: review of ``THE PREIMAGE OF RECTIFIER NETWORK ACTIVITIES''

final_score: rating: null
review_content: Hi, 
Sorry for the delay. The paper reads as unfinished. I'm wondering if the authors meant to submit a revision and didn't get a chance to do it? I mean, is not that the paper is not carefully written, is more that it feels not ready for ICLR in terms of content. 

First question I have is if the authors are aware of this work which seems relevant to current work (https://arxiv.org/abs/1402.1869). It was accepted at NIPS if I'm not mistaken. The papers don't overlap, but use the same kind of constructions so I thought it makes sense to point it out. 

After eq 4, you say this is a generalization of convnets. Isn't this just a standard MLP ? Also you mention this construction will be described for convolutional nets as well, though is never done in the paper. By later you mean in a different paper? (this are the kind of signs that make me think that the real intend was to upload a revision that I'd love to read). 

Finally why I understand what the authors are going for, their construction of a new coordinate system happens at each layer. How do you put all of these together? It feels you don't get something easy to work with when you do this (e.g. given an arbitrary MLP can I easily characterize the preimage  at some intermediary layer in a closed form and maybe reason about it?). Is not clear to me what the results of this work are and what implications it has. Any comments of the authors explaining their intend will be useful. I think is a very interesting (and full of potential) direction, it just seems to be the authors might still need some time to figure out what is their main result. 
review_title: Pre-review questions

final_score: rating: null
review_content: Kernel is referring to the nullspace? If so, should it not pass through the origin and what is a transformation kernel? 
What is the considered x^{(l+1)} in Figure 1? 


review_title: terminology

final_score: rating: 4: Ok but not good enough - rejection
review_content: I really appreciate the directions the authors are taken and I think something quite interesting can come out of it. I hope the authors continue on this path and are able to come up with something quite interesting soon. However I feel this paper right now is not quite ready. Is not clear to me what the results of this work are yet. The preimage construction is not obviously (at least not to me) helpful. It feels like the right direction, but it didn't got to a point where we can use it to identify the underlying mechanism behind our models. We know relu models need to split apart and unite different region of the space, and I think we can agree that we can construct such mechanism (it comes from the fact that relu models are universal approximators) .. though this doesn't speak to what happens in practice.  All in all I think this work needs a bit more work yet. 
review_title: not ready yet

final_score: rating: 4: Ok but not good enough - rejection
review_content: Summary:
This paper looks at the structure of the preimage of a particular activity at a hidden layer of a network. It proves that any particular activity has a preimage of a piecewise linear set of subspaces.

Pros:
Formalizing the geometry of the preimages of a particular activity vector would increase our understanding of networks

Cons:
Analysis seems quite preliminary, and no novel theoretical results or clear practical conclusions.

The main theoretical conclusion seems to be the preimage being this stitch of lower dimensional subspaces? Would a direct inductive approach have worked? (e.g. working backwards from the penultimate layer say?) This is definitely an interesting direction, and it would be great to see more results on it (e.g. how does the depth/width, etc affect the division of space, or what happens during training) but it doesn't seem ready yet.
review_title: Nice premise but somewhat incomplete?

Meta-review: This paper studies the invertibility properties of deep rectified networks, and more generally the piecewise linear structure that they implicitly define. The authors introduce a 'pseudocode' to compute preimages of (generally non-invertible) half-rectified layers, and discuss potential implications of their method with manifold-type models for signal classes. 
 
 The reviewers agreed that, while this is an interesting and important question, the paper is currently poorly organized, and leaves the reader a bit disoriented, since the analysis is incomplete. The AC thus recommends rejection of the manuscript. 
 
 As an addendum, the AC thinks that the authors should make an effort to reorganize the paper and clearly state the contributions, and not expect the reader to find them out on their own. In this field of machine learning, I see contributions as being either (i) theoretical, in which case we expect to see theorems and proofs, (ii) algorithmical, in which case an algorithmic is presented, studied, extensively tested, and laid out in such a way that the interested reader can use it, or (iii) experimental, in which case we expect to see improved numerical performance in some dataset. Currently the paper has none of these contributions. 
 Also, a very related reference seems to be missing: "Signal Recovery from Pooling Representations", Bruna ,Szlam and Lecun, ICML'14, in which the invertibility of ReLU layers is precisely characterized (proposition 2.2).
Decision: Reject