NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID: 7244 Learning nonlinear level sets for dimensionality reduction in function approximation

### Reviewer 1

The paper is well-writen, there are a few flaws in notation and typos though, such as: - (62) rho should map to \mathbb{R}_{\geq 0}, since it maps values to 0 - (133) remove paranthesis - (150) the partial derivatives are somewhat misleading, yet formal correct; i suggest you replace \frac{\partial x_1}{\partial z_i}(z) with \frac{\partial g^{-1}(z)}{\partial z_i}. By equation (6) it is clear, that this is meant, but i find this explicit formulation more comprehensible. - Figure 5: i think you mean epoch; the caption reads 'epcho' General comments: - All figures are too small. Without the ability to zoom, e.g. when you print out your paper, it is hardly possible to comprehend what is depicted. - the statement in line 153 and below is misleading and also wrong, if one does insist on formal correctness, equation 8 is a tautology: in Euclidean space every Orthogonal is a Perpendicular, and every Perpendicular is an Orthogonal, so this equivalence holds independent of the choice of x (assuming that z = g(x)); what you want to say is: f(x) does not change with z_i in the neighbourhood of z <=> \langle J_i(z), \nabla f(x) \rangle = 0 - Figure 2 is a bit hard to understand, though very expressive - in line 218 "sensitivity" is defined different than in Figure 2 - in equation 12 and 13 f_1 and f_2 are defined over [0,1] x [0,1] but in figure 2 we see that z_1 takes negative values - how is the domain of g being determined? Bottom line: I think this is a good and solid work which, except from the aforementioned flaws, supplies innovative methodology endowed with comprehensive thoughtful experiments, which do not only support the theoretical aspects of the methodology but also show the viability in real-world applications. #### After reading the rebuttal, I thank the authors for the clarifications and incorporating my suggestions.

### Reviewer 2

The paper well structured, easy to read and technically sound. The problem under consideration is of importance in engineering and the proposed solution appears to be a first step towards a new general tool for more efficient function approximation. The use of RevNets in this context is an original and novel idea. On the other one could argue that the major novelty of this work is a bit shallow (a new problem specific loss function) and that, as the authors admit, use of the method is still restricted to rather specific scenarios. However, on the plus side, this paper makes headway into a direction of considerable industrial interest and demonstrates the potential of neural networks beyond their common use cases.