Separating mixed distributions is a long standing challenge for machine learning and signal processing. Applications include: single-channel multi-speaker separation (cocktail party problem), singing voice separation and separating reflections from images. Most current methods either rely on making strong assumptions on the source distributions (e.g. sparsity, low rank, repetitiveness) or rely on having training samples of each source in the mixture. In this work, we tackle the scenario of extracting an unobserved distribution additively mixed with a signal from an observed (arbitrary) distribution. We introduce a new method: Neural Egg Separation -an iterative method that learns to separate the known distribution from progressively finer estimates of the unknown distribution. In some settings, Neural Egg Separation is initialization sensitive, we therefore introduce GLO Masking which ensures a good initialization. Extensive experiments show that our method outperforms current methods that use the same level of supervision and often achieves similar performance to full supervision.
This paper describes a signal separation method called neural egg separation (NES).
The separation problem is tackled in a semi-supervised setting where the observed mixture contains a target signal and a background noise, with access to the distributions of target and mixture signals.
The strength of the paper is that it describes the importance of the problem setup for practical use with some motivating examples. 
However, some unclear notations weaken the claim of the paper.
Specific comments follow.
* The loss in (1) is unclear.
Assuming latex grammar, \| \| is usually used to denote a vector norm, but (1) has two values inside. 
I would write \ell(T(y_i), b_i) to show a loss function, instead of the \| \| style.
More importantly, the loss should be explicitly defined. Does this mean the l2 error?
* The iterative separation process of (2) is even unclear.
Does T^m(b_j + x_i^m) share the parameter of that from previous iterations like T^{m-1}?
Or are the parameters fixed throughout the iterations?
* Use of \cdot.
There may be a confusion between the inner product and element-wise product with the \cdot operator.
Right after (5), there is an inequality z \cdot z \leq 1, which is meant to be the inner product.
On the other hand, the use of \cdot in (8) looks like the element-wise product to describe a masking operation.
Clarifying the objective and overall procedures is necessary for presenting the proposed method.
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EDIT: I confirmed the revisions regarding the notation issues, but there still have confusing parts.
* Definitions of norm operator \| \| is unclear.
  * L_1 is mentioned below (1), and used other parts (3) or Algorithm 1. Equation (12) in Appendix uses |W1|_1^2, which looks like the l1 norm as well. Use consistent notations.
  * Equations (12, 13, 14) uses \|\|_2 or \|\|_1 to specify the type of norm, whereas (5), (6), (7) and other parts after (15) use \|\|. This confuses me. What do you mean by \|\| without subscript?
  * \|\| operator taking to symbols is a weird notation for me. Usually, norm is defined for a single vector (or a matrix). For example in (5), I would write \| b - G(z_b) \|, if you want to measure the difference between b and G(z_b).
The experimental result is impressive, as the other reviewers mention. I strongly recommend clarifying the notation to better deliver the method.