Profiling cellular phenotypes from microscopic imaging can provide meaningful biological information resulting from various factors affecting the cells. One motivating application is drug development: morphological cell features can be captured from images, from which similarities between different drugs applied at different dosages can be quantified. The general approach is to find a function mapping the images to an embedding space of manageable dimensionality whose geometry captures relevant features of the input images. An important known issue for such methods is separating relevant biological signal from nuisance variation. For example, the embedding vectors tend to be more correlated for cells that were cultured and imaged during the same week than for cells from a different week, despite having identical drug compounds applied in both cases. In this case, the particular batch a set of experiments were conducted in constitutes the domain of the data; an ideal set of image embeddings should contain only the relevant biological information (e.g. drug effects). We develop a general framework for adjusting the image embeddings in order to 'forget' domain-specific information while preserving relevant biological information. To do this, we minimize a loss function based on distances between marginal distributions (such as the Wasserstein distance) of embeddings across domains for each replicated treatment. For the dataset presented, the replicated treatment is the negative control. We find that for our transformed embeddings (1) the underlying geometric structure is not only preserved but the embeddings also carry improved biological signal (2) less domain-specific information is present.
The authors present a method that aims to remove domain-specific information while preserving the relevant biological information between biological data measured in different experiments or "batches". A network is trained to learn the transformations that minimize the Wasserstein distance between distributions. The wasserstein distance is also called the "earth mover distance" and is traditionally formulated as the cost it takes for an optimal transport plan to move one distribution to another. In this paper they have a neural network compute the wasserstein distance using a different formulation that was used in Arjovsky et al. 2017, finds a lipschitz function f, which shows the maximal difference when evaluated on samples from the two distributions. Here these functions are formulated as affine transforms of the data with parameters theta that are computed by a neural network. Results are examined mainly by looking at the first two PCA components of the data.  
The paper presents an interesting idea and is fairly well written. However I have a few concerns:
1. Most of the ideas presented in the paper rely on works by Arjovsky et al. (2017), Gulrajani et al. (2017), and Gulrajani et al. (2017). Some selections, which are presented in the papers are not explained, for example, the gradient penalty, the choice of \lambda and the choice of points for gradient computation.
2. The experimental results are not fully convincing, they simply compare the first two PC components on this Broad Bioimage benchmark collection. This section could be improved by demonstrating the approach on more datasets.
3. There is a lack comparison to other methods such as Shaham et al. (2017). Why is using earth mover distance better than MMD based distance? They only compare it to a method named CORAL and to Typical
Variation Normalization (TVN). What about comparison to other batch normalization methods in biology such as SEURAT? 
4. Why is the affine transform assumption valid in biology? There can definitely be non-linear effects that are different between batches, such as ion detection efficiency differences. 
5. Only early stopping seems to constrain their model to be near identity. Doesn't this also prevent optimal results ? How does this compare to the near-identity constraints in resnets in Shaham et al. ?