We thank the reviewers for their helpful comments. We will primarily address reviewer_8's concern that algorithmic details were not clear enough. We firmly believe in improving reproducibility of results in computer science. The most useful step we can take in this regard is to release open source code implementing the algorithms. In the camera ready version of the paper, we will include a link to a github repository including not just the main algorithm, but also the code for estimating "remainder information" and code for generating the figures in the paper (we could not include this link in the submission without violating the double blind policy). The subroutine for optimizing TC is described only partially in Appendix A because this optimization is a special case of an existing algorithm. However, we agree with reviewer_8's sentiment about the difficulty of reconstructing the results from previous work. Indeed, the special case that appears here is so much simpler that a more complete description could be instructive. To that end, we will include an algorithm box and discussion about complexity in the appropriate section of the appendix (O(#variables * #samples * max cardinality of discrete variables)). $
Other comments:
- We thank the reviewers for pointing out several minor issues which we will correct. 
- Reviewer_7 makes a good point that this looks more like an iterative procedure than a "deep" one. We will make some rhetorical adjustments. 
- Reviewer_7 asks "[Is it correct that] iterating the process removes dependencies in the remainder information, but not in the already specified Y variables[?]" Not true! The Y's themselves may have dependencies that are captured at subsequent layers. This might not have been clear because we define X^k to include Y_1…Y_{k-1} within it. We will call attention to this shortcoming in the notation. 
- Reviewer_6: One way to look at this is as a kind of subspace clustering. Each new latent factor induces a new clustering that captures dependence in some subspace. We'll add a discussion about that in related work if there's room.