NIPS 2018
Sun Dec 2nd through Sat the 8th, 2018 at Palais des Congrès de Montréal
Paper ID: 3779 Rectangular Bounding Process

Reviewer 1

The authors define a new random partition of product spaces, called Rectangular Bounding Process (RBP). Stochastic partition models are useful to divide a product space into a certain number of regions, in such a way that data in each region are \textit{homogeneous}. A well-known an example of these types of models is the Mondrian process defined in Roy and Teh (2009). The authors state that many models in existing literature produce unnecessary cuts in regions where data are rare, and the RBP tries to overcome such an issue. In Section 2 the authors review some previous works on the subject, in Section 3 they define the RBP, describing some of its properties, among which the self--consistency. At the end of the paper they provide some applications (regression trees and relational modeling). I think that the paper is not sufficiently clear and not well written, furthermore the definition of the RBP is not rigorous and precise. There are many obscure points that are not appropriately explained and discussed, and too many inaccuracies throughout the manuscript. I suggest to reject the paper, and below I list the main points which support my conclusion and help the authors to rewrite the whole paper for possible publication elsewhere. 1) In the introduction the authors have to underline and explain why stochastic partition model are important tools in real applications. 2) Section 2 is not well organized, it seems to me a long list of previous contributions and related works without a clear focus. The authors should rewrite the section in a more systematic way, limiting themselves to underline the difference between the RBP with respect to the other existing stochastic partition processes. 3) Is there a big difference in terms of flexibility between the Mondrian process and the proposed RBP? What does the RBP bring with respect to the Mondrian process? 4) Pg. 2, line 42. What does it mean expected total volume''? Please, specify the definition. 5) Pg. 3, line 110. What does it mean significant regions''? How do you define a significant region? 6) Pg. 3, line 112-113. $\lambda, \tau$ are introduced here, the authors say that they are parameters, do they belong to $\R$ or to a subset of $\R$? The domain must be specified. 7) Pg. 3, line 118. Is $P (s_k^{(d)})$ a probability distribution? It seems that this is not a probability distribution, indeed \begin{align*} P(s_{k}^{(d)} \in [0, L^{(d)}]) &= P(s_{k}^{(d)}=0)+P(s_{k}^{(d)} \in (0, L^{(d)}]) = \frac{1}{1+\lambda L^{(d)}} + \frac{\lambda}{1+\lambda L^{(d)}}\int_0^{L^{(d)}}\frac{1}{L^{(d)}} \D s\& = \frac{1+\lambda}{1+\lambda L^{(d)}} \not = 1 \end{align*} Furthermore the notation $P (s_k^{(d)})$ has not actual probabilistic meaning, one should use $\Lcr(s_{k}^{(d)})$ to denote the law of the random variable, otherwise $P(s_{k}^{(d)} \in A)$ to denote the probability that the random variable $s_{k}^{(d)}$ belongs to the Borel set $A$. This comment applies throughout the manuscript. 8) Pg. 3, line 121. The role of the cost of $\square_k$ is not clear. 9) Pg. 4, line 122. What does it mean The generative process defines a measurable space''? This sentence has no actual meaning, a random variable cannot define a measurable space. 10) Pg. 4, line 122. $\mathcal{F}(\R^D)$ has not been defined. 11) Pg.4, line 123. Each element in $\Omega_X$ denotes a partition $\boxplus_X$'' a partition of what? Probably a partition of $X$. 12) Pg. 4, line 125. $\square_k$ was a box, now is a product of functions. What do the authors mean with the notation $\bigotimes_{d=1}^D u_k^{(d)} ([0, L^{(d)}])$? It seems to me a product of indicator functions, but the authors speak of an outer product. I think that they have to clarify this point. 13) Pg. 4, Proposition 1. What do you mean with the phrase the value of $\tau$''? In the proof of Proposition 1 (see the supplementary material) the authors evaluate $\E[l_k^{(d)}]$, which is not the expected total volume of the bounding boxes mentioned in the statement of the proposition. The final step of the proof is given just after the statement of Proposition 1. This is very confusing. 14) Pg. 5, Figure 3. This figure has to be removed and the self-consistency has to be explained by words. Some minor comments are listed below: 1) Pg. 1, line 26. Replace As'' with as''; 2) Pg. 3, line 121. The notation $\square_k$ is not elegant; 3) Pg. 4, line 124. Teplace boxs'' with boxes''; 4) Pg. 5, line 183. Teplace Combining1'' with Combining 1''. References. D.M. Roy and Y.W. Teh (2009). The Mondrian process. In \textit{NIPS}, pp. 1377--1384