Paper ID: | 4215 |
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Title: | On two ways to use determinantal point processes for Monte Carlo integration |

Update: Thanks for the detailed feedback. I appreciate the theoretical and methodological exploration and improvements, but I hope the presentation can be made more clear to eliminate possible confusions. ====== I believe the quality of the paper is good, as the technical development and proof are sound and theoretical concerns are widely covered. I am not satisfied with the presentation. I cannot distinguish clearly the novel contributions from existing results, e.g., I cannot tell if Theorem 1 has new results or is a known result listed for a modern proof, if it is a novel idea to use the multivariate orthonormal polynomials in EZ, and if Section 3.1 contains novel results. I also expect more details on the methodological modification over BH for the proposed sampling method for multivariate Jacobi ensemble. I expect at least one experimental result is provided in the main context for each of sections 4.2 and 4.3. Based on the confusion, I cannot tell precisely the originality and significance. But I think a modern analysis on an old method in parallel with a recent one aligns important methods in the field, providing more connections and comparisons of the two methods, and making a foundation for more methods and analysis in the field.

I believe that the paper contains extremely interesting material. It is very interesting from a theoretical and practical point of view. Quadrature techniques (deterministic or stochastic) are fundamental tools for several applications. However, in my opinion, the key- underlying point of the ideas contained in the paper is the procedure (an efficient one) for drawing from DPPs. If there is one sampling scheme described in the current version of the paper, it must be explained better and explicitly remarked with a table, for instance. If the authors consider that is just a background part contained in other previous papers, I disagree. I believe this is the core, the key point, in order to use DPPs then it should appear in a background section at least. Moreover, this part should be very clean and clear in order to guide and motivate the reader inside the more complex and theoretical parts of your contribution. If the reader acquires the ability of generating a realization of a DPP, surely your paper increases its impact. This is also true that the provided code helps substantially in this sense.

Determinantal point processes (DPP) are currently a very active research topic. They provide a law on point processes that favors repulsion. The use of DPP in Monte Carlo methods is tempting, but rather difficult because simulation algorithms are not obvious. The present paper studies two Monte Carlo algorithms based on DPP: Ermakov & Zolotukhin (EZ, 1960) and Bardenet & Hardy (2016, BH). The latter paper provides a fast CLT of the estimator, while there is no such results about the EZ estimate in the literature. After proving the consistency of the EZ estimate via DPP, the paper discusses the possibility of a normal asymptotic behavior on simulation. DPP based Monte Carlo algorithms are certainly new and might represent a major improvement in this domain, since repulsion induces a faster speed of convergence than iid sampling or importance sampling or Markov chain Monte Carlo. The present paper highlights the key features of the EZ and BH estimators based on DPP.