Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This paper studies nonparametric MLE inference of continuous determinantal point processes (DPPs). The authors analyze various representations between discrete DPPs and continuous DPPs. In particular, they observe that the difference of normalization of DPP is bounded by the trace of a matrix. Then, they consider a new optimization problem penalizing the trace and propose a method of estimating likelihood/correlation DPP kernels. Moreover, they study an iterative method (a.k.a Picard iteration) for solving the MLE problem with the regularization term. Statistical guarantees of the MLE objective and correlation kernel are also studied. 
main_review: This work tries to bridge the gap between continuous DPPs and discrete DPPs and provides some rigorous results. However, some derivation is unnatural and various analysis methods are already known. Moreover, as DPPs are used in many ML applications and mostly they are given by discrete settings, one should justify importance of continuous DPPs rather than discrete DPPs.

The equation (6) comes from penalizing the difference of normalization term as shown in Theorem 1. However, this only considers the error of normalization term and one needs to consider the first term (i.e., numerator in probability of DPP) in equation (5). Does the Theorem 1 similarly validate without identity operator?

In Theorem 6, the authors provide a statistical guarantee of correlation kernel. Is it possible to provide similar result in terms of the likelihood kernel in Algorithm 1?

Overall, it needs to be improved the justification of the contributions of this paper. Also, it would be better to provide more concrete results (e.g., error estimation) rather than visualization.

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I have read authors response and other reviews and noticed there was a misunderstanding regarding continuous DPPs. Authors clarified this in their response. I update my score.
limitations_and_societal_impact: - In line 192, it seems that m <= |C| + n since samples in \mathcal{I} can contain points in C_l.
- In Theorem 6, what is the \kappa?
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper introduces an algorithm to learn non-parametric, continuous determinantal point processes (DPPs). This algorithm relies on recent representer results for nonnegative functions in a RKHS and on a previous, related algorithm to learn discrete DPP parameterizations. 

The authors prove that the proposed algorithm is guaranteed to increase the MLE objective function at each iteration, as well as prove statistical guarantees for the resulting solution: distance to an optimal parameterization, sample complexity.
main_review: Originality: The proposed method to learn a non-parameteric, continuous DPP is new; it is, to the extent of my knowledge, the first result on learning non-parametric, continuous DPPs.

Quality: This paper is of high quality; the proofs are detailed, the theory meticulous (to the extent that I can judge it, being unfamiliar with results in Marteau-Ferey et al., 2020, and Rudi et al., 2020.). Furthermore, the theoretical analysis is regularly tied back to core concepts in DPPs (e.g., the effective sample size), and key arguments are regularly highlighted to support the provided analysis. 

Clarity: With the caveat that I am not familiar with continuous DPPs (but familiar with their discrete counterparts), I found this paper very well written and accessible. The authors carefully frame their contributions in terms of previously known results within continuous and discrete DPP landscapes, draw the reader's attention to crucial arguments in their proofs, and describe in detail the assumptions and potential limitations of their work. Overall, this is a very well written paper.

Significance: The results are significant to the community, in particular given growing interest in the applications of continuous DPPs. This is the first result on learning non-parametric continuous DPPs, and also has interesting implications for potential extensions to learning algorithms for discrete DPPs. One potential limitation of this work is that the empirical verification focuses only on exponentiated quadratic kernels.


Questions & Comments:
- You allude in the final section that the regularized objective might transfer to the finite DPP setting. Do you mean that by introducing a trace-based regularizer to the discrete Picard iteration, you may be able to recover a guarantee of the style of Theorem 4?
- Have you tried learning L-ensembles with correlation kernels that are not of the same form as the RKHS kernel? Would you expect this to influence the learning process?
limitations_and_societal_impact: The authors clearly describe the limitations of their work; I do not see a societal impact that requires discussing.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper presents a new approach for maximum likelihood estimation (MLE) for continuous determinantal point processes (DPPs), which leverages recent work on kernel methods.  This approach involves nonparametric learning for a restricted class of the MLE problem, which is shown to fall within the scope of recent work on a representer theorem for nonnegative functions in an RKHS.  An optimization problem over matrices is characterized, and then a fixed point algorithm for solving the finite-dimensional optimization problem is proposed.  Statistical guarantees for the MLE approximation are presented.  Finally, the authors provide an empirical validation of their proposed method using experiments run on synthetic data sampled from a DPP.
main_review: The MLE learning approach for continuous DPPs proposed in this paper is novel, and appears to be a significant contribution.  As noted by the authors, substantial prior work exists on discrete or finite DPPs, but much less attention has been paid to continuous DPPs within the machine learning community.  This paper is reasonably well written, and appears to be one of the first contributions in the area of nonparametric learning for continuous DPPs.  The main theoretical contributions in Sections 3 - 5 of the paper appear to be sound, although I have not carefully checked the proofs.  Furthermore, the statistical guarantees in Section 5 establish approximation bounds on the MLE solution and correlation kernel that appear to be reasonably tight under realistic assumptions, with the exception of the lower bound on $p$ in Theorem 6.

A few aspects of this paper could be improved.  First, the authors should discuss the computational complexity of the algorithms for estimating the DPP likelihood kernel (Alg. 1) and correlation kernel (Alg. 2).  What is the computational complexity of the fixed point algorithm in Sec. 4?  A very brief discussion of space and time complexity is given in Appendix S4.3, but this discussion should be significantly expanded upon.  Second, experiments that involve using the proposed approach to learn a continuous DPP from a real dataset should be performed.  Lastly, it would be helpful for the authors to provide a more detailed comparison of their nonparametric learning approach with prior work on parametric learning for continuous DPPs.
limitations_and_societal_impact: The authors have included a short, but adequate, discussion of the limitations of their work in Section 7 of the paper.  Since this work involves very general algorithmic and theoretical contributions, a discussion of potential negative societal impact does not appear to be applicable.
review_title: null

Meta-review: This paper studies nonparametric maximum likelihood estimation for continuous DPPs. In contrast to discrete DPPs, for continuous ones there are many open questions and new challenges. The main results are based on recent representer theorems for continuous DPPs as nonnegative functions in an RKHS. The paper uses this machinery to develop a new fixed point algorithm which is guaranteed to increase the MLE objective after each step. Furthermore they give strong statistical guarantees – bounds on the error of their estimator in the appropriate metric, depending on the sample complexity. All the reviewers agreed that the results were interesting and original, and that the paper was well-written and accessible. 
Decision: Accept (Poster)