1. The proposed method only gets convergence rate in expectation (i.e. only variance bound), not with high probability. Though Chebyshev's inequality gives bound in probability from the variance bound, this is still weaker than that of Bach [3]. 2. The method description lacks necessary details and intuition: - It's not clear how to get/estimate the mean element mu_g for different kernel spaces. - It's not clear how to sample from the DPP if the eigenfunctions e_n's are inaccessible (Eq (10) line 130). This seems to be the same problem with sampling from the leverage score in [3], so I'm not sure how sampling from the DPP is easier than sampling from the leverage score. - There is no intuition why DPP with that particular repulsion kernel is better than other sampling schemes. 3. The empirical results are not presented clearly: - In Figure 1: what is "quadrature error"? Is it the sup of error over all possible integrand f in the RKHS, or for a specific f? If it's the sup over all f, how does one get that quantity for other methods such as Bayesian quadrature (which doesn't have theoretical guarantee). If it's for a specific f, which function is it, and why is the error on that specific f representative of other functions? Other comment: - Eq (18), definition of principal angle: seems to be missing absolute value on the right hand side, as it could be negative. Minor: - Reference for Kernel herding is missing [?] - Line 205: Getting of the product -> Getting rid of the product - Please ensure correct capitalization in the references (e.g., [1] tsp -> TSP, [39] rkhss -> RKHSs) [3] F. Bach. On the equivalence between kernel quadrature rules and random feature expansions. The Journal of Machine Learning Research, 18(1):714â751, 2017. ===== Update after rebuttal: My questions have been adequately addressed. The main comparison in the paper seems to be the results of F. Bach [3]. Compared to [3], I do think the theoretical contribution (better convergence rate) is significant. However, as the other reviews pointed out, the theoretical comparison with Bayesian quadrature is lacking. The authors have agreed to address this. Therefore, I'm increasing my score.
- It's not clear how to sample from the DPP if the eigenfunctions e_n's are inaccessible (Eq (10) line 130). This seems to be the same problem with sampling from the leverage score in [3], so I'm not sure how sampling from the DPP is easier than sampling from the leverage score.