Robust density estimation refers to the consistent estimation of the density function even when the data is contaminated by outliers. We find that existing forest density estimation at a certain point is inherently resistant to the outliers outside the cells containing the point, which we call \textit{non-local outliers}, but not resistant to the rest \textit{local outliers}. To achieve robustness against all outliers, we propose an ensemble learning algorithm called \textit{medians of forests for robust density estimation} (\textit{MFRDE}), which adopts a pointwise median operation on forest density estimators fitted on subsampled datasets. Compared to existing robust kernel-based methods, MFRDE enables us to choose larger subsampling sizes, sacrificing less accuracy for density estimation while achieving robustness. On the theoretical side, we introduce the local outlier exponent to quantify the number of local outliers. Under this exponent, we show that even if the number of outliers reaches a certain polynomial order in the sample size, MFRDE is able to achieve almost the same convergence rate as the same algorithm on uncontaminated data, whereas robust kernel-based methods fail. On the practical side, real data experiments show that MFRDE outperforms existing robust kernel-based methods. Moreover, we apply MFRDE to anomaly detection to showcase a further application.

Post-rebuttal: The authors have promised to incorporate an exposition of the sampler in the revised paper, I believe that will make the paper a more self-contained read. I maintain my rating of strong accept (8). I think this paper makes very nice contributions to the fundamental question of estimating the MLE distribution given a bunch of observations. I think the key contributions can be broken up into two key parts: - A bunch of simple but elegant structural results for the MLE distribution in terms of'tent distributions' -- distributions such that its log-density is piecewise linear, and is supported over subdivisions of the convex hull of the datapoints. This allows them to write a convex program for optimizing over tent distributions.

The submission provides a polynomial-time approximation algorithm for finding the maximum-likelihood log-concave density for a given set of data points in R d, for arbitrary d. The work is theoretical in nature, with proofs and no experiments. The problem is very interesting, since log-concave distributions include may of the commonly used parametric families (such as Gaussian), and the log-concave MLE has also other interesting properties. Previously the sample-complexity of learning a log-concave distribution has been studied, but a polynomial-time algorithm has been lacking. The present work provides such an algorithm.

The paper targets the problem of measuring the representation power of Graph neural networks (GNNs), an interesting and important topic, that has become popular recently (partially due to two prominent works (Xu et al. There are three main contributions: 1. Establishing the equivalence between two methods for measuring GNN representation power: (i) their ability to approximate permutation invariant functions (ii) their ability to distinguish non-isomorphic graphs. Although not very surprising, this is a nice observation. The authors show that these sigma algebras are an equivalent way to measure representation power of GNNs, for instance, the inclusion of sigma algebras originating from two models is equivalent to saying one model is more powerful than the other. This is a potentially useful observation.

This paper leverages the graph isomorphism problem to study the expressive power of GNNs. In addition, a measure of expressiveness is formalized using sigma-algebras and the authors propose a novel variant of GNN, RING-GNN, that is evaluated in an experimental study where it shows competitive results. The reviewers agree that this is a nice contribution, the theoretical results are interesting (though somehow expected) and that the proposed extension of G-invariant networks is relevant. However, all reviewers agree that the experimental comparison with RING-GNN-SVD is unfair and MUST BE REMOVED in a published version of the paper (that is removing the last line from table 1). One of the reviewer also note that a comparison with LanczosNet should be included (though the lack of comparison is not ground for rejection).

Since everything is parametric, I'd expect explicit rates of convergence involvind all probalem complexity parameters (n, m, p, etc.) To make the rest of my points clear, let me recall the following notations are used in the paper: - n: the dimensionality of the covariate (i.e feature vector) X. Thus X is random vector in R n. BTW, in the context of ML or stats, I'd use another notation here, as n conventionally stands for "sample size".

This paper proposes a method for distributionally robust optimization under KL ambiguity sets for exponential families. Although KL ambiguity sets have their drawbacks, in particular not covering any changes in the inputs x, the present work produces a standard conic problem for a wide problem class via a novel analysis, provides good theoretical analysis, and yields good numerical results for a variety of small-scale classification problems. With the various clarifications that came up in the reviews, this paper makes a solid contribution to the DRO literature and will be quite welcome to the NeurIPS audience.

The main contributions of this work are pulling these ideas together into a practical framework that works on a real large-scale simulator. The original challenges that are addressed include: how to apply PPL to an existing code base? The other strength of the paper is the sheer depth of related work that is considered and explained, while being smooth to read at the same time. Ideally, we would have had more detail on the specific contributions of this paper, particularly on the "prior inflation" scheme and the protocol. The limitations of the writing come mainly from needing further explanation and discussion for why various ideas are being used, e.g., why do you consider LMH, RMH, IC? why would you "like to employ deep neural networks" in this context?

The paper presents a new probabilistic programming framework that makes Bayesian inference applicable to simulation code at scale. A large scale high energy physics application is presented. Probabilistic inference can be applied to an existing simulation code bass, allowing for'plug-and-play' inference. A large-scale particle physics application was provided. On the downside, the involved inference approaches themselves have already been published before.

This paper develops a novel method to infer directional relationships between cortical areas of the brain based on simultaneously acquired EEG and fMRI data. Specifically, the fMRI activations are used to select ROIs related to the paradigm of interest. This information is used in a coupled state-space and forward propagation model to identify robust spatial sources and directional connectivity. The authors use a variational Bayesian framework to infer the latent posteriors and noise covariances. They demonstrate the power of joint EEG/fMRI analysis using two simulated experiments and a real-world dataset.