Chromatic Correlation Clustering (CCC) (introduced by Bonchi et al. [6]) is a natural generalization of the celebrated Correlation Clustering (CC) problem, introduced by Bonchi et al. [6]. It models objects with categorical pairwise relationships by an edge-colored graph, and has many applications in data mining, social networks and bioinformatics. We show that there exists a $2.5$-approximation to the CCC problem based on a Linear Programming (LP) approach, thus improving the best-known approximation ratio of 3 achieved by Klodt et al. [21] . We also present an efficient heuristic algorithm for CCC leveraging a greedy clustering strategy, and conduct extensive experiments to demonstrate the effectiveness and efficiency of our proposed algorithm.

We improve the effectiveness of propagation-and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is significantly stronger than the commonly used univariate-input relaxation which has been proposed as a natural convex relaxation barrier for verification. While our description of the relaxation may require an exponential number of inequalities, we show that they can be separated in linear time and hence can be efficiently incorporated into optimization algorithms on an as-needed basis. Based on this novel relaxation, we design two polynomial-time algorithms for neural network verification: a linear-programming-based algorithm that leverages the full power of our relaxation, and a fast propagation algorithm that generalizes existing approaches. In both cases, we show that for a modest increase in computational effort, our strengthened relaxation enables us to verify a significantly larger number of instances compared to similar algorithms.

Designing spectral convolutional networks is a challenging problem in graph learning. ChebNet, one of the early attempts, approximates the spectral graph convolutions using Chebyshev polynomials.

Summary and Contributions: ***Update after author feedback*** Authors have addressed some of my concerns in feedback. After some reflection, I tend to *conditionally* increase my score (from 6 to 7) because of the following reasons: 1. they have provided partial evidence of assessment of variance in the experiments, showing that results are significant. It is crucial that such variance estimates are added to the paper and mentioned. There are strong arguments from a majority in the community, that not assessing variance arising from randomness is an experimental flaw. This is the *conditional* part of my score increase 2. The fact that they provide a separation oracle for the *tightest* convex relaxation of ReLU, helps to close down the chapter on such type of methods based on convex relaxations of activations.

Chromatic Correlation Clustering (CCC) (introduced by Bonchi et al. [6]) is a natural generalization of the celebrated Correlation Clustering (CC) problem, introduced by Bonchi et al. [6]. It models objects with categorical pairwise relationships by an edge-colored graph, and has many applications in data mining, social networks and bioinformatics. We show that there exists a 2.5 -approximation to the CCC problem based on a Linear Programming (LP) approach, thus improving the best-known approximation ratio of 3 achieved by Klodt et al. [21] . We also present an efficient heuristic algorithm for CCC leveraging a greedy clustering strategy, and conduct extensive experiments to demonstrate the effectiveness and efficiency of our proposed algorithm.

We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is significantly stronger than the commonly used univariate-input relaxation which has been proposed as a natural convex relaxation barrier for verification. While our description of the relaxation may require an exponential number of inequalities, we show that they can be separated in linear time and hence can be efficiently incorporated into optimization algorithms on an as-needed basis.

Designing spectral convolutional networks is a challenging problem in graph learning. ChebNet, one of the early attempts, approximates the spectral graph convolutions using Chebyshev polynomials. GPR-GNN and BernNet demonstrate that the Monomial and Bernstein bases also outperform the Chebyshev basis in terms of learning the spectral graph convolutions. Such conclusions are counter-intuitive in the field of approximation theory, where it is established that the Chebyshev polynomial achieves the optimum convergent rate for approximating a function. In this paper, we revisit the problem of approximating the spectral graph convolutions with Chebyshev polynomials.

This paper provides a clever method for tightening a well-known convex relaxation.

We study properties of popular near–uniform (Dirichlet) priors for learn- ing undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam–style phase space argument expands the priors into their infinite mixture and resolves most of the observed problems. This leads to a surprisingly good estimator of entropies of discrete distributions. Learning a probability distribution from examples is one of the basic problems in data analysis. Common practical approaches introduce a family of parametric models, leading to questions about model selection.

Designing spectral convolutional networks is a challenging problem in graph learning. ChebNet, one of the early attempts, approximates the spectral convolution using Chebyshev polynomials. GCN simplifies ChebNet by utilizing only the first two Chebyshev polynomials while still outperforming it on real-world datasets. GPR-GNN and BernNet demonstrate that the Monomial and Bernstein bases also outperform the Chebyshev basis in terms of learning the spectral convolution. Such conclusions are counter-intuitive in the field of approximation theory, where it is established that the Chebyshev polynomial achieves the optimum convergent rate for approximating a function. In this paper, we revisit the problem of approximating the spectral convolution with Chebyshev polynomials. We show that ChebNet's inferior performance is primarily due to illegal coefficients learnt by ChebNet approximating analytic filter functions, which leads to over-fitting. We then propose ChebNetII, a new GNN model based on Chebyshev interpolation, which enhances the original Chebyshev polynomial approximation while reducing the Runge phenomena. We conducted an extensive experimental study to demonstrate that ChebNetII can learn arbitrary graph spectrum filters and achieve superior performance in both full- and semi-supervised node classification tasks.