First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper presents a novel algorithm to solve the balanced k-cut problem. The k-cut criterion is defined by using the Lovasz extension of a set function. A relaxation of balanced k-cut functionals can be formulated using Lovasz extensions of cuts and of normalizing constraints. The appealing feature of this relaxation is the connection between convex Lovasz extensions and submodular set functions. The authors propose an algorithm that minimizes a sum of auxiliary variables that are lower-bounded by ratios.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k -cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced k -cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced k -cut criteria.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced k-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced k-cut criteria.

Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efficient first-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately.

Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced k-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced k-cut criteria.

In this paper we develop the first algorithms for online submodular minimization that preserve differential privacy under full information feedback and bandit feedback. A sequence of $T$ submodular functions over a collection of $n$ elements arrive online, and at each timestep the algorithm must choose a subset of $[n]$ before seeing the function. The algorithm incurs a cost equal to the function evaluated on the chosen set, and seeks to choose a sequence of sets that achieves low expected regret. Our first result is in the full information setting, where the algorithm can observe the entire function after making its decision at each timestep. We give an algorithm in this setting that is $\epsilon$-differentially private and achieves expected regret $\tilde{O}\left(\frac{n^{3/2}\sqrt{T}}{\epsilon}\right)$. This algorithm works by relaxing submodular function to a convex function using the Lovasz extension, and then simulating an algorithm for differentially private online convex optimization. Our second result is in the bandit setting, where the algorithm can only see the cost incurred by its chosen set, and does not have access to the entire function. This setting is significantly more challenging because the algorithm does not receive enough information to compute the Lovasz extension or its subgradients. Instead, we construct an unbiased estimate using a single-point estimation, and then simulate private online convex optimization using this estimate. Our algorithm using bandit feedback is $\epsilon$-differentially private and achieves expected regret $\tilde{O}\left(\frac{n^{3/2}T^{3/4}}{\epsilon}\right)$.

We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem arises in many learning on graphs and hypergraphs settings and is closely related to decomposable submodular function minimization. We approach the problem via a new dual strategy and describe an objective that may be optimized via random coordinate descent (RCD) methods and projections onto cones. We also establish the linear convergence rate of the RCD algorithm and develop efficient projection algorithms with provable performance guarantees. Numerical experiments in transductive learning on hypergraphs confirm the efficiency of the proposed algorithm and demonstrate the significant improvements in prediction accuracy with respect to state-of-the-art methods.

Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case optimal submodular extension for the more general class of metric labeling; and (iii) efficiently compute the marginals for the widely used dense CRF model with the help of a recently proposed Gaussian filtering method. Using synthetic and real data, we show that our approach provides comparable upper bounds on the log-partition function to those obtained using tree-reweighted message passing (TRW) in cases where the latter is computationally feasible. Importantly, unlike TRW, our approach provides the first practical algorithm to compute an upper bound on the dense CRF model.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced $k$-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced $k$-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced $k$-cut criteria.