We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.
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Given an undirected graph $G=(V,E)$, the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset $W\subseteq V$ such that $W$ induces a simple path in $G$. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its solution. While the first formulation (cec) is based on constraints that explicitly eliminate cycles, the second one (cut) ensures connectivity via cutset constraints. We compare, both theoretically and experimentally, the newly proposed approaches with a state-of-the-art formulation recently proposed in the literature. More specifically, we show that the polyhedra defined by formulation cut and that of the formulation available in the literature are the same. Besides, we show that these two formulations are stronger in theory than cec. We also propose a new branch-and-cut procedure using the new formulations. Computational experiments show that the newly proposed formulation cec, although less strong from a theoretical point of view, is the best performing approach as it can solve all but one of the 1065 benchmark instances used in the literature within the given time limit. In addition, our newly proposed approaches outperform the state-of-the-art formulation when it comes to the median times to solve the instances to optimality. Furthermore, we perform extended computational experiments considering more challenging and hard-to-solve larger instances and evaluate the impacts on the results when offering initial feasible solutions (warm starts) to the formulations.
Multiple antenna arrays play a key role in wireless networks for communications but also localization and sensing. The use of large antenna arrays pushes towards a propagation regime in which the wavefront is no longer plane but spherical. This allows to infer the position and orientation of an arbitrary source from the received signal without the need of using multiple anchor nodes. To understand the fundamental limits of large antenna arrays for localization, this paper fusions wave propagation theory with estimation theory, and computes the Cram{\'e}r-Rao Bound (CRB) for the estimation of the three Cartesian coordinates of the source on the basis of the electromagnetic vector field, observed over a rectangular surface area. To simplify the analysis, we assume that the source is a dipole, whose center is located on the line perpendicular to the surface center, with an orientation a priori known. Numerical and asymptotic results are given to quantify the CRBs, and to gain insights into the effect of various system parameters on the ultimate estimation accuracy. It turns out that surfaces of practical size may guarantee a centimeter-level accuracy in the mmWave bands.
Until recently, applications of neural networks in machine learning have almost exclusively relied on real-valued networks. It was recently observed, however, that complex-valued neural networks (CVNNs) exhibit superior performance in applications in which the input is naturally complex-valued, such as MRI fingerprinting. While the mathematical theory of real-valued networks has, by now, reached some level of maturity, this is far from true for complex-valued networks. In this paper, we analyze the expressivity of complex-valued networks by providing explicit quantitative error bounds for approximating $C^n$ functions on compact subsets of $\mathbb{C}^d$ by complex-valued neural networks that employ the modReLU activation function, given by $\sigma(z) = \mathrm{ReLU}(|z| - 1) \, \mathrm{sgn} (z)$, which is one of the most popular complex activation functions used in practice. We show that the derived approximation rates are optimal (up to log factors) in the class of modReLU networks with weights of moderate growth.
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It has been proven to be NP-complete on general graphs, as well as on split and planar graphs. In this paper, we enrich its complexity study with two new results. First, we prove that it remains NP-complete even when restricted on the class of bipartite graphs. This is unlike what it is expected, given that the K-way vertex cut problem is a generalization of the Maximum Independent set problem which is polynomially solvable on bipartite graphs. We also provide its equivalence to the wellknown problem, namely the Critical Node Problem (CNP), On split graphs. Therefore, any solving algorithm for the CNP on split graphs is a solving algorithm for the K-way vertex cut problem and vice versa.
This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization, distributionally robust optimization). However, most of the existing algorithms are slow in practice, and their analysis revolves around the convergence to a nearly stationary point. We consider leveraging the Polyak-\L ojasiewicz (PL) condition to design faster stochastic algorithms with stronger convergence guarantee. Although PL condition has been utilized for designing many stochastic minimization algorithms, their applications for non-convex min-max optimization remain rare. In this paper, we propose and analyze a generic framework of proximal epoch-based method with many well-known stochastic updates embeddable. Fast convergence is established in terms of both {\bf the primal objective gap and the duality gap}. Compared with existing studies, (i) our analysis is based on a novel Lyapunov function consisting of the primal objective gap and the duality gap of a regularized function, and (ii) the results are more comprehensive with improved rates that have better dependence on the condition number under different assumptions. We also conduct deep and non-deep learning experiments to verify the effectiveness of our methods.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with their Gromov-Wasserstein discrepancy provides the correspondence between their nodes and achieves graph matching. When one of the graphs has isolated but self-connected nodes ($i.e.$, a disconnected graph), the optimal transport indicates the clustering structure of the other graph and achieves graph partitioning. Using this concept, we extend our method to multi-graph partitioning and matching by learning a Gromov-Wasserstein barycenter graph for multiple observed graphs; the barycenter graph plays the role of the disconnected graph, and since it is learned, so is the clustering. Our method combines a recursive $K$-partition mechanism with a regularized proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is the first attempt to make Gromov-Wasserstein discrepancy applicable to large-scale graph analysis and unify graph partitioning and matching into the same framework. It outperforms state-of-the-art graph partitioning and matching methods, achieving a trade-off between accuracy and efficiency.
Developing classification algorithms that are fair with respect to sensitive attributes of the data has become an important problem due to the growing deployment of classification algorithms in various social contexts. Several recent works have focused on fairness with respect to a specific metric, modeled the corresponding fair classification problem as a constrained optimization problem, and developed tailored algorithms to solve them. Despite this, there still remain important metrics for which we do not have fair classifiers and many of the aforementioned algorithms do not come with theoretical guarantees; perhaps because the resulting optimization problem is non-convex. The main contribution of this paper is a new meta-algorithm for classification that takes as input a large class of fairness constraints, with respect to multiple non-disjoint sensitive attributes, and which comes with provable guarantees. This is achieved by first developing a meta-algorithm for a large family of classification problems with convex constraints, and then showing that classification problems with general types of fairness constraints can be reduced to those in this family. We present empirical results that show that our algorithm can achieve near-perfect fairness with respect to various fairness metrics, and that the loss in accuracy due to the imposed fairness constraints is often small. Overall, this work unifies several prior works on fair classification, presents a practical algorithm with theoretical guarantees, and can handle fairness metrics that were previously not possible.
The Jaccard index, also referred to as the intersection-over-union score, is commonly employed in the evaluation of image segmentation results given its perceptual qualities, scale invariance - which lends appropriate relevance to small objects, and appropriate counting of false negatives, in comparison to per-pixel losses. We present a method for direct optimization of the mean intersection-over-union loss in neural networks, in the context of semantic image segmentation, based on the convex Lov\'asz extension of submodular losses. The loss is shown to perform better with respect to the Jaccard index measure than the traditionally used cross-entropy loss. We show quantitative and qualitative differences between optimizing the Jaccard index per image versus optimizing the Jaccard index taken over an entire dataset. We evaluate the impact of our method in a semantic segmentation pipeline and show substantially improved intersection-over-union segmentation scores on the Pascal VOC and Cityscapes datasets using state-of-the-art deep learning segmentation architectures.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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