A Sparse Completely Positive Relaxation of the Modularity Maximization for Community Detection

In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated stochastic block model (DCSBM) assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR) type block coordinate descent (BCD) algorithm to solve the relaxation and prove an $\mathcal{O}(1/\sqrt{N})$ convergence rate to a stationary point where $N$ is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from the solution. Non-asymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a quite competitive alternative for community detection on sparse networks with over 50 million nodes

KL property of exponent 1/21/2 for zero-norm composite quadratic functions

This paper is concerned with a class of zero-norm regularized and constrained composite quadratic optimization problems, which has important applications in the fields such as sparse eigenvalue problems, sparse portfolio problems, and nonnegative matrix factorizations. For this class of nonconvex and nonsmooth problems, we establish the KL property of exponent 1/2 of its objective function under a suitable assumption, and provide some examples to illustrate that the assumption holds

Influence-based Community Partition with Sandwich Method for Social Networks

Community partition is an important problem in many areas such as biology network, social network. The objective of this problem is to analyse the relationships among data via the network topology. In this paper, we consider the community partition problem under IC model in social networks. We formulate the problem as a combinatorial optimization problem which aims at partitioning a given social network into disjoint M communities. The objective is to maximize the sum of influence propagation of a social network through maximizing it within each community. The existing work shows the influence maximization for community partition problem (IMCPP) to be NP hard. We first prove that the objective function of IMCPP under IC model is neither submodular nor supermodular. Then both supermodular upper bound and submodular lower bound are constructed and proved so that the sandwich framework can be applied. A continuous greedy algorithm and a discrete implementation are designed for upper bound and lower bound problems and the algorithm for both of the two problems gets a 1-1/e approximation ratio. We also devise a simply greedy to solve the original objective function and apply the sandwich approximation framework to it to guarantee a data dependent approximation factor. Finally, our algorithms are evaluated on two real data sets, which clearly verifies the effectiveness of our method in community partition problem, as well as the advantage of our method against the other methods.Comment: 32 pages. arXiv admin note: text overlap with arXiv:2002.0855

Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis

In this paper we study nonconvex and nonsmooth multi-block optimization over Riemannian manifolds with coupled linear constraints. Such optimization problems naturally arise from machine learning, statistical learning, compressive sensing, image processing, and tensor PCA, among others. We develop an ADMM-like primal-dual approach based on decoupled solvable subroutines such as linearized proximal mappings. First, we introduce the optimality conditions for the afore-mentioned optimization models. Then, the notion of $\epsilon$-stationary solutions is introduced as a result. The main part of the paper is to show that the proposed algorithms enjoy an iteration complexity of $O(1/\epsilon^2)$ to reach an $\epsilon$-stationary solution. For prohibitively large-size tensor or machine learning models, we present a sampling-based stochastic algorithm with the same iteration complexity bound in expectation. In case the subproblems are not analytically solvable, a feasible curvilinear line-search variant of the algorithm based on retraction operators is proposed. Finally, we show specifically how the algorithms can be implemented to solve a variety of practical problems such as the NP-hard maximum bisection problem, the $\ell_q$ regularized sparse tensor principal component analysis and the community detection problem. Our preliminary numerical results show great potentials of the proposed methods

A Brief Introduction to Manifold Optimization

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization are also presented.Comment: 43 page