A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization

• URL: http://arxiv.org/abs/2112.13199v1
• Date: Sat, 25 Dec 2021 07:38:14 GMT
• Title: A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization
• Authors: Yifeng Fan, Yuehaw Khoo, and Zhizhen Zhao
• Abstract summary: We propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization (CPQR) The proposed algorithm is efficient and scales linearly with the number of data points. We also leverage the recently developed leave-one-out' technique to establish a near-optimal guarantee for exact recovery of the cluster memberships.
• Score: 20.413250472034143
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Community detection and orthogonal group synchronization are both fundamental problems with a variety of important applications in science and engineering. In this work, we consider the joint problem of community detection and orthogonal group synchronization which aims to recover the communities and perform synchronization simultaneously. To this end, we propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization (CPQR). The proposed algorithm is efficient and scales linearly with the number of data points. We also leverage the recently developed `leave-one-out' technique to establish a near-optimal guarantee for exact recovery of the cluster memberships and stable recovery of the orthogonal transforms. Numerical experiments demonstrate the efficiency and efficacy of our algorithm and confirm our theoretical characterization of it.

Related papers

• ALEXR: An Optimal Single-Loop Algorithm for Convex Finite-Sum Coupled Compositional Stochastic Optimization [53.14532968909759]
  We introduce an efficient single-loop primal-dual block-coordinate algorithm, dubbed ALEXR. We establish the convergence rates of ALEXR in both convex and strongly convex cases under smoothness and non-smoothness conditions. We present lower complexity bounds to demonstrate that the convergence rates of ALEXR are optimal among first-order block-coordinate algorithms for the considered class of cFCCO problems.
  arXiv  Detail & Related papers  (2023-12-04T19:00:07Z)
• An Optimization-based Deep Equilibrium Model for Hyperspectral Image Deconvolution with Convergence Guarantees [71.57324258813675]
  We propose a novel methodology for addressing the hyperspectral image deconvolution problem. A new optimization problem is formulated, leveraging a learnable regularizer in the form of a neural network. The derived iterative solver is then expressed as a fixed-point calculation problem within the Deep Equilibrium framework.
  arXiv  Detail & Related papers  (2023-06-10T08:25:16Z)
• Unrolled algorithms for group synchronization [7.969977930633441]
  Group synchronization problem involves estimating a collection of group elements from noisy measurements of their pairwise ratios. Standard methods to estimate the group elements are based on iteratively applying linear and non-linear operators. Motivated by the structural similarity to deep neural networks, we adopt the concept of algorithm unrolling.
  arXiv  Detail & Related papers  (2022-07-19T17:25:56Z)
• Multi-Frequency Joint Community Detection and Phase Synchronization [22.683901672480353]
  This paper studies the joint community detection and phase synchronization problem on the textitstochastic block model with relative phase We show this problem exhibits a textitmulti-frequency'' structure by closely examining its maximum likelihood estimation (MLE) formulation. Two simple yet efficient algorithms that leverage the MLE formulation and benefit from the information across multiple frequencies are proposed.
  arXiv  Detail & Related papers  (2022-06-16T23:08:27Z)
• Perfect Spectral Clustering with Discrete Covariates [68.8204255655161]
  We propose a spectral algorithm that achieves perfect clustering with high probability on a class of large, sparse networks. Our method is the first to offer a guarantee of consistent latent structure recovery using spectral clustering.
  arXiv  Detail & Related papers  (2022-05-17T01:41:06Z)
• First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems [88.58409977434269]
  We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) Our contribution is to provide two simple first-order algorithmic frameworks based on the quadratic penalty method and the augmented Lagrangian method. We provide nonasymptotic theoretical guarantees for these algorithms.
  arXiv  Detail & Related papers  (2022-04-07T00:11:05Z)
• Gradient Based Clustering [72.15857783681658]
  We propose a general approach for distance based clustering, using the gradient of the cost function that measures clustering quality. The approach is an iterative two step procedure (alternating between cluster assignment and cluster center updates) and is applicable to a wide range of functions.
  arXiv  Detail & Related papers  (2022-02-01T19:31:15Z)
• Joint Community Detection and Rotational Synchronization via Semidefinite Programming [17.845257705485533]
  In the presence of heterogeneous data, where randomly rotated objects fall into multiple underlying categories, it is challenging to simultaneously classify them into clusters and synchronize them based on pairwise relations. We propose a series of semidefinite relaxations, and prove their exact recovery when extending the celebrated block model to this new setting. Numerical experiments demonstrate the efficacy of our proposed algorithms and confirm our theoretical result which indicates a sharp phase transition for exact recovery.
  arXiv  Detail & Related papers  (2021-05-13T01:40:20Z)
• Near-Optimal Performance Bounds for Orthogonal and Permutation Group Synchronization via Spectral Methods [0.7614628596146599]
  Group synchronization asks to recover group elements from their pairwise measurements. spectral methods have enjoyed great popularity due to their efficiency and convenience. For permutation group synchronization under random corruption, we show that the widely-used two-step procedure can recover all the group elements exactly.
  arXiv  Detail & Related papers  (2020-08-12T14:20:16Z)
• Message Passing Least Squares Framework and its Application to Rotation Synchronization [16.650654530240566]
  We first describe our theoretically guaranteed message passing algorithm that estimates the corruption levels of the measured group ratios. We then propose a novel reweighted least squares method to estimate the group elements, where the weights are iteratively updated using the estimated corruption levels. We demonstrate the superior performance of our algorithm over state-of-the-art methods for rotation synchronization using both synthetic and real data.
  arXiv  Detail & Related papers  (2020-07-27T15:39:19Z)
• IDEAL: Inexact DEcentralized Accelerated Augmented Lagrangian Method [64.15649345392822]
  We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds.
  arXiv  Detail & Related papers  (2020-06-11T18:49:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.