{"title":"局部几何决定全局景观的低秩分解同步","authors":"Shuyang Ling","doi":"10.1007/s10208-025-09707-9","DOIUrl":null,"url":null,"abstract":"<p>The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, <span>\\(\\mathbb {Z}_2\\)</span>-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer–Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the rank of the factorization exceeds twice the condition number of the “Laplacian\" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer–Monteiro factorization is robust to “monotone adversaries\", mirroring the resilience of the SDR. In other words, introducing “favorable\" adversaries into the data will not result in the emergence of new spurious local minimizers.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"2 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Local Geometry Determines Global Landscape in Low-Rank Factorization for Synchronization\",\"authors\":\"Shuyang Ling\",\"doi\":\"10.1007/s10208-025-09707-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, <span>\\\\(\\\\mathbb {Z}_2\\\\)</span>-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer–Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the rank of the factorization exceeds twice the condition number of the “Laplacian\\\" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer–Monteiro factorization is robust to “monotone adversaries\\\", mirroring the resilience of the SDR. In other words, introducing “favorable\\\" adversaries into the data will not result in the emergence of new spurious local minimizers.</p>\",\"PeriodicalId\":55151,\"journal\":{\"name\":\"Foundations of Computational Mathematics\",\"volume\":\"2 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2025-04-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Computational Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10208-025-09707-9\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10208-025-09707-9","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Local Geometry Determines Global Landscape in Low-Rank Factorization for Synchronization
The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, \(\mathbb {Z}_2\)-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer–Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the rank of the factorization exceeds twice the condition number of the “Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer–Monteiro factorization is robust to “monotone adversaries", mirroring the resilience of the SDR. In other words, introducing “favorable" adversaries into the data will not result in the emergence of new spurious local minimizers.
期刊介绍:
Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer.
With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles.
The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.