局部几何决定全局景观的低秩分解同步

IF 2.5 1区 数学 Q2 COMPUTER SCIENCE, THEORY & METHODS
Shuyang Ling
{"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}
引用次数: 0

摘要

正交群同步问题侧重于从其损坏的双测量中恢复正交群元素,包括一般签名网络上的高维Kuramoto模型,\(\mathbb {Z}_2\) -同步,随机块模型下的社区检测和正交Procrustes问题等例子。半定松弛(SDR)在解决这一问题上已经证明了它的力量;然而,其昂贵的计算成本阻碍了其广泛的实际应用。我们考虑了Burer-Monteiro分解方法来解决正交群同步问题,这是一种有效的、可扩展的低秩分解方法。尽管这种分解方法在经验上取得了显著的成功,但理解非凸优化景观何时是良性的仍然是一项具有挑战性的任务,即优化景观只有一个局部最小化器,这也是全局的。在这项工作中,我们证明了如果分解的秩超过“拉普拉斯”(证书矩阵)在全局最小值处的条件数的两倍,则优化景观不存在虚假的局部最小值。我们的主要定理是纯代数的和通用的,它无缝地适用于前面提到的所有例子:在几乎相同的条件下,非凸景观仍然是良性的,这使得SDR的成功。此外,我们说明Burer-Monteiro分解对“单调对手”具有鲁棒性,反映了SDR的弹性。换句话说,在数据中引入“有利的”对手不会导致出现新的虚假的局部最小值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Foundations of Computational Mathematics
Foundations of Computational Mathematics 数学-计算机:理论方法
CiteScore
6.90
自引率
3.30%
发文量
46
审稿时长
>12 weeks
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信