{"title":"解决矩阵斯宾塞猜想，直至多对数等级","authors":"Nikhil Bansal, Haotian Jiang, Raghu Meka","doi":"10.1137/23m1592201","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric [math] matrices [math] each with [math] and rank at most [math], one can efficiently find [math] signs [math] such that their signed sum has spectral norm [math]. This result also implies a [math] qubit lower bound for quantum random access codes encoding [math] classical bits with advantage [math]. Our proof uses the recent refinement of the noncommutative Khintchine inequality due to Bandeira, Boedihardjo, and van Handel [Invent. Math., 234 (2023), pp. 419–487] for random matrices with correlated Gaussian entries.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Resolving Matrix Spencer Conjecture up to Poly-Logarithmic Rank\",\"authors\":\"Nikhil Bansal, Haotian Jiang, Raghu Meka\",\"doi\":\"10.1137/23m1592201\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric [math] matrices [math] each with [math] and rank at most [math], one can efficiently find [math] signs [math] such that their signed sum has spectral norm [math]. This result also implies a [math] qubit lower bound for quantum random access codes encoding [math] classical bits with advantage [math]. Our proof uses the recent refinement of the noncommutative Khintchine inequality due to Bandeira, Boedihardjo, and van Handel [Invent. Math., 234 (2023), pp. 419–487] for random matrices with correlated Gaussian entries.\",\"PeriodicalId\":49532,\"journal\":{\"name\":\"SIAM Journal on Computing\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Computing\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1137/23m1592201\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Computing","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1137/23m1592201","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
SIAM 计算期刊》，提前印刷。 摘要我们给出了矩阵斯宾塞猜想到多对数秩的一个简单证明：给定对称[数学]矩阵[数学]，每个矩阵[数学]的秩最多[数学]，我们可以有效地找到[数学]符号[数学]，使得它们的符号和具有谱规范[数学]。这一结果也意味着以[数学]优势编码[数学]经典比特的量子随机存取码的[数学]比特下限。我们的证明使用了班代拉、布埃迪哈卓和范-汉德尔（Bandeira, Boedihardjo, and van Handel）[《发明数学》，234 (2023)，第 419-487 页]最近对具有相关高斯条目的随机矩阵的非交换 Khintchine 不等式的改进。
Resolving Matrix Spencer Conjecture up to Poly-Logarithmic Rank
SIAM Journal on Computing, Ahead of Print. Abstract. We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric [math] matrices [math] each with [math] and rank at most [math], one can efficiently find [math] signs [math] such that their signed sum has spectral norm [math]. This result also implies a [math] qubit lower bound for quantum random access codes encoding [math] classical bits with advantage [math]. Our proof uses the recent refinement of the noncommutative Khintchine inequality due to Bandeira, Boedihardjo, and van Handel [Invent. Math., 234 (2023), pp. 419–487] for random matrices with correlated Gaussian entries.
期刊介绍：
The SIAM Journal on Computing aims to provide coverage of the most significant work going on in the mathematical and formal aspects of computer science and nonnumerical computing. Submissions must be clearly written and make a significant technical contribution. Topics include but are not limited to analysis and design of algorithms, algorithmic game theory, data structures, computational complexity, computational algebra, computational aspects of combinatorics and graph theory, computational biology, computational geometry, computational robotics, the mathematical aspects of programming languages, artificial intelligence, computational learning, databases, information retrieval, cryptography, networks, distributed computing, parallel algorithms, and computer architecture.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
