{"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}
引用次数: 0
Abstract
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.
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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.
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