{"title":"Hardness of Reconstructing Multivariate Polynomials over Finite Fields","authors":"Parikshit Gopalan, Subhash Khot, Rishi Saket","doi":"10.1137/070705258","DOIUrl":null,"url":null,"abstract":"We study the polynomial reconstruction problem, for low-degree multivariate polynomials over F[2]. In this problem, we are given a set of points x epsi {0, 1}n and target values f(x) epsi {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 - epsiv fraction of the points. Our goal is to find agree d polynomial that has good-agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 - 2-d + delta fraction of the points for any epsiv, delta > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, wherejis the algorithm is allowed to find a polynomial of degree d. Previously the only known, hardness of approximation (or even NP-completeness) was for the case when d = I, which follows from a celebrated result of Has tad. In the setting of computational learning, our result shows the hardness of (non-proper) agnostic learning of parities, where the learner is allowed, a low-degree polynomial over F[2] as a hypothesis. This is the first non-proper hardness result for this central problem in computational learning. Our results extend-to multivariate polynomial reconstruction over any finite field.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"88","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/070705258","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 88
Abstract
We study the polynomial reconstruction problem, for low-degree multivariate polynomials over F[2]. In this problem, we are given a set of points x epsi {0, 1}n and target values f(x) epsi {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 - epsiv fraction of the points. Our goal is to find agree d polynomial that has good-agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 - 2-d + delta fraction of the points for any epsiv, delta > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, wherejis the algorithm is allowed to find a polynomial of degree d. Previously the only known, hardness of approximation (or even NP-completeness) was for the case when d = I, which follows from a celebrated result of Has tad. In the setting of computational learning, our result shows the hardness of (non-proper) agnostic learning of parities, where the learner is allowed, a low-degree polynomial over F[2] as a hypothesis. This is the first non-proper hardness result for this central problem in computational learning. Our results extend-to multivariate polynomial reconstruction over any finite field.