Robust Output Sensitive Algorithms for Subanalytic Geometry

亚解析几何的鲁棒输出敏感算法

• 批准号: 0211458
• 负责人: J Maurice Rojas
• 金额: $ 9.88万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0211458&HistoricalAwards=false
• 关键词: Robust; Output; Sensitive; Algorithms; Subanalytic

The investigator aims to complete the nascent theory ofoutput-sensitive algorithms in real algebraic geometry.Output-sensitive in this context means that the complexity of theunderlying algorithm depends mainly on intrinsic geometricparameters, e.g., the number of connected components of theunderlying solution set, as opposed to extrinsic parameters likethe degrees of the input polynomials. Such algorithms are fasterthan the traditional methods of computational algebra by a factorexponential in the dimension, but have so far been discoveredonly in various isolated contexts. So a unified algorithmicapproach has a broad impact. Furthermore, the underlyingapproach takes numerical conditioning into account from theoutset, thus providing algorithms that are certifiably preciseeven when applied to approximate data. Another novelty is thatthe underlying theory applies in the even broader arena of realand p-adic analytic functions. The algorithmic aspects of p-adicanalytic functions are almost completely unexplored, so asecondary focus of this project is to elaborate and apply thisnew theory to equation-solving over finite fields and motivicintegration. The investigator combines advanced techniques from numericalanalysis and algebraic geometry to provide a new approach to afundamental problem occuring in many applications: solvinganalytic inequalities. For example, finding the optimalallocation of resources in a large organization (e.g., an army,an airline, or a large business) has long been known to reduce tosolving linear inequalities. From a different direction, it isknown that the complexity of certain neural net architectures(which are useful in training automated bomb-sniffers and patternrecognition systems) depends critically on understanding thesolutions of nonlinear polynomial inequalities. Both theseexamples are special cases of analytic inequalities, and thisproject provides new algorithms for their solution that aremagnitudes faster than current algorithms. Furthermore, thesenew algorithms provide certifiably precise solutions --- afeature which is especially important when facing uncertainphysical data. Another novel aspect is the principalinvestigator's recent discovery that the underlying techniquesapply to an even broader context, which can provide new solutionsto many problems in the design of cryptosystems.

研究者旨在完成实际代数几何形状中对外的敏感算法的新生理论。在此上下文中对output敏感性的敏感性意味着，在这种情况下，列出算法的复杂性主要取决于内在的几何图形参数，例如，跨性别的溶液与反对序列的参数相关的组件数量多项式。 这种算法是通过在维度中的Factorexpenential的传统计算代数方法更重要的，但是到目前为止，在各种孤立的环境中都被发现。 因此，统一的算法接触具有广泛的影响。 此外，基础接口将数值条件从theOutset中考虑到了，因此在应用于近似数据时提供了确切的算法。 另一个新颖性是，基本理论适用于更广泛的p-ad-Analytic函数领域。 p-亚丙分析功能的算法方面几乎是完全未探索的，因此该项目的高压焦点是详细说明并将本新理论应用于有限领域和动机整体的方程式解决方案。 研究者结合了数值分析和代数几何形状的先进技术，以在许多应用中提供一种新的方法来解决临时问题：解决分析不平等。 例如，在大型组织（例如，陆军，航空公司或大型企业）中找到最佳资源的最佳分配已知可以减少线性不平等。 从不同的方向来看，已经知道某些神经网架构的复杂性（在训练自动化的炸弹弹头和模式识别系统中很有用）取决于理解非线性多项式不平等的示例。 这两个示例都是分析性不平等的特殊情况，并且ThisProject为其解决方案提供了比当前算法更快的解决方案的新算法。 此外，这些算法提供了确切的精确解决方案 - - AFEATURE，这在面对不确定的数据时尤其重要。 另一个新颖的方面是主要分类者最近发现的，即基础技术将其应用于更广泛的环境，这可以在密码系统设计中提供许多新的解决方案。