Collaborative Research: Algebraic Geometry of Tensors

合作研究：张量的代数几何

• 批准号: 0901770
• 负责人: Christopher Peterson
• 金额: $ 19.53万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901770&HistoricalAwards=false
• 关键词: Collaborative; Research; Algebraic; Geometry; Tensors

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PIs will study higher secant varieties of classically studied varieties such as Segre varieties, Grassmann varieties, and Segre-Veronese varieties. These varieties correspond to parameter spaces for rank one tensors, alternating tensors, and hybrids of regular tensors and symmetric tensors, and their (closed) higher secant varieties correspond to compactifications of the parameter spaces for higher rank tensors. The main goal of the research is the classification of defective secant varieties of Segre varieties, Grassmann varieties and Segre-Veronese varieties. This is analogous to the celebrated theorem of Alexander and Hirschowitz, which asserts that higher secant varieties of Veronese varieties have the expected dimension (modulo a fully described list of exceptions). This work completed the Waring problem for polynomials which had stood for some time as an outstanding unsolved problem. There is a corresponding, conjectural complete list of defective secant varieties for Segre varieties and for Grassmann varieties. The first component of the project is on the refinement of existing methods and the development of new theoretical and algorithmic methods towards the solution of this classification problem. The second component of the project is concerned with decomposition of tensors.In many applications, it is natural to represent a collection of data as a multi-indexed list. Alternatively, one can think of the data as a multidimensional array (sometimes called a multi-way array). For example, a digital grayscale picture can be stored as a matrix of numbers where each pixel location in the picture corresponds to a location in the matrix and the number in the matrix corresponds to the darkness of the pixel. In a similar manner, a digital color picture can be stored as a three dimensional array of numbers. A mathematical framework that includes the study of multi-way arrays, and their representations as sums of more basic objects, is through parameter spaces of tensors. This project explores problems related to tensors, tensor decomposition, tensor rank and tensor border rank from an algebro-geometric point viewpoint. These subjects have significant applications in fields as diverse as signal processing, data analysis, computational biology, combinatorics, algebraic geometry and statistics. It is the expectation, therefore, that techniques developed through this research will advance our knowledge and understanding across multiple disciplines.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。PI 将研究经典研究品种的较高割线品种，例如 Segre 品种、Grassmann 品种和 Segre-Veronese 品种。这些变体对应于一阶张量、交替张量以及规则张量和对称张量的混合的参数空间，并且它们的（封闭的）更高割线变体对应于更高阶张量的参数空间的压缩。研究的主要目标是对 Segre 品种、Grassmann 品种和 Segre-Veronese 品种的缺陷割线品种进行分类。这类似于亚历山大和赫肖维茨的著名定理，该定理断言维罗内簇的更高割线簇具有预期的维度（模完全描述的例外列表）。这项工作完成了多项式的韦林问题，该问题长期以来一直是一个悬而未决的问题。对于 Segre 品种和 Grassmann 品种，有一个相应的、推测性的完整的缺陷割线品种列表。该项目的第一个组成部分是完善现有方法并开发新的理论和算法方法来解决该分类问题。该项目的第二个组成部分涉及张量的分解。在许多应用中，很自然地将数据集合表示为多索引列表。或者，可以将数据视为多维数组（有时称为多路数组）。例如，数字灰度图片可以存储为数字矩阵，其中图片中的每个像素位置对应于矩阵中的位置，并且矩阵中的数字对应于像素的暗度。以类似的方式，数字彩色图片可以存储为三维数字数组。包括多路数组及其作为更基本对象的和的表示的研究的数学框架是通过张量的参数空间。该项目从代数几何点的角度探讨与张量、张量分解、张量秩和张量边界秩相关的问题。这些学科在信号处理、数据分析、计算生物学、组合学、代数几何和统计学等领域具有重要的应用。因此，我们期望通过这项研究开发的技术将增进我们跨多个学科的知识和理解。