Invariant Theory, Tensors, and Applications

在许多科学和工程领域，包括信号处理、数据挖掘、神经科学和化学计量学，数据经常以多维数组的形式出现。这样的多维数组称为张量。为了理解手头数据的内部结构，将给定的张量分解为更简单张量的和是有用的。然而，这种分解的经典方法存在着诸如数值不稳定性等缺陷。该研究项目的主要目标之一是开发一种新的张量分解方法的理论基础，该方法将在应用中表现得更好，特别是在生物医学信号处理中。该项目涉及研究生和本科生。本研究项目研究张量和不变量理论。该项目旨在开发确定张量的上下界的方法。张量的凸分解理论将得到发展和推广。这种新的张量分解理论框架是低阶(PARAFAC)分解的另一种选择，并有望在数值上表现得更好。研究人员将研究箭图和有位势箭图的(半)不变量环。他还将探索张量不变量及其与扭曲交换代数的关系。

In many areas of science and engineering, including signal processing, data mining, neuroscience, and chemometrics, data often appear in multidimensional arrays. Such multidimensional arrays are known as tensors. To understand the internal structure of the data at hand, it is useful to decompose a given tensor as a sum of simpler tensors. However, the classical method for such decompositions has flaws, such as numerical instability. One of the main goals of this research project is to develop the theoretical underpinning of a new method for tensor decomposition that will perform better in applications, particularly in biomedical signal processing. The project involves graduate and undergraduate students in the research.This research project studies tensors and invariant theory. The project aims to develop methods for lower and upper bounds for the rank of a tensor. The theory of convex decompositions of tensors will be developed and generalized. This new theoretical framework for tensor decompositions is an alternative to the low rank (PARAFAC) decomposition and is expected to behave better numerically. The investigator will study rings of (semi-)invariants for quivers and quivers with potentials. He will also explore tensor invariants and their relations to twisted commutative algebras.