RUI: Commutativity in Numerical Computation

这个项目涉及科学推论的一致性。对同一现象的多次观察很少产生相同的结果。统计噪声可能是第一个想到的复杂问题，但也有很多其他可能的差异：不同的天线布置；不同的拍摄角度；不同的形式，例如图像和文本；和更多。推断真正的潜在现象需要复杂的数学。这个项目的重点是矩阵方法。在矩阵的简单数值项中隐藏着称为特征向量的复杂模式，这些模式与乐器的谐波有关。尽管不同类型的测量可能产生不同的数值，但在观察相同的物体时，它们可能具有相同的特征向量。该项目的目标之一是从不一致的数据中提取一致性。从理论上讲，不同仪器对同一物体的测量应该在某些方面一致，但由于统计噪声和其他误差和不确定性来源，每种仪器提供的世界观都不精确，两种仪器之间也不会完全一致。其目标是利用数学方法使测量结果一致，从而可能更好地符合宇宙的真实状态。该项目将在一所主要本科院校完成，主要目标是让本科生参与STEM领域的研究。该项目的数学重点是交换厄米矩阵的同时对角化。理论上，交换厄米矩阵有一个共同的特征向量矩阵。然而，在病态条件下，数值计算的特征向量可能相差很大。同时对角化的一种数值方法，要么显式地，要么隐式地寻求输入矩阵精确交换的小扰动。项目目标包括对理论、方法和应用的贡献。首先，计划重新考虑“近交换性”，其中特征值固定在适当位置，并且注意力从厄米矩阵空间转移到特征向量矩阵的酉群。其次，该项目将寻求同时对角化的新数值方法，部分动机是相对较新的光谱分治方法。第三，新方法将应用于独立成分分析和多模态数据的无监督机器学习等应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns consistency in scientific inferences. Multiple observations of the same phenomenon rarely produce identical results. Statistical noise may be the first complication that comes to mind, but there are plenty of other possible discrepancies as well: different antenna arrangements; different camera angles; different modalities, e.g., images and text; and more. Inferring the true underlying phenomenon requires sophisticated mathematics. This project focuses on matrix methods. Hiding within the simple numerical entries of a matrix are intricate patterns called eigenvectors that are connected to the harmonics of musical instruments. Even though different kinds of measurements may produce different numerical values, they may share the same eigenvectors when observing the same objects. One of the goals of the project is to pull consistency out of inconsistent data. In theory, measurements on the same object from different instruments should agree in certain ways, but because of statistical noise and other sources of error and uncertainty, each instrument provides an inexact view of the world, and the two instruments will not agree perfectly with each other. The goal is to bring the measurements into alignment using mathematics and therefore likely into better agreement with the true state of the universe. The project will be completed at a Primarily Undergraduate Institution, and introducing undergraduate students to research in STEM fields is a primary objective.The mathematical focus of the project is simultaneous diagonalization of commuting Hermitian matrices. In theory, commuting Hermitian matrices share a common eigenvector matrix. However, numerically-computed eigenvectors may differ wildly in the presence of ill conditioning. A numerical method for simultaneous diagonalization seeks, either explicitly or implicitly, small perturbations of the input matrices that commute exactly. The project objectives include contributions to theory, method, and application. First, a reconsideration of "near commutativity" is planned, in which eigenvalues are fixed in place and attention shifts from the space of Hermitian matrices to the unitary group of eigenvector matrices. Second, the project will pursue new numerical methods for simultaneous diagonalization, motivated in part by relatively recent spectral divide-and-conquer methods. Third, the new approaches will be applied to applications such as independent component analysis and unsupervised machine learning with multimodal data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.