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会产生相同的结果。首先想到的可能是统计噪声，但也有很多其他可能的差异：不同的天线布置；不同的摄像头角度；不同的模式，例如图像和文本；等等。推断真实的潜在现象需要复杂的数学计算。这个项目的重点是矩阵方法。隐藏在矩阵的简单数字条目中的是被称为特征向量的复杂模式，这些模式与乐器的谐波相关联。尽管不同类型的测量可能产生不同的数值，但当观察相同的对象时，它们可能共享相同的特征向量。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在某些方面应该是一致的，但由于统计噪声和其他误差和不确定性来源，每个仪器提供的世界视图都不准确，这两个仪器不会彼此完全一致。其目标是使用数学方法使测量结果保持一致，从而可能更好地符合宇宙的真实状态。该项目将在一个以本科为主的机构完成，介绍本科生从事STEM领域的研究是一个主要目标。该项目的数学重点是交换厄米特矩阵的同时对角化。理论上，交换厄米特矩阵共享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会有很大的不同。一种同时对角化的数值方法，显式或隐式地寻找精确交换的输入矩阵的微小扰动。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”，其中特征值是固定的，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，该项目将寻求用于同时对角化的新的数值方法，部分原因是相对较新的频谱分而治之方法。第三，新的方法将应用于独立成分分析和多模式数据的无监督机器学习等应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.