Closing the Gap Between Matrix and Tensor Computation

可以充分证明张量计算是数值分析中的“下一件大事”。高维建模变得越来越普遍，它需要对巨大的多维数组进行操作和分析。研究人员和他的同事将通过追求四个基本研究方向来丰富矩阵计算和张量计算之间的相互作用。他们将（1）开发基于具有低克罗内克积秩的矩阵的张量逼近技术，（2）实现一对基本张量代数子程序，一个展示斯特拉森乘法的新收缩级推广，一个演示如何有效计算具有对称性的张量之间的收缩，（3）通过张量网络分析巨大向量的数据稀疏表示，以及（4）通过涉及对称张量的嵌入思想。 数据表是二维的，存在许多矩阵计算技术用于从行和列中出现的数字中提取信息。张量可以被认为是一个表，其条目是其他表。例如，一个 10 行 8 列的表有 80 个“单元格”。如果每个单元格本身就是一个 5×4 表，则整个数据集可以被视为一个 10×8×5×4 张量。这种类型的数据集在工程和科学中越来越重要，因为它是构建与取决于许多因素的模型相关的信息的自然方式。该研究计划旨在帮助科学界建立基础设施，使基于张量的计算像基于矩阵的计算一样自然和容易。矩阵计算领域提供的成功的问题解决和问题分析工具将得到拓宽和推广。外展议程包括制作教育材料，这将有助于确保精通张量的科学界的发展。这些材料包括关于张量计算的在线十讲座短期课程、参与针对四年制大学的客座讲师计划，以及在 Golub 和 Van Loan 即将出版的高被引用矩阵计算教科书第四版中添加张量计算章节。

A strong case can be made that tensor computation is the ``next big thing'' in numerical analysis. High-dimensional modeling is becoming commonplace and it requires the manipulation and analysis of huge multidimensional arrays. The investigator and his colleagues will enrich the interplay between matrix computations and tensor computations by pursuing four basic directions of research. They will(1) develop tensor approximation techniques based on matrices that have low Kronecker product rank, (2) implement a pair of basic tensor algebra subprograms, one that showcases a new contraction-level generalization of Strassen multiplication and one that demonstrates how to compute effectively contractions between tensors that have symmetry, (3) analyze the data sparse representation of huge vectors through tensor networks, and (4) develop a unifying framework for SVD-like tensor decompositions through an embedding idea that involves symmetric tensors. A table of data is 2-dimensional and many matrix computation techniques exist for extracting information from the numbers that appear in the rows and columns. A tensor can be thought of as a table whose entries are other tables. For example, a table having 10 rows and 8 columns has 80 ``cells''. If each of those cells is itself a 5-by-4 table, then the entire data set can be thought of as a 10-by-8-by-5-by-4 tensor. Data sets of this variety are increasingly prominent in engineering and the sciences because it is the natural way to structure the information associated with a model that depends upon many factors.The research plan is to help build an infrastructure for the scientific community that makes tensor-based computation as natural and easy as matrix-based computation. The successful problem-solving and problem-analysis tools provided by the matrix computation field will be broadened and generalized. The outreach agenda includes the production of educational materials that will help ensure the development of a tensor-savvy scientific community. These materials include an online, ten-lecture short course on tensor computation, participation in a Visiting Lecturer program that targets 4-year colleges, and the addition of a tensor computation chapter in the upcoming fourth edition of the highly-cited textbook on matrix computations by Golub and Van Loan.