标题： 可分希尔伯特空间上的拟管张量代数 显示英文标题

标题： Quasitubal Tensor Algebra Over Separable Hilbert Spaces

摘要： 管状张量框架提供了一个清晰且有效的代数环境来进行张量计算，支持类似于矩阵的特性，比如奇异值分解和Eckart-Young类最优性结果。 管状张量框架的基础在于将张量视为有限大小管组成的矩阵。 在这项工作中，我们为处理具有无限大小管的张量奠定了数学和计算基础：这些张量的元素来自可分Hilbert空间的矩阵。 一个关键挑战是，管状张量的重要矩阵类似特征的存在依赖于管环中单位元的存在。 对于作为无限维Hilbert空间元素的管来说，这种单位元不可能存在。 我们通过将管状空间嵌入到交换幺C*-代数的有界算子中来规避这个问题。 由此产生的准管状代数恢复了分解和低秩逼近所需的结构属性。 除了为处理具有无限维管的管状张量奠定理论基础外，我们还讨论了我们的构造的计算方面，并提供了数值实例，在该实例中使用我们的理论计算了无限大小合成张量的一个有限维近似。 我们认为，我们的理论为在本质上无限维问题的背景下应用矩阵类似的张量框架开辟了新的令人兴奋的方向。 显示英文摘要

摘要： The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.