论文标题
随机双张量积分
Random Double Tensors Integrals
论文作者
论文摘要
在这项工作中，我们尝试建立一个随机双张量积分（DTI）的理论。我们从DTI的定义开始，并讨论如何基于DTI的随机性结构。然后，建立了随机DTI的单位不变标准的尾巴结合，并且该结合可以帮助我们得出各种两种张量的均值，例如算术平均值，几何平均值，和谐平均值和一般平均值的单位不变标准的尾部界限。通过将DTI与扰动公式相关联，即将张量值函数差异与函数输入张量的差异联系起来的公式，对于张量量的LIPSchitz估计值的函数输入张量的差异，随机张紧函数的lipschitz估计值与随机张力的参数是为Vanilla case and Quasi-Comportor衍生而成的。我们还以随机张量均值的融合意义建立了随机DTI的连续性属性，并应用此连续性属性以获得张量值函数的衍生功能的单位不变标准的尾巴结合。
In this work, we try to build a theory for random double tensor integrals (DTI). We begin with the definition of DTI and discuss how randomness structure is built upon DTI. Then, the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means, e.g., arithmetic mean, geometric mean, harmonic mean, and general mean. By associating DTI with perturbation formula, i.e., a formula to relate the tensor-valued function difference with respect the difference of the function input tensors, the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case, respectively. We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean, and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.
