TY - JOUR
T1 - Random Double Tensors Integrals
AU - Chang , Shih Yu
AU - Wei , Yimin
JO - Annals of Applied Mathematics
VL - 1
SP - 1
EP - 28
PY - 2023
DA - 2023/04
SN - 39
DO - http://doi.org/10.4208/aam.OA-2023-0004
UR - https://global-sci.org/intro/article_detail/aam/21630.html
KW - Einstein product, double tensor integrals (DTI), random DTI, tail bound, Lipschitz estimate, convergence in the random tensor mean, derivative of tensor-valued function.
AB - <p style="text-align: justify;">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.</p>