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| Random Double Tensors Integrals |
| Shih Yu Chang,Yimin Wei |
| (Department of Applied Data Science, San Jose State University, San
Jose, CA, USA;;School of Mathematical Sciences and Shanghai Key Laboratory of
Contemporary Applied Mathematics, Fudan University, Shanghai 200433,
China) |
| DOI： |
| Abstract: |
| 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. |
| Key words: Prey-predator system, T-S model, stability, eco-epidemiology |