  The essential task of tensor data analysis focuses on the tensor
decomposition and the corresponding notion of rank. In this paper, by
introducing the notion of tensor Singular Value Decomposition (t-SVD), we
establish a Regularized Tensor Nuclear Norm Minimization (RTNNM) model for
low-tubal-rank tensor recovery. As we know that many variants of the Restricted
Isometry Property (RIP) have proven to be crucial analysis tools for sparse
recovery. In the t-SVD framework, we initiatively define a novel tensor
Restricted Isometry Property (t-RIP). Furthermore, we show that any third-order
tensor $\boldsymbol{\mathcal{X}}$ can stably be recovered from few linear noise
measurements under some certain t-RIP conditions via the RTNNM model. We note
that, as far as the authors are aware, such kind of result has not previously
been reported in the literature.
