  This paper studies the Tensor Robust Principal Component (TRPCA) problem
which extends the known Robust PCA (Candes et al. 2011) to the tensor case. Our
model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and
Martin 2011) and its induced tensor tubal rank and tensor nuclear norm.
Consider that we have a 3-way tensor ${\mathcal{X}}\in\mathbb{R}^{n_1\times
n_2\times n_3}$ such that ${\mathcal{X}}={\mathcal{L}}_0+{\mathcal{E}}_0$,
where ${\mathcal{L}}_0$ has low tubal rank and ${\mathcal{E}}_0$ is sparse. Is
that possible to recover both components? In this work, we prove that under
certain suitable assumptions, we can recover both the low-rank and the sparse
components exactly by simply solving a convex program whose objective is a
weighted combination of the tensor nuclear norm and the $\ell_1$-norm, i.e.,
$\min_{{\mathcal{L}},\ {\mathcal{E}}} \
\|{{\mathcal{L}}}\|_*+\lambda\|{{\mathcal{E}}}\|_1, \ \text{s.t.} \
{\mathcal{X}}={\mathcal{L}}+{\mathcal{E}}$, where $\lambda=
{1}/{\sqrt{\max(n_1,n_2)n_3}}$. Interestingly, TRPCA involves RPCA as a special
case when $n_3=1$ and thus it is a simple and elegant tensor extension of RPCA.
Also numerical experiments verify our theory and the application for the image
denoising demonstrates the effectiveness of our method.
