*Appendix A.2. Proof of Theorem 2*

**Proof.** The key idea is to rewrite Problem (29) into a standard two-block ADMM problem. For notational simplicity, let:

$$f(\mathbf{x}) = \frac{1}{2} \|\mathbf{L} + \mathbf{S} - \mathbf{X}\|\_{\mathbf{F}}^2 \qquad g(\mathbf{z}) = \gamma \|\mathbf{K}\| \text{tr} \mathbf{N} + \gamma \lambda R(\mathbf{S}) \lambda$$

where **x**, **y**, **z** and **A** are defined as follows:

$$\mathbf{x} = \begin{bmatrix} \text{vec}(\underline{\mathbf{L}}) \\ \text{vec}(\underline{\mathbf{S}}) \end{bmatrix}, \ \mathbf{y} = \begin{bmatrix} \text{vec}(\underline{\mathbf{Y}}\_1) \\ \text{vec}(\underline{\mathbf{Y}}\_2) \end{bmatrix}, \ \mathbf{z} = \begin{bmatrix} \text{vec}(\underline{\mathbf{K}}) \\ \text{vec}(\underline{\mathbf{R}}) \end{bmatrix}, \ \mathbf{A} = \begin{bmatrix} \text{diag}(\text{vec}(\underline{\mathbf{B}})) & \mathbf{0} \\ \mathbf{0} & \text{diag}(\text{vec}(\underline{\mathbf{B}})) \end{bmatrix}',$$

and vec(·) denotes an operation of tensor vectorization (see [40]).

It can be verified that *f*(·) and *g*(·) are closed, proper convex functions. Then, Problem (29) can be re-written as follows:

$$\min\_{\mathbf{x}, \mathbf{z}} \quad f(\mathbf{x}) + \mathbf{g}(\mathbf{z})$$
 
$$\text{s.t.} \quad \mathbf{A}\mathbf{x} - \mathbf{z} = \mathbf{0}.$$

According to the convergence analysis in [48], we have:


where *f* -, *g* are the optimal values of *f*(**x**), *g*(**z**), respectively. Variable **y** is a dual optimal point defined as:

$$\mathbf{y}^\* = \begin{bmatrix} \operatorname{vec}(\underline{\chi}\_1^\*) \\ \operatorname{vec}(\underline{\chi}\_2^\*) \end{bmatrix} \text{ \textit{\textquotedblleft}}$$

where (Y- 1,Y- <sup>2</sup> ) is the dual component of a saddle point (L-, S-, K-,R-,Y- 1,Y- <sup>2</sup> ) of the unaugmented Lagrangian *L*(L, S, K, R, Y1, Y2).

### *Appendix A.3. Proof of Lemma 4*

**Proof.** Let the full t-SVD of <sup>X</sup> be <sup>X</sup> <sup>=</sup> <sup>U</sup> <sup>∗</sup> <sup>Λ</sup> <sup>∗</sup> <sup>V</sup> , where U, <sup>V</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> are orthogonal tensors and <sup>Λ</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> is *<sup>f</sup>*-diagonal. Then:

$$\|\underline{\mathbf{X}}\|\_{\text{TNN}} = \|\overline{\underline{\mathbf{U}} \* \underline{\mathbf{A}}} \* \underline{\mathbf{V}}^{\top}\|\_{\ast} = \|\underline{\underline{\mathbf{U}}} \cdot \overline{\underline{\mathbf{A}}} \cdot \overline{\underline{\mathbf{V}}^{\top}}\|\_{\ast} = \|\underline{\underline{\mathbf{A}}}\|\_{\ast}.\tag{A38}$$

Then Q ∗ X = (Q ∗ U) ∗ Λ ∗ V . Since

$$(\underline{\mathbf{Q}} \ast \underline{\mathbf{U}})^\top \ast (\underline{\mathbf{Q}} \ast \underline{\mathbf{U}}) = \underline{\mathbf{U}}^\top \ast \underline{\mathbf{Q}}^\top \ast \underline{\mathbf{Q}} \ast \underline{\mathbf{U}} = \mathbf{I} \tag{A39}$$

we obtain that:

$$\begin{aligned} \|\underline{\mathbf{Q}} \* \underline{\mathbf{X}}\|\_{\text{TNN}} &= \|\underline{\overline{\mathbf{Q}} \* \underline{\mathbf{X}}}\|\_{\*} \\ &= \|\underline{(\underline{\mathbf{Q}} \* \underline{\mathbf{U}}) \* \underline{\mathbf{A}} \underline{\mathbf{V}}^{\top}}\|\_{\*} \\ &= \|\underline{(\underline{\mathbf{Q}} \* \underline{\mathbf{U}}) \cdot \underline{\mathbf{A}}} \cdot \underline{\mathbf{V}}^{\top}\|\_{\*} \\ &= \|\underline{\overline{\mathbf{A}}}\|\_{\*} .\end{aligned} \tag{A40}$$

Thus, Q ∗ XTNN = XTNN.

*Appendix A.4. Proof of Theorem 3*

**Proof.** Please note that (Q<sup>∗</sup> <sup>∗</sup> <sup>X</sup>∗, S∗) is a feasible point of Problem (28), then we have:

$$\begin{split} &\frac{1}{2}||\mathbb{B}\odot(\mathbb{L}^{\star}+\mathbb{S}^{\star}-\mathbb{M})||\_{\mathrm{F}}^{2}+\gamma(\|\mathbb{L}^{\star}\|\_{\mathrm{TNN}}+\lambda\|\mathbb{S}^{\star}\|\_{1}) \\ &\leq\frac{1}{2}||\mathbb{B}\odot(\underline{\mathbf{Q}}\_{\star}\*\underline{\mathbf{X}}\_{\star}+\underline{\mathbf{S}}\_{\star}-\underline{\mathbf{M}})||\_{\mathrm{F}}^{2}+\gamma(\|\underline{\mathbf{Q}}\_{\star}\*\underline{\mathbf{X}}\_{\star}\|\_{\mathrm{TNN}}+\lambda\|\underline{\mathbf{S}}\_{\star}\|\_{1}) \\ &=\frac{1}{2}||\mathbb{B}\odot(\underline{\mathbf{Q}}\_{\star}\*\underline{\mathbf{X}}\_{\star}+\underline{\mathbf{S}}\_{\star}-\underline{\mathbf{M}})||\_{\mathrm{F}}^{2}+\gamma(\|\underline{\mathbf{X}}\_{\star}\|\_{\mathrm{TNN}}+\lambda\|\underline{\mathbf{S}}\_{\star}\|\_{1}) \end{split} \tag{A41}$$

By the assumption that *r*tubal(L-) <sup>≤</sup> *<sup>r</sup>*, there exists a decomposition <sup>L</sup>- = Q- <sup>∗</sup> <sup>X</sup>-, such that (Q-, X-, S-) is also a feasible point of Problem (39).

Moreover, since (Q∗, X∗, S∗) is a global optimal solution to Problem (39), then we have that

$$\begin{split} &\frac{1}{2}||\underline{\mathbf{B}}\odot(\underline{\mathbf{Q}}\_{\ast}\ast\underline{\mathbf{X}}\_{\ast}+\underline{\mathbf{S}}\_{\ast}-\underline{\mathbf{M}})||\_{\mathbb{F}}^{2}+\gamma(||\underline{\mathbf{X}}\_{\ast}||\_{\text{TNN}}+\lambda||\underline{\mathbf{S}}\_{\ast}||\_{1}) \\ &\leq\frac{1}{2}||\underline{\mathbf{B}}\odot(\underline{\mathbf{Q}}^{\star}\ast\underline{\mathbf{X}}^{\star}+\underline{\mathbf{S}}^{\star}-\underline{\mathbf{M}})||\_{\mathbb{F}}^{2}+\gamma(||\underline{\mathbf{X}}^{\star}||\_{\text{TNN}}+\lambda||\underline{\mathbf{S}}^{\star}||\_{1}). \end{split}$$

By L- = Q- <sup>∗</sup> <sup>X</sup>-, we have:

$$\|\underline{\mathbf{L}}^{\star}\|\_{\text{TNN}} = \|\underline{\mathbf{Q}}^{\star} \ast \underline{\mathbf{X}}^{\star}\|\_{\text{TNN}} = \|\underline{\mathbf{X}}^{\star}\|\_{\text{TNN}}.\tag{A42}$$

Thus, we deduce:

$$\begin{split} &\frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{Q}}\_{\star} \ast \underline{\mathbf{X}}\_{\star} + \underline{\mathbf{S}}\_{\star} - \underline{\mathbf{M}}) \|\_{\sf F}^{2} + \gamma (\| \underline{\mathbf{X}}\_{\star} \|\_{\rm TNN} + \lambda \| \underline{\mathbf{S}}\_{\star} \|\_{1}) \\ &\leq \frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{\star} + \underline{\mathbf{S}}^{\star} - \underline{\mathbf{M}}) \|\_{\sf F}^{2} + \gamma (\| \underline{\mathbf{L}}^{\star} \|\_{\rm TNN} + \lambda \| \underline{\mathbf{S}}^{\star} \|\_{1}) . \end{split} \tag{A43}$$

According to Equations (A41) and (A43), we further have:

$$\begin{split} &\frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{Q}}\_{\star} \ast \underline{\mathbf{X}}\_{\star} + \underline{\mathbf{S}}\_{\star} - \underline{\mathbf{M}}) \|\_{\sf F}^{2} + \gamma (\| \underline{\mathbf{X}}\_{\star} \|\_{\sf TN} + \lambda \| \underline{\mathbf{S}}\_{\star} \|\_{1}) \\ &\leq \frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{\star} + \underline{\mathbf{S}}^{\star} - \underline{\mathbf{M}}) \|\_{\sf F}^{2} + \gamma (\| \underline{\mathbf{L}}^{\star} \|\_{\sf TN} + \lambda \| \underline{\mathbf{S}}^{\star} \|\_{1}) . \end{split} \tag{A44}$$

In this way, (Q<sup>∗</sup> <sup>∗</sup> <sup>X</sup>∗, S∗) is also the optimal solution to Problem (28).
