*Appendix A.1. The Proof of Theorem 1*

Appendix A.1.1. Key Lemmas for the Proof of Theorem 1

Before Proving Theorem 1, we should define some notations and operators first.

Suppose L0 <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> with tubal rank *<sup>r</sup>* has the skinny t-SVD L0 <sup>=</sup> <sup>U</sup> <sup>∗</sup> <sup>Λ</sup> <sup>∗</sup> <sup>V</sup> , where <sup>U</sup> <sup>∈</sup> <sup>R</sup>*n*1×*r*×*n*<sup>3</sup> , <sup>V</sup> <sup>∈</sup> <sup>R</sup>*r*×*n*2×*n*<sup>3</sup> are orthogonal tensors, and <sup>Λ</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> is an *<sup>f</sup>*-diagonal tensor. Define the following set:

$$\mathcal{T} := \left\{ \underline{\mathsf{U}} \ast \underline{\mathsf{A}} + \underline{\mathsf{B}} \ast \underline{\mathsf{V}}^{\top} \mid \underline{\mathsf{A}} \in \mathbb{R}^{r \times n\_2 \times n\_3}, \underline{\mathsf{B}} \in \mathbb{R}^{n\_1 \times r \times n\_3} \right\} \subset \mathbb{R}^{n\_1 \times n\_2 \times n\_3}.\tag{A1}$$

Then, define the projector onto <sup>T</sup> for any tensor T <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> as follows:

$$\begin{split} \mathcal{P}\_{\mathsf{T}}(\underline{\mathbf{T}}) & \coloneqq \underline{\mathsf{U}} \ast \underline{\mathsf{U}}^{\top} \ast \underline{\mathsf{T}} + \underline{\mathsf{T}} \ast \underline{\mathsf{V}} \ast \underline{\mathsf{V}}^{\top} - \underline{\mathsf{U}} \ast \underline{\mathsf{U}}^{\top} \ast \underline{\mathsf{T}} \ast \underline{\mathsf{V}} \ast \underline{\mathsf{V}}^{\top}, \\ \mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{T}}) & \coloneqq (\underline{\mathsf{I}} - \underline{\mathsf{U}} \ast \underline{\mathsf{U}}^{\top}) \ast \underline{\mathsf{T}} \ast (\underline{\mathsf{I}} - \underline{\mathsf{V}} \ast \underline{\mathsf{V}}^{\top}). \end{split} \tag{A2}$$

Let Ω<sup>⊥</sup> be the complement of Ω ⊂ [*n*1] × [*n*2] × [*n*3] which is the support of S0. Then, define two operators PΩ,PΩ<sup>⊥</sup> as follows:

$$\mathcal{P}\_{\Omega}(\underline{\mathsf{T}}) := \sum\_{(i,j,k) \in \Omega} \left\langle \underline{\mathsf{T}}\_{i} \mathbb{A}\_{i} \* \mathfrak{e}\_{k} \* \mathfrak{k}\_{\dot{j}}^{\top} \right\rangle, \quad \mathcal{P}\_{\Omega^{\perp}}(\underline{\mathsf{T}}) := \sum\_{(i,j,k) \in \Omega^{\perp}} \left\langle \underline{\mathsf{T}}\_{i} \mathbb{A}\_{i} \* \mathfrak{e}\_{k} \* \mathfrak{k}\_{\dot{j}}^{\top} \right\rangle,\tag{A3}$$

for any T <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> .

Define two sets Γ and Γ⊥ as follows:

$$\Gamma = \{ (\underline{\mathbf{A}}, \underline{\mathbf{A}}) \mid \underline{\mathbf{A}} \in \mathbb{R}^{n\_1 \times n\_2 \times n\_3} \}, \quad \Gamma^\perp = \{ (\underline{\mathbf{A}}, -\underline{\mathbf{A}}) \mid \underline{\mathbf{A}} \in \mathbb{R}^{n\_1 \times n\_2 \times n\_3} \}. \tag{A4}$$

*Sensors* **2019**, *19*, 5335

Then, for any tensors <sup>X</sup>*ι*,X*<sup>s</sup>* <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> , the projectors of the tensor **<sup>X</sup>** = (X*ι*,X*s*) into the sets <sup>Γ</sup> and Γ⊥ are given as follows, respectively:

$$\mathcal{P}\_{\Gamma}(\underline{\mathbf{X}}) = \left(\frac{\underline{\chi}\_{\iota} + \underline{\chi}\_{s}}{2}, \frac{\underline{\chi}\_{\iota} + \underline{\chi}\_{s}}{2}\right), \quad \mathcal{P}\_{\Gamma^{\perp}}(\underline{\mathbf{X}}) = \left(\frac{\underline{\chi}\_{\iota} - \underline{\chi}\_{s}}{2}, \frac{\underline{\chi}\_{s} - \underline{\chi}\_{s}}{2}\right). \tag{A5}$$

For any tensors X*ι*, X*<sup>s</sup>* <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> , define two operators on **<sup>X</sup>** = (X*ι*, X*s*) as follows:

$$(\mathcal{P}\_{\mathsf{T}} \times \mathcal{P}\_{\mathsf{D}})(\underline{\mathbf{X}}) = (\mathcal{P}\_{\mathsf{T}}(\underline{\mathbf{X}}), \mathcal{P}\_{\mathsf{D}}(\underline{\mathbf{X}}\_{\mathsf{s}})), \quad (\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\Omega^{\perp}})(\underline{\mathbf{X}}) = (\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{X}}\_{\mathsf{t}}), \mathcal{P}\_{\Omega^{\perp}}(\underline{\mathbf{X}}\_{\mathsf{s}})).\tag{A6}$$

Also define two norms as follows:

$$\|\underline{\mathbf{X}}\|\_{\mathrm{F}} = \sqrt{\|\underline{\mathbf{X}}\_{\mathrm{i}}\|\_{\mathrm{F}}^{2} + \|\underline{\mathbf{X}}\_{\mathrm{s}}\|\_{\mathrm{F}'}^{2}} \quad \|\underline{\mathbf{X}}\|\_{\mathrm{F},\mu} = \sqrt{\|\underline{\mathbf{X}}\_{\mathrm{i}}\|\_{\mathrm{F}}^{2} + \mu^{2} \|\underline{\mathbf{X}}\_{\mathrm{s}}\|\_{\mathrm{F}}^{2}}.\tag{A7}$$

where *μ* is a constant that will be determined afterwards.

We first give Lemma A1 which can be seen as a modified version of Lemma C.1 in [2].

**Lemma A1.** *Assume that* PΩPT ≤ <sup>1</sup> <sup>2</sup> *, and <sup>λ</sup>* <sup>≤</sup> <sup>1</sup> 2 <sup>√</sup>*n*<sup>3</sup> *. Suppose there exists a tensor* G∗ *satisfying the following conditions:*

$$\begin{cases} \mathcal{P}\tau(\underline{\mathbf{G}}^{\star}) = \underline{\mathbf{U}} \ast \underline{\mathbf{V}}^{\top}, \\ \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{G}}^{\star})\|\_{\mathrm{sp}} \leq \frac{1}{2}, \\ \|\mathcal{P}\_{\Omega}(\underline{\mathbf{G}}^{\star} - \lambda \text{sign}(\underline{\mathbf{S}}\_{0}))\|\_{\mathrm{F}} \leq \frac{\lambda}{4}, \\ \|\mathcal{P}\_{\Omega^{\perp}}(\underline{\mathbf{G}}^{\star})\|\_{\infty} \leq \frac{\lambda}{2}. \end{cases} \tag{A8}$$

*Then for any perturbation* <sup>Δ</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> *, one has:*

$$\begin{split} & \|\mathsf{L}\_{0} + \underline{\mathsf{A}}\|\_{\mathsf{TNN}} + \lambda \|\underline{\mathsf{S}}\_{0} - \underline{\mathsf{A}}\|\_{1} \\ & \geq \|\mathsf{L}\_{0}\|\_{\mathsf{TNN}} + \lambda \|\underline{\mathsf{S}}\_{0}\|\_{1} + \left(\frac{3}{4} - \|\mathscr{P}\_{\mathsf{T}^{\perp}}(\underline{\mathsf{C}})\|\_{\mathsf{S}\mathsf{P}}\right) \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathsf{A}})\|\_{\mathsf{TNN}} + \left(\frac{3}{4}\lambda - \mathscr{P}\_{\Omega^{\perp}}(\|\underline{\mathsf{C}}\|\_{\infty})\right) \|\mathcal{P}\_{\Omega^{\perp}}(\Lambda)\|\_{1}. \end{split} \tag{A9}$$

**Proof.** Let G*<sup>ι</sup>* ∈ *∂*L0TNN, i.e., any sub-gradient of ·TNN at L0, then it satisfies:

$$\|\mathcal{P}\_{\mathsf{T}}(\mathsf{Q}\_{\mathsf{T}}) = \mathsf{U} \* \underline{\mathsf{Y}}^{\top}, \quad \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\mathsf{Q}\_{\mathsf{T}})\|\_{\mathsf{SP}} \le 1. \tag{A10}$$

G*<sup>ι</sup>* ∈ *∂*L0TNN and G*<sup>s</sup>* ∈ *∂*(*λ*S01). According to the convexity of ·TNN and ·1, we have:

$$\|\underline{\mathbf{L}}\_{0} + \underline{\boldsymbol{\Delta}}\|\_{\text{TNN}} \ge \|\underline{\mathbf{L}}\_{0}\|\_{\text{TNN}} + \langle \underline{\mathbf{G}}\_{\nu}\underline{\boldsymbol{\Delta}}\rangle,\quad \lambda\|\underline{\mathbf{S}}\_{0} - \underline{\boldsymbol{\Delta}}\|\_{1} \ge \lambda\|\underline{\mathbf{S}}\_{0}\|\_{1} - \langle \underline{\mathbf{G}}\_{\nu}\underline{\boldsymbol{\Delta}}\rangle.\tag{A11}$$

By choosing G*<sup>ι</sup>* = U ∗ V + P ∗ Q , where P and Q comes from the skinny t-SVD of PT<sup>⊥</sup> (Δ) = P ∗ Σ ∗ Q , one has:

$$
\begin{split}
\langle\underline{\mathsf{C}}\underline{\mathsf{C}}\,\Delta\rangle &= \langle\underline{\mathsf{C}}\,\Delta\rangle + \langle\underline{\mathsf{C}}\,-\underline{\mathsf{C}}\,\Delta\rangle = \langle\underline{\mathsf{C}}\,\Delta\rangle + \langle\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathsf{C}}\_{\mathsf{I}}), \mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\Delta})\rangle - \langle\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathsf{C}}), \mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\Delta})\rangle \\&= \langle\underline{\mathsf{C}}\,\Delta\rangle - (1 - \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathsf{C}})\|\_{\mathsf{SP}})\|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\Delta})\|\_{\mathsf{TNN}}.
\end{split}
\tag{A12}
$$

Also, by choosing G*<sup>s</sup>* = *λ*sign(S0) − sign(PΩ<sup>⊥</sup> (Δ)), one has:

$$\begin{split}-\langle\mathsf{\mathsf{\mathsf{\overline{L}}}},\mathsf{\mathsf{\mathsf{\varPi}}}\rangle &= -\langle\mathsf{\mathsf{\mathsf{\overline{L}}}},\mathsf{\mathsf{\varDelta}}\rangle - \langle\mathsf{\mathsf{\mathsf{\overline{L}}}},-\mathsf{\mathsf{\mathsf{\mathsf{C}}}}\,\mathsf{\mathsf{\varDelta}}\rangle \\ &= -\langle\mathsf{\mathsf{\mathsf{\overline{L}}}},\mathsf{\mathsf{\varDelta}}\rangle - \langle\mathsf{\mathsf{P}}\_{\Omega}(\lambda\text{sign}(\underline{\mathsf{S}}\_{\mathsf{D}})-\underline{\mathsf{L}}),\mathsf{\varDelta}\_{\Omega}(\underline{\mathsf{A}})\rangle - \langle\mathsf{\mathcal{P}}\_{\Omega^{\perp}}(\underline{\mathsf{S}}\_{\mathsf{d}}),\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{A}})\rangle + \langle\mathsf{\mathsf{P}}\_{\Omega^{\perp}}(\underline{\mathsf{C}}),\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{A}})\rangle \\ &\geq -\langle\underline{\mathsf{C}},\underline{\mathsf{A}}\rangle - \|\mathsf{P}\_{\Omega}(\lambda\text{sign}(\underline{\mathsf{S}}\_{\mathsf{D}}) - \underline{\mathsf{C}})\|\_{\mathbb{F}}\|\mathsf{P}\_{\Omega}(\underline{\mathsf{A}})\|\_{\mathbb{F}} + \|\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{A}})\|\_{1} - \|\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{C}})\|\_{\mathbb{S}^{\mathsf{d}}}\|\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{A}})\|\_{1} \\ &\geq -\langle\mathsf{G},\underline{\mathsf{A}}\rangle - \frac{\lambda}{4} \|\mathsf{P}\mathbf{2}(\underline{\mathsf{A}})\|\mathsf{I} + (1 - \|\mathsf{P}\_{\Omega^{\perp}}(\underline{\mathsf{G}})\|\_{\mathbb{R}^{$$

Also note that:

$$\begin{split} \|\mathcal{P}\Omega(\Delta)\|\_{\mathbb{F}} &\leq \|\mathcal{P}\mathcal{Q}\mathcal{T}\{\Delta\}\|\|\_{\mathbb{F}} + \|\mathcal{P}\mathcal{Q}\mathcal{P}\_{\mathsf{T}^{\perp}}(\Delta)\|\|\_{\mathbb{F}} \\ &\leq \|\mathcal{P}\_{\Omega}\mathcal{P}\_{\mathsf{T}}(\Delta)\|\|\_{\mathbb{F}} + \|\mathcal{P}\_{\Omega}\mathcal{P}\_{\mathsf{T}^{\perp}}(\Delta)\|\|\_{\mathbb{F}} \\ &\leq \frac{1}{2} \|\Delta\|\_{\mathbb{F}} + \|\mathcal{P}\_{\Omega}\mathcal{P}\_{\mathsf{T}^{\perp}}(\Delta)\|\_{\mathbb{F}} \\ &\leq \frac{1}{2} \|\mathcal{P}\_{\Omega}(\Delta)\|\|\_{\mathbb{F}} + \frac{1}{2} \|\mathcal{P}\_{\Omega^{\perp}}(\Delta)\|\|\_{\mathbb{F}} + \|\mathcal{P}\_{\Omega}\mathcal{P}\_{\mathsf{T}^{\perp}}(\Delta)\|\|\_{\mathbb{F}} \end{split} \tag{A14}$$

which leads to:

$$\|\mathcal{P}\_{\Omega}(\underline{\Lambda})\|\_{\mathbb{F}} \le \|\mathcal{P}\_{\Omega^{\perp}}(\underline{\Lambda})\|\_{\mathbb{F}} + 2\|\mathcal{P}\_{\Omega}\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\Lambda})\|\_{\mathbb{F}} \le \|\mathcal{P}\_{\Omega^{\perp}}(\underline{\Lambda})\|\_{1} + 2\sqrt{n\_{3}}\|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\Lambda})\|\_{\mathbb{T}\mathbb{N}\mathbb{N}}.\tag{A15}$$

Putting things together, we have:

$$\begin{split} & \|\underline{\boldsymbol{\Delta}}\_{0} + \underline{\boldsymbol{\Delta}}\|\_{\text{TNN}} + \lambda \|\underline{\boldsymbol{\Delta}}\_{0} - \underline{\boldsymbol{\Delta}}\|\_{1} - (\|\underline{\boldsymbol{\Delta}}\boldsymbol{\rho}\|\_{\text{TNN}} + \lambda \|\underline{\boldsymbol{\Delta}}\_{0}\|\_{1}) \\ & \geq \left(1 - \frac{\lambda\sqrt{n\_{3}}}{2} - \|\boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}}(\underline{\boldsymbol{\Omega}})\|\_{\text{sp}}\right) \|\boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}}(\underline{\boldsymbol{\Delta}})\|\_{\text{TNN}} + \left(\frac{3}{4}\lambda - \boldsymbol{\mathcal{P}}\_{\mathsf{Q}^{\perp}}(\|\underline{\boldsymbol{\Omega}}\|\_{\infty})\right) \|\boldsymbol{\mathcal{P}}\_{\mathsf{Q}^{\perp}}(\underline{\boldsymbol{\Delta}})\|\_{1}. \end{split} \tag{A16}$$

Since *<sup>λ</sup>* <sup>≤</sup> <sup>1</sup> 2 <sup>√</sup>*n*<sup>3</sup> , it holds that:

$$\begin{split} & \| \| \underline{\boldsymbol{\Delta}}\_{0} + \underline{\boldsymbol{\Delta}} \| \| \boldsymbol{\rm T} \boldsymbol{\rm N} \boldsymbol{\lambda} + \boldsymbol{\lambda} \| \underline{\boldsymbol{\Delta}}\_{0} - \boldsymbol{\Delta} \| \| \boldsymbol{\rm T} \\ & \geq \| \underline{\boldsymbol{\Delta}}\_{0} \| \boldsymbol{\rm T} \boldsymbol{\rm N} \boldsymbol{\lambda} + \boldsymbol{\lambda} \| \underline{\boldsymbol{\Delta}}\_{0} \| \| + \left( \frac{3}{4} - \| \boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}} (\underline{\boldsymbol{\Omega}}) \| \| \boldsymbol{\rm p} \right) \| \boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}} (\underline{\boldsymbol{\Delta}}) \| \| \boldsymbol{\rm T} \boldsymbol{\rm N} \boldsymbol{\rm N} + \left( \frac{3}{4} \lambda - \boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}} (\| \underline{\boldsymbol{\rm G}} \| \| \boldsymbol{\rm s} \|) \right) \| \boldsymbol{\mathcal{P}}\_{\mathsf{T}^{\perp}} (\boldsymbol{\Delta}) \| \| \boldsymbol{\rm N} \boldsymbol{\rm S} \| \end{split}$$

for any perturbation <sup>Δ</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> .

**Lemma A2.** *Suppose that* PΩPT ≤ 1/2*, then for any* **X** = (X*ι*, X*s*)*, we have:*

$$\|\mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}}\times\mathcal{P}\_{\mathsf{D}})(\underline{\mathbf{X}})\|\_{\mathsf{F},\mu}^{2}\geq\frac{1+\mu^{2}}{8}\|\mathcal{P}\_{\mathsf{T}}\times\mathcal{P}\_{\mathsf{D}}(\underline{\mathbf{X}})\|\_{\mathsf{F}}^{2}.\tag{A17}$$

**Proof.** According to the definitions of P<sup>Γ</sup> and P<sup>T</sup> × PΩ, we have:

$$\mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}} \times \mathcal{P}\_{\mathsf{D}})(\underline{\mathbf{X}}) = \left(\frac{\mathcal{P}\_{\mathsf{T}}(\underline{\mathbf{X}}\_{i}) + \mathcal{P}\_{\mathsf{D}}(\underline{\mathbf{X}}\_{s})}{2}, \frac{\mathcal{P}\_{\mathsf{T}}(\underline{\mathbf{X}}\_{i}) + \mathcal{P}\_{\mathsf{D}}(\underline{\mathbf{X}}\_{s})}{2}\right). \tag{A18}$$

Then, we have:

PΓ(P<sup>T</sup> × PΩ)(**X**)<sup>2</sup> F,*<sup>μ</sup>* = (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) · 1 4 · PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> <sup>F</sup> + 2 PT(X*ι*),PΩ(X*s*) <sup>=</sup> (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) 4 PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> <sup>F</sup> + 2 PΩPTPT(X*ι*),PΩ(X*s*) <sup>≥</sup> (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) 4 PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> <sup>F</sup> − 2PΩPTPT(X*ι*)FPΩ(X*s*)<sup>F</sup> <sup>≥</sup> (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) 4 PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> <sup>F</sup> − 2 · 1 2 PT(X*ι*)FPΩ(X*s*)<sup>F</sup> <sup>≥</sup> (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) 4 PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> <sup>F</sup> <sup>−</sup> PT(X*ι*)<sup>2</sup> <sup>F</sup> <sup>+</sup> PΩ(X*s*)<sup>2</sup> F 2 <sup>=</sup> (<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*2) <sup>8</sup> P<sup>T</sup> × PΩ(**X**)<sup>2</sup> F. (A19)

Hence completes the proof.

### Appendix A.1.2. Proof of Theorem 1

**Proof.** For **<sup>X</sup>** = (L, <sup>S</sup>), define **X** <sup>=</sup> LTNN <sup>+</sup> *<sup>λ</sup>*S1. Let **Xˆ** = (Lˆ, Sˆ),**X**<sup>∗</sup> = (L0, S0). According to the optimality of (Lˆ, Sˆ) and the feasibility of (L0, S0), we directly have:

$$\|\hat{\mathbf{X}}\|\_{\diamond} \le \|\mathbf{X}^\*\|\_{\diamond} \tag{A20}$$

$$\|\underline{\mathsf{L}} + \underline{\mathsf{S}} - \underline{\mathsf{M}}\|\_{\mathbb{F}} \le \delta,\tag{A21}$$

$$\|\underline{\mathbf{L}}\_{0} + \underline{\mathbf{S}}\_{0} - \underline{\mathbf{M}}\|\_{\mathcal{F}} \le \delta. \tag{A22}$$

Let <sup>Δ</sup>*<sup>ι</sup>* <sup>=</sup> Lˆ <sup>−</sup> L0, <sup>Δ</sup>*<sup>s</sup>* <sup>=</sup> Sˆ <sup>−</sup> S0. Then, we have:

$$\|\Delta + \Delta\_{\mathsf{F}}\|\_{\mathsf{F}} = \|\widehat{\mathsf{L}} + \widehat{\mathsf{S}} - \mathsf{M} - (\mathsf{L}\_{0} + \mathsf{S}\_{0} - \mathsf{M})\|\_{\mathsf{F}} \leq \|\widehat{\mathsf{L}} + \widehat{\mathsf{S}} - \mathsf{M}\|\_{\mathsf{F}} + \|\mathsf{L}\_{0} + \mathsf{S}\_{0} - \mathsf{M}\|\_{\mathsf{F}} \leq 2\delta. \tag{A23}$$

Define the pair of error tensors **<sup>Δ</sup>** <sup>=</sup> **Xˆ** <sup>−</sup> **<sup>X</sup>**<sup>∗</sup> = (Δ*ι*, <sup>Δ</sup>*s*). The goal is to bound **Δ**F,*μ*.

First, we use the decomposition **<sup>Δ</sup>** <sup>=</sup> <sup>P</sup>Γ(**Δ**) + <sup>P</sup>Γ<sup>⊥</sup> (**Δ**), and let **<sup>Δ</sup>**<sup>Γ</sup> <sup>=</sup> <sup>P</sup>Γ(**Δ**)=(Δ<sup>Γ</sup> *<sup>ι</sup>* , <sup>Δ</sup><sup>Γ</sup> *<sup>s</sup>* ) = ( <sup>Δ</sup>*ι*+Δ*<sup>s</sup>* <sup>2</sup> , <sup>Δ</sup>*ι*+Δ*<sup>s</sup>* <sup>2</sup> ), **<sup>Δ</sup>**Γ<sup>⊥</sup> <sup>=</sup> <sup>P</sup>Γ<sup>⊥</sup> (**Δ**)=(ΔΓ<sup>⊥</sup> *<sup>ι</sup>* , <sup>Δ</sup>Γ<sup>⊥</sup> *<sup>s</sup>* )=( <sup>Δ</sup>*ι*−Δ*<sup>s</sup>* <sup>2</sup> , <sup>Δ</sup>*s*−Δ*<sup>ι</sup>* <sup>2</sup> ) for simplicity. Then, we have:

$$\|\|\mathbf{A}\|\|\_{\mathbb{F}\_{\succ}^{\mu}} = \|\mathbf{A}^{\mathsf{T}} + \mathbf{A}^{\mathsf{T}^{\bot}}\|\|\_{\mathbb{F}\_{\succ}^{\mu}} \le \|\mathbf{A}^{\mathsf{T}}\|\|\_{\mathbb{F}\_{\succ}^{\mu}} + \|\mathbf{A}^{\mathsf{T}^{\bot}}\|\|\_{\mathbb{F}\_{\succ}^{\mu}}.\tag{A24}$$

Please note that Δ<sup>Γ</sup> *<sup>ι</sup>* = <sup>Δ</sup><sup>Γ</sup> *<sup>s</sup>* <sup>=</sup> <sup>Δ</sup>*ι*+Δ*<sup>s</sup>* <sup>2</sup> , thus **Δ**ΓF,*<sup>μ</sup>* can be bounded easily as follows:

$$\|\underline{\boldsymbol{\Delta}}^{\mathsf{T}}\|\_{\mathrm{F},\mathbb{H}} = \sqrt{\|\underline{\boldsymbol{\Delta}}^{\mathsf{T}}\|\_{\mathrm{F}}^{2} + \mu^{2} \|\underline{\boldsymbol{\Delta}}^{\mathsf{T}}\|\_{\mathrm{F}}^{2}} = \frac{\sqrt{1 + \mu^{2}}}{2} \|\underline{\boldsymbol{\Delta}} + \underline{\boldsymbol{\Delta}}\_{s}\|\_{\mathrm{F}} \leq \delta \sqrt{1 + \mu^{2}}.\tag{A25}$$

Then, it remains to bound **Δ**Γ<sup>⊥</sup> F,*μ*. Due to the triangular inequality:

$$\|\underline{\boldsymbol{\Delta}}^{\mathsf{T}^{\perp}}\|\_{\mathsf{F},\mu} \leq \| (\mathcal{P}\_{\mathsf{T}} \times \mathcal{P}\_{\Omega})\underline{\boldsymbol{\Delta}}^{\mathsf{T}^{\perp}}\|\_{\mathsf{F},\mu} + \| (\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\Omega^{\perp}})\underline{\boldsymbol{\Delta}}^{\mathsf{T}^{\perp}}\|\_{\mathsf{F},\mu} \tag{A26}$$

**(A) bound** (PT<sup>⊥</sup> × PΩ<sup>⊥</sup> )**Δ**Γ<sup>⊥</sup> F,*μ*. According to the convexity of · we have

$$\|\underline{\mathbf{X}}^{\*} + \underline{\mathbf{A}}\|\_{\diamond} = \|\underline{\mathbf{X}}^{\*} + \underline{\mathbf{A}}^{\mathsf{T}} + \underline{\mathbf{A}}^{\mathsf{T}^{\bot}}\|\_{\diamond} \ge \|\underline{\mathbf{X}}^{\*} + \underline{\mathbf{A}}^{\mathsf{T}^{\bot}}\|\_{\diamond} - \|\underline{\mathbf{A}}^{\mathsf{T}}\|\_{\diamond}.\tag{A27}$$

Using Lemma A1, we have:

$$\begin{split} \|\underline{\mathbf{X}}^{\star} + \underline{\mathbf{A}}^{\mathrm{I}^{\perp}}\|\_{\diamond} &\geq \|\underline{\mathbf{X}}^{\star}\|\_{\diamond} + \left(\frac{3}{4} - \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{G}})\|\_{\mathsf{op}}\right) \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{A}}^{\mathrm{I}^{\perp}})\|\_{\mathsf{T}\mathsf{N}\mathsf{N}} + \left(\frac{3}{4}\lambda - \mathcal{P}\_{\mathsf{T}^{\perp}}(\|\underline{\mathbf{G}}\|\_{\mathsf{op}})\right) \|\mathcal{P}\_{\mathsf{T}^{\perp}}(\underline{\mathbf{A}}^{\mathrm{I}^{\perp}})\|\_{\mathsf{I}} \\ &\geq \|\underline{\mathbf{X}}^{\star}\|\_{\diamond} + \frac{1}{4} \|(\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\mathsf{T}^{\perp}})\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}\|\_{\diamond}. \end{split} \tag{A28}$$

Combining Equations (A20), (A27) and (A28), we have:

$$\|\underline{\mathbf{A}}^{\mathsf{T}}\|\_{\circ} \geq \frac{1}{4} \|(\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\Omega^{\perp}})\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}\|\_{\circ} \tag{A29}$$

Then, with *<sup>μ</sup>* <sup>=</sup> <sup>√</sup>*n*3*λ*, we reach a bound on (PT<sup>⊥</sup> × PΩ<sup>⊥</sup> )**Δ**Γ<sup>⊥</sup> F,*<sup>μ</sup>* as follows:

(PT<sup>⊥</sup> × PΩ<sup>⊥</sup> )**Δ**Γ<sup>⊥</sup> F,*<sup>μ</sup>* ≤ PT<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>ι</sup>* )<sup>F</sup> <sup>+</sup> *<sup>μ</sup>*PΩ<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>s</sup>* )<sup>F</sup> <sup>≤</sup> <sup>√</sup>*n*3PT<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>ι</sup>* )TNN <sup>+</sup> *<sup>μ</sup>*PΩ<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>s</sup>* )<sup>1</sup> <sup>≤</sup> <sup>√</sup>*n*<sup>3</sup> PT<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>ι</sup>* )TNN <sup>+</sup> *<sup>λ</sup>*PΩ<sup>⊥</sup> (ΔΓ<sup>⊥</sup> *<sup>s</sup>* )<sup>1</sup> <sup>≤</sup> <sup>√</sup>*n*3(PT<sup>⊥</sup> × PΩ<sup>⊥</sup> )**Δ**Γ<sup>⊥</sup> ≤ 4 <sup>√</sup>*n*3**Δ**Γ ≤ 4 <sup>√</sup>*n*<sup>3</sup> Δ<sup>Γ</sup> *<sup>ι</sup>* TNN <sup>+</sup> *<sup>λ</sup>*Δ<sup>Γ</sup> *s* 1 ≤ 4 <sup>√</sup>*n*<sup>3</sup> min{*n*1, *<sup>n</sup>*2}Δ<sup>Γ</sup> *<sup>ι</sup>* <sup>F</sup> + *λ* <sup>√</sup>*n*1*n*2*n*3Δ<sup>Γ</sup> *s* F = 4 <sup>√</sup>*n*<sup>3</sup> min{*n*1, *n*2} + *λ* <sup>√</sup>*n*1*n*2*n*<sup>3</sup> Δ<sup>Γ</sup> *ι* F ≤ 4 <sup>√</sup>*n*<sup>3</sup> min{*n*1, *n*2} + *λ* <sup>√</sup>*n*1*n*2*n*<sup>3</sup> *δ*. (A30)

**(B) bound** (P<sup>T</sup> × PΩ)**Δ**Γ<sup>⊥</sup> F,*μ*. Please note that:

$$\mathcal{P}\_{\mathsf{T}}(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}) = \underline{\mathbf{0}} = \mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}} \times \mathcal{P}\_{\mathsf{D}})(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}) + \mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\mathsf{D}^{\perp}})(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}),\tag{A31}$$

which means:

$$\|\mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}}\times\mathcal{P}\_{\mathsf{D}})(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}})\|\_{\mathsf{F},\mu} = \|\mathcal{P}\_{\mathsf{T}}(\mathcal{P}\_{\mathsf{T}^{\perp}}\times\mathcal{P}\_{\mathsf{D}^{\perp}}(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}))\|\_{\mathsf{F},\mu} \leq \|\mathcal{P}\_{\mathsf{T}^{\perp}}\times\mathcal{P}\_{\mathsf{D}^{\perp}}(\underline{\mathbf{A}}^{\mathsf{T}^{\perp}})\|\_{\mathsf{F},\mu}.\tag{A.32}$$

According to Lemma A2, we have:

$$\begin{split} \| (\mathcal{P}\mathbb{1} \times \mathcal{P}\boldsymbol{\alpha})(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \|\_{\mathrm{F},\boldsymbol{\mu}} &\leq \| \mathcal{P}\boldsymbol{\pi}(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \| \| \boldsymbol{\mathrm{v}} + \boldsymbol{\mu} \| \| \boldsymbol{\mathcal{P}}\boldsymbol{\alpha}(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \| \| \boldsymbol{\mathrm{F}} \\ &\leq \sqrt{1 + \mu^{2}} \sqrt{\| \mathcal{P}\boldsymbol{\pi}(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \|\_{\mathrm{F}}^{2} + \| \boldsymbol{\mathcal{P}}\boldsymbol{\alpha}(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \|\_{\mathrm{F}}^{2}} \\ &= \sqrt{1 + \mu^{2}} \| \boldsymbol{\mathcal{P}}\boldsymbol{\pi} \times \boldsymbol{\mathcal{P}}\boldsymbol{\alpha}(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \| \| \boldsymbol{\mathrm{F}} \\ &\leq \sqrt{1 + \mu^{2}} \cdot \sqrt{\frac{8}{\sqrt{1 + \mu^{2}}}} \cdot \| \boldsymbol{\mathcal{P}}\boldsymbol{\pi}(\boldsymbol{\mathcal{P}}\boldsymbol{\pi} \times \boldsymbol{\mathcal{P}}\boldsymbol{\alpha})(\underline{\boldsymbol{\Delta}}^{\mathsf{T}}) \| \| \boldsymbol{\mathrm{F}}. \end{split} \tag{A33}$$

According to Equations (A32) and (A33), we obtain:

$$\| (\mathcal{P}\_{\mathsf{T}} \times \mathcal{P}\_{\Omega}) (\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}) \|\_{\mathsf{F},\mathsf{H}} \leq 2\sqrt{2} \| (\mathcal{P}\_{\mathsf{T}^{\perp}} \times \mathcal{P}\_{\Omega^{\perp}}) (\underline{\mathbf{A}}^{\mathsf{T}^{\perp}}) \|\_{\mathsf{F},\mathsf{H}}.\tag{A34}$$

Thus, combing Equations (A24), (A25), (A30) and (A34), and setting *<sup>μ</sup>* <sup>=</sup> <sup>√</sup>*n*3*λ*, we obtain:

$$\|\underline{\mathbf{A}}\|\_{\mathbb{F},\mu} \le \left(\sqrt{1+n\_3\lambda^2} + 4(1+2\sqrt{2})\left(\sqrt{\min\{n\_1,n\_2\}n\_3} + n\_3\lambda\sqrt{n\_1n\_2}\right)\right)\delta.\tag{A35}$$

Since *λ* = √ <sup>1</sup> max{*n*1,*n*2}*n*<sup>3</sup> , we have:

$$\|\underline{\Delta}\|\_{\mathcal{F},\mu} \le \left(\sqrt{1 + \frac{1}{\max\{n\_1, n\_2\}}} + 8(1 + 2\sqrt{2})\sqrt{\min\{n\_1, n\_2\}n\_3}\right)\delta,\tag{A36}$$

*Sensors* **2019**, *19*, 5335

which indicates that:

$$\begin{split} \|\underline{\mathbf{L}} - \underline{\mathbf{L}}\_{0}\|\_{\mathrm{F}} &\leq \left(\sqrt{1 + \frac{1}{\max\{n\_{1}, n\_{2}\}}} + 8(1 + 2\sqrt{2})\sqrt{\min\{n\_{1}, n\_{2}\}n\_{3}}\right)\delta \\ \|\underline{\mathbf{S}} - \underline{\mathbf{S}}\_{0}\|\_{\mathrm{F}} &\leq \left(\sqrt{1 + \max\{n\_{1}, n\_{2}\}} + 8(1 + 2\sqrt{2})\sqrt{n\_{1}n\_{2}n\_{3}}\right)\delta. \end{split} \tag{A37}$$

Moreover, according to the analysis in [2], the conditions PΩPT ≤ <sup>1</sup> <sup>2</sup> and Equation (A8) in Lemma A1 hold with probability at least 1 − *<sup>c</sup>*1(*n*<sup>3</sup> max{*n*1, *<sup>n</sup>*2})−*c*<sup>2</sup> , where *<sup>c</sup>*<sup>1</sup> and *<sup>c</sup>*<sup>2</sup> are positive constants.

In this way, the proof of Theorem 1 is completed.
