*3.1. The Proposed STPCP*

As for the measurement Model (1), we further assume that the noise tensor E0 has bounded energy measured in F-norm, i.e., E0<sup>F</sup> ≤ *δ*. Please note that the limited energy assumption is very mild, since most signals are of limited energy.

To recover the low-rank tensor L0 and the sparse tensor S0, we first produce the following optimization problem:

$$\mathbf{u}\left(\underline{\mathbf{L}},\underline{\mathbf{S}}\right) = \underset{\mathbf{L}\,\mathbf{S}}{\operatorname{argmin}} \,\|\underline{\mathbf{L}}\|\_{\operatorname{TNN}} + \lambda\|\underline{\mathbf{S}}\|\_{1\prime} \quad \text{s.t.} \; \|\underline{\mathbf{M}} - \underline{\mathbf{L}} - \underline{\mathbf{S}}\|\_{\mathbf{F}} \le \delta\,\tag{18}$$

where *λ* is a positive parameter balancing the two regularizers. The motivation is to use TNN as a low-rank regularization term to exploit the low-dimensional structure in the signal tensor, whereas tensor *l*1-norm is used to impose sparsity in the corruption tensor (since we assumes it to be sparse).

The relationship between Model (18) and existing models are discussed in Remark 1 and Remark 2.

### **Remark 1.** *The following models can be seen as special cases as the proposed STPCP Model (18);*


$$\min\_{\mathbf{L,S}} \quad \|\mathbf{L}\|\_{\*} + \lambda \|\mathbf{S}\|\_{1}, \quad \text{s.t.} \; \|\mathbf{M} - \mathbf{L} - \mathbf{S}\|\_{\mathbf{F}} \le \delta. \tag{19}$$

*(III). When n*<sup>3</sup> = 1 *and δ* = 0*, the proposed STPCP further degenerates to Robust Principal Component Analysis (RPCA) [46] given as follows*

$$\min\_{\mathbf{L,S}} \quad \|\mathbf{L}\|\_{\*} + \lambda \|\mathbf{S}\|\_{1}, \quad \text{s.t.} \; \mathbf{L} + \mathbf{S} = \mathbf{M}. \tag{20}$$

**Remark 2.** *The differences from the proposed Model (18) and TNN-based RTD Model ((17) [37]) is as follows. First, our model does not need to upper estimate the l*∞*-norm of the underlying tensor. Second, our model is a constrained optimization problem, whereas Model (17) is an unconstrained optimization problem.*

### *3.2. A Theorem for Stable Recovery*

To analyze the statistical performance of Model (18), we should assume on the underlying low-rank tensor L0 that it is not sparse. Only by this assumption, L0 can be identified from its mixture with sparse S0. Such an assumption can be described by the tensor incoherence condition [2,47], which is used to provide an identifiablility for low-rank L0.

**Definition 12** (Tensor incoherence condition [2,47])**.** *Given a 3-way tensor* <sup>T</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> *with tubal rank r, suppose it has the skinny t-SVD* <sup>T</sup> <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 f-diagonal tensor. Then,* <sup>T</sup> *is said to satisfy the tensor incoherent condition (TIC) with parameter μ*(T) *if the following inequalities hold:*

$$\max\_{i \in \lfloor n\_1 \rfloor} \| \underline{\mathbf{U}}^{\top} \ast \boldsymbol{\mathfrak{k}}\_i \|\_{\mathbf{F}} \quad \leq \sqrt{\frac{r \mu(\underline{\mathbf{T}})}{n\_1 n\_3}},\tag{21}$$

$$\max\_{j \in \lfloor n\_2 \rfloor} \| \underline{\mathbf{V}}^\top \ast \mathfrak{k}\_j \|\_F \quad \le \sqrt{\frac{r \mu(\underline{\mathbf{T}})}{n\_2 n\_3}},\tag{22}$$

$$\|\underline{\mathsf{T}}\*\underline{\mathsf{Y}}^{\top}\|\_{\infty} \leq \sqrt{\frac{r\mu(\underline{\mathsf{T}})}{n\_1 n\_2 n\_3}}.\tag{23}$$

*where* <sup>e</sup>˚*<sup>i</sup>* <sup>∈</sup> <sup>R</sup>*n*1×1×*n*<sup>3</sup> *is a tensor column basis with only the* (*i*, 1, 1)*-th element being 1 and all the others being 0, and* <sup>e</sup>˚*<sup>j</sup>* <sup>∈</sup> <sup>R</sup>*n*2×1×*n*<sup>3</sup> *is also a tensor column basis with only the* (*j*, 1, 1)*-th element being 1 and all the others being 0.*

**Assumption 1.** *Suppose the true tensor* L0 *in the measurement model (1) satisfies tensor incoherence condition with parameter μ.*

Assumption 1 intrinsically ensures that the row bases and column bases of L0 do not align well with the canonical row and column bases. Thus, the low-rank L0 is not sparse, which avoids the ambiguity that low-rank component can also be sparse in the measurement Model (1).

We should also force the sparse component in Model (1) is not low rank.

**Assumption 2.** *Assume the support* Ω *of* S0 *is drawn uniformly at random.*

Now we can establish an upper bound on the estimation error of Lˆ and Sˆ in Problem (18).

**Theorem 1** (An Upper Bound on the Estimation Error)**.** *Suppose* L0 *and* S0 *satisfy Assumption 1 and Assumption 2, respectively. If the tubal rank r of* L0 *and the sparsity (i.e., the l*0*-norm) s of* S0 *are respectively upper bounded as follows:*

$$r \le \frac{c\_7 \min\{n\_1, n\_2\}}{\mu \log^2(n\_3 \max\{n\_1, n\_2\})}, \quad \text{and} \quad s \le c\_8 n\_1 n\_2 n\_3 \tag{24}$$

*where cl and cs are two sufficiently small numerical constants independent on the dimensions n*1*, n*<sup>2</sup> *and n*3*. Then the estimator defined in Model (18) satisfy the following inequalities:*

$$\begin{split} \|\hat{\mathbf{L}} - \mathbf{L}\_{0}\|\_{\mathbb{F}} &\leq \left(\sqrt{1 + \frac{1}{\max\{n\_{1}, n\_{2}\}}} + 8(1 + 2\sqrt{2})\sqrt{\min\{n\_{1}, n\_{2}\}n\_{3}}\right)\delta \\ \|\underline{\mathbf{S}} - \underline{\mathbf{S}}\_{0}\|\_{\mathbb{F}} &\leq \left(\sqrt{1 + \max\{n\_{1}, n\_{2}\}} + 8(1 + 2\sqrt{2})\sqrt{n\_{1}n\_{2}n\_{3}}\right)\delta \end{split} \tag{25}$$

*with probability at least* <sup>1</sup> − *<sup>c</sup>*1(*n*<sup>3</sup> max{*n*1, *<sup>n</sup>*2})−*c*<sup>2</sup> *(over the choice of support of* S0*), where <sup>c</sup>*<sup>1</sup> *and <sup>c</sup>*<sup>2</sup> *are positive constants independent on the dimensions n*1*, n*<sup>2</sup> *and n*3*.*

The proof of Theorem 1 are given in the appendix. In Theorem 1, estimation errors on L0 and S0 are separately established. It indicates that the estimation error scales linearly with the noise level *δ*, which is in consistence with the result in [37].

**Remark 3.** *A significant progress over [37] is that in the noiseless setting where* E0 *vanishes, our analysis can provide exact recovery guarantee of* L0 *and* S0*. This is because the tensor incoherence condition adopted in our analysis intrinsically ensures that the low-rank tensor* L0 *is not sparse and thus can be separated from the sparse corruption tensor, whereas the non-spiky condition adopted in [37] fails to provide identifiability in the measurement Model (1).*

For Theorem 1, we also give the following remark.

**Remark 4.** *The error bounds established in Theorem 1 are consistent with the theoretical analysis for the special cases shown in Remark 1.*


### **4. Algorithms**

In this section, we design two algorithms. The first algorithm is based on the framework of ADMM [48] which has been extensively used in convex optimization with good convergence behavior. However, ADMM requires full SVDs on large matrices in each iteration which is high computational burden in high-dimensional settings. Thus, the second algorithm is proposed to solve this issue by using a factorization trick which can instead conducting SVDs on much smaller matrices.

### *4.1. An ADMM Algorithm*

The proposed estimator (18) is equivalent to the following unconstrained problem:

$$\min\_{\mathbf{L}, \mathbf{S}} \frac{1}{2} \|\underline{\mathbf{L}} + \underline{\mathbf{S}} - \underline{\mathbf{M}}\|\_{\mathbf{F}}^2 + \gamma (\|\underline{\mathbf{L}}\|\_{\text{TNN}} + \lambda \|\underline{\mathbf{S}}\|\_1),\tag{26}$$

where *γ* is a positive parameter balancing the data fidelity term and the regularization term.

Besides being corrupted by noises and outliers, the observed tensor M may also suffer from missing entries which can be taken as outliers with known positions in many applications. Thus, it is more practical to consider the recovery of L0 against outliers S0, noises E0 and missing entries shown in the following measurement model:

$$
\underline{\mathbf{M}} = \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}\_0 + \underline{\mathbf{S}}\_0 + \underline{\mathbf{E}}\_0),
\tag{27}
$$

where tensor <sup>B</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> denote the missing mask where <sup>B</sup>*ijk* <sup>=</sup> 1, if the (*i*, *<sup>j</sup>*, *<sup>k</sup>*)-th entry of <sup>L</sup> is observed and B*ijk* = 0 otherwise, and denotes element-wise multiplication. Taking into consideration of missing entries, Model (26) can be further modified as:

$$\min\_{\mathbf{L}\in\mathbb{Z}}\frac{1}{2}||\mathbb{B}\odot(\mathbf{L}+\mathbb{S}-\mathbb{M})||\_{\mathbb{F}}^{2}+\gamma(||\mathbf{L}||\mathbb{T}\mathbb{N}\mathbb{N}+\lambda||\mathbb{S}||\mathbb{1}).\tag{28}$$

By adding auxiliary variables to Problem (28), we obtain:

$$\begin{aligned} \min\_{\mathbf{K}, \mathbf{L}, \mathbf{E}, \mathbf{\underline{S}}} & \frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{L}} + \underline{\mathbf{S}} - \underline{\mathbf{M}}) \|\_{\mathbf{F}}^2 + \gamma \| \underline{\mathbf{K}} \|\_{\mathbf{T} \text{NN}} + \gamma \lambda \| \underline{\mathbf{B}} \|\_{1} \\ \text{s.t. } & \underline{\mathbf{K}} = \underline{\mathbf{L}}, \; \underline{\mathbf{B}} = \underline{\mathbf{S}}. \end{aligned} \tag{29}$$

The Augmented Lagrangian (AL) of Problem (29) is given as follows:

$$\begin{split} \|L\_{\rho}(\underline{\mathbf{L}}, \underline{\mathbf{S}}, \underline{\mathbf{K}}, \underline{\mathbf{R}}, \underline{\mathbf{Y}}\_{1}, \underline{\mathbf{Y}}\_{2}) - \frac{1}{2} \|\underline{\mathbf{B}} \odot (\underline{\mathbf{L}} + \underline{\mathbf{S}} - \underline{\mathbf{M}})\|\_{\rm F}^{2} + \gamma \|\underline{\mathbf{K}}\|\_{\rm TN} + \gamma \lambda \|\underline{\mathbf{R}}\|\_{1} \\ &+ \langle \underline{\mathbf{X}}\_{1}, \underline{\mathbf{K}} - \underline{\mathbf{L}} \rangle + \frac{\rho}{2} \|\underline{\mathbf{K}} - \underline{\mathbf{L}}\|\_{\rm F}^{2} + \langle \underline{\mathbf{X}}\_{2}, \underline{\mathbf{R}} - \underline{\mathbf{S}} \rangle + \frac{\rho}{2} \|\underline{\mathbf{R}} - \underline{\mathbf{S}}\|\_{\rm F}^{2} . \end{split} \tag{30}$$

where Y1, Y2 <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> are Lagrangian multipliers and *<sup>ρ</sup>* is a penalty parameter.

According the strategy of ADMM, we update prime variables (L, S) and (K,R) by alternative minimization of AL in Problem (29) as follows:

• Update (L, S). We update (L, S) by minimizing *L<sup>ρ</sup>* with other variables fixed as follows:

$$\begin{split} & (\underline{\mathsf{L}}^{t+1}, \underline{\mathsf{S}}^{t+1}) \\ &= \operatorname{argmin}\_{\left(\underline{\mathsf{L}}, \underline{\mathsf{S}}\right)} L\_{\boldsymbol{\theta}}(\underline{\mathsf{L}}, \underline{\mathsf{S}}, \underline{\mathsf{K}}^{t}, \underline{\mathsf{K}}^{t}, \underline{\mathsf{N}}^{t}\_{1}, \underline{\mathsf{N}}^{t}\_{2}) \\ &= \operatorname{argmin}\_{\left(\underline{\mathsf{L}}, \underline{\mathsf{S}}\right)} \frac{1}{2} \|\underline{\mathsf{B}} \circ \left(\underline{\mathsf{L}} + \underline{\mathsf{S}} - \underline{\mathsf{M}}\right)\|\_{\mathrm{F}}^{2} + \left(\underline{\mathsf{Y}}\_{1}^{t}, \underline{\mathsf{K}}^{t} - \underline{\mathsf{L}}\right) + \frac{\rho}{2} \|\underline{\mathsf{K}}^{t} - \underline{\mathsf{L}}\|\_{\mathrm{F}}^{2} + \left(\underline{\mathsf{T}}\_{2}^{t}, \underline{\mathsf{R}}^{t} - \underline{\mathsf{S}}\right) + \frac{\rho}{2} \|\underline{\mathsf{R}}^{t} - \underline{\mathsf{S}}\|\_{\mathrm{F}}^{2}. \end{split} \tag{31}$$

Taking derivatives of the right-hand side of Equation (31) with respect to L and S respectively, and setting the results zero, we obtain:

$$\begin{split} \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{t+1} + \underline{\mathbf{S}}^{t+1}) - \underline{\mathbf{B}} \odot \underline{\mathbf{M}} - \underline{\mathbf{Y}}\_{1}^{t} + \rho (\underline{\mathbf{L}}^{t+1} - \underline{\mathbf{K}}^{t}) &= \underline{\mathbf{0}} \\ \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{t+1} + \underline{\mathbf{S}}^{t+1}) - \underline{\mathbf{B}} \odot \underline{\mathbf{M}} - \underline{\mathbf{Y}}\_{2}^{t} + \rho (\underline{\mathbf{S}}^{t+1} - \underline{\mathbf{R}}^{t}) &= \underline{\mathbf{0}}. \end{split} \tag{32}$$

Resolving the above equation group yields:

$$\begin{array}{l} \mathbf{L}^{t+1} = \left(\rho(\underline{\mathbf{B}} + \rho\underline{\mathbf{1}}) \odot \underline{\mathbf{K}}^{t} + \rho\underline{\mathbf{B}} \odot \underline{\mathbf{M}} + (\underline{\mathbf{B}} + \rho\underline{\mathbf{1}}) \odot \underline{\mathbf{Y}}\_{1}^{t} - \underline{\mathbf{B}} \odot \underline{\mathbf{Y}}\_{2}^{t} - \rho\underline{\mathbf{B}} \odot \underline{\mathbf{R}}^{t}\right) \odot \left(\rho(2\underline{\mathbf{B}} + \rho\underline{\mathbf{1}})\right), \\\underline{\mathbf{S}}^{t+1} = \left(\rho(\underline{\mathbf{B}} + \rho\underline{\mathbf{1}}) \odot \underline{\mathbf{R}}^{t} + \rho\underline{\mathbf{B}} \odot \underline{\mathbf{M}} + (\underline{\mathbf{B}} + \rho\underline{\mathbf{1}}) \odot \underline{\mathbf{Y}}\_{2}^{t} - \underline{\mathbf{B}} \odot \underline{\mathbf{Y}}\_{1}^{t} - \rho\underline{\mathbf{B}} \odot \underline{\mathbf{K}}^{t}\right) \odot \left(\rho(2\underline{\mathbf{B}} + \rho\underline{\mathbf{1}})\right), \end{array} \tag{33}$$

where denotes entry-wise division and 1 denotes the tensor all whose entries are 1. • Update (K, R). We update (K, R) by minimizing *L<sup>ρ</sup>* with other variables fixed as follows:

(K*t*<sup>+</sup>1, R*t*+1) <sup>=</sup> argmin(K,R) *<sup>L</sup>ρ*(L*t*<sup>+</sup>1, S*t*<sup>+</sup>1, K, R, Y*<sup>t</sup>* 1, Y*<sup>t</sup>* 2) (34)

$$\mathbf{H} = \operatorname{argmin}\_{\{\mathbf{Z}\_{\mathsf{T}}\}} \gamma \|\mathbf{K}\| \|\mathbf{r}\mathbf{N}\mathbf{N} + \gamma \lambda \|\|\mathbf{R}\|\|\_{1} + \left\langle \mathbf{X}\_{\mathsf{T}}^{\mathsf{t}}, \mathbf{K} - \mathbf{L}^{\mathsf{t}+1} \right\rangle + \frac{\rho}{2} \|\|\mathbf{K} - \mathbf{L}^{\mathsf{t}+1}\|\_{\mathsf{F}}^{2} + \left\langle \mathbf{X}\_{\mathsf{T}}^{\mathsf{t}}, \mathbf{R} - \mathbf{S}^{\mathsf{t}+1} \right\rangle + \frac{\rho}{2} \|\|\mathbf{R} - \mathbf{S}^{\mathsf{t}+1}\|\_{\mathsf{F}}^{2}$$

Please note that Problem (34) can further be solved separately as follows:

$$\begin{split} \underline{\mathsf{K}}^{t+1} &= \underset{\stackrel{\mathsf{K}}{\mathsf{K}}}{\operatorname{argmin}} \, \gamma \| \underline{\mathsf{K}} \|\_{\mathsf{TNN}} + \left\langle \underline{\mathsf{Y}}\_{1}^{t}, \underline{\mathsf{K}} - \underline{\mathsf{L}}^{t+1} \right\rangle + \frac{\rho}{2} \| \underline{\mathsf{K}} - \underline{\mathsf{L}}^{t+1} \|\_{\mathsf{F}}^{2} \\ &= \mathfrak{S}\_{\gamma\rho^{-1}}^{\|\cdot\|\_{\mathsf{TNN}}} \left( \underline{\mathsf{L}}^{t+1} - \rho^{-1} \underline{\mathsf{Y}}\_{1}^{t} \right) . \end{split} \tag{35}$$

and

$$\begin{split} \underline{\mathbf{R}}^{t+1} &= \underset{\underline{\mathbf{R}}}{\operatorname{argmin}} \ \gamma \lambda || \underline{\mathbf{R}} ||\_1 + \left< \underline{\mathbf{Y}}\_1^t, \underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1} \right> + \frac{\rho}{2} || \underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1} ||\_{\mathbf{F}}^2 \\ &= \underline{\mathbf{S}}^{\parallel \cdot \parallel\_1}\_{\gamma \lambda \rho^{-1}} \left( \underline{\mathbf{S}}^{t+1} - \rho^{-1} \underline{\mathbf{Y}}\_2^t \right) . \end{split} \tag{36}$$

where <sup>S</sup>·TNN *<sup>τ</sup>* (·) is the proximity operator of TNN [5]. and <sup>S</sup>·<sup>1</sup> *<sup>τ</sup>* (·) is the proximity operator of tensor *l*1-norm given as follows [49]:

$$\mathfrak{S}\_{\tau}^{\parallel \cdot \parallel\_{1}}(\underline{\mathbf{A}}) := \operatorname\*{argmin}\_{\underline{\mathbf{A}}} \tau \|\underline{\mathbf{A}}\|\_{1} + \frac{1}{2} \|\underline{\mathbf{A}} - \underline{\mathbf{A}}\|\_{\overline{\mathbf{F}}}^{2} = \operatorname\*{sign}(\underline{\mathbf{A}}) \odot \max\{ (|\underline{\mathbf{A}}| - \tau, 0) \},$$

In [5], a closed-form expression of S*τ*(·) is given as follows:

**Lemma 3** (Proximity operator of TNN [5])**.** *For any 3D tensor* <sup>A</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> *with reduced t-SVD* <sup>A</sup> <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> *and* <sup>V</sup> <sup>∈</sup> <sup>R</sup>*n*2×*r*×*n*<sup>3</sup> *are orthogonal tensors and* <sup>Λ</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> *is the f-diagonal tensor of singular tubes, the proximity operator* S·TNN *<sup>τ</sup>* (A) *at* A *can be computed by:*

$$\mathfrak{S}\_{\tau}^{\parallel \cdot \parallel \text{TNN}}(\mathsf{A}) := \operatorname\*{argmin}\_{\mathsf{A}} \tau \|\mathsf{X}\|\_{\mathsf{TNN}} + \frac{1}{2} \|\mathsf{X} - \mathsf{A}\|\_{\mathsf{F}}^{2} = \mathsf{L} \* \|\mathsf{f}\|\_{\mathsf{F}}^{2} (\max(\mathsf{f}\mathsf{f}\mathsf{3}(\Delta) - \tau, 0)) \* \underline{\mathsf{Y}}^{\top},$$

• Update (Y1, Y2). The Lagrangian multipliers are updated by gradient ascent as follows:

$$\begin{aligned} \underline{\mathsf{Y}}\_{1}^{t+1} &= \underline{\mathsf{Y}}\_{1}^{t} + \rho(\underline{\mathsf{K}}^{t+1} - \underline{\mathsf{L}}^{t+1}), \\ \underline{\mathsf{Y}}\_{2}^{t+1} &= \underline{\mathsf{Y}}\_{2}^{t} + \rho(\underline{\mathsf{R}}^{t+1} - \underline{\mathsf{S}}^{t+1}). \end{aligned} \tag{37}$$

The algorithm is summarized in Algorithm 1. The convergence analysis of Algorithm 1 is established in Theorem 2.

### **Algorithm 1** Solving Problem (29) using ADMM.

**Input:** The observed tensor M, the parameters *γ*, *λ*, *ρ*, *δ*. 1: Initialize *t* = 0, L0 = S<sup>0</sup> = K<sup>0</sup> = R0 = Y<sup>0</sup> <sup>1</sup> = <sup>Y</sup><sup>0</sup> <sup>2</sup> <sup>=</sup> <sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> 2: **for** *t* = 0, ··· , *T*max **do** 3: Update (L*t*<sup>+</sup>1, S*t*+1) by Equation (33); 4: Update (K*t*<sup>+</sup>1, R*t*+1) by Equations (35)–(36); 5: Update (Y*t*+<sup>1</sup> <sup>1</sup> , Y*t*+<sup>1</sup> <sup>2</sup> ) by Equation (37); 6: Check the convergence criteria: (i) convergence of variables: A*t*+<sup>1</sup> <sup>−</sup> <sup>A</sup>*<sup>t</sup>* <sup>∞</sup> ≤ *δ*, ∀A ∈ {L, S, K, R}, (ii) convergence of constraints: max{K*t*+<sup>1</sup> <sup>−</sup> <sup>L</sup>*<sup>t</sup>* ∞, R*t*+<sup>1</sup> <sup>−</sup> <sup>S</sup>*t*+1∞} ≤ *<sup>δ</sup>*. 7: **end for Output:** (Lˆ, Sˆ)=(L*t*<sup>+</sup>1, S*t*+1).

**Theorem 2** (Convergence of Algorithm 1)**.** *For any ρ* > 0*, if the unaugmented Lagrangian L*(L, S, K,R,Y1,Y2) *has a saddle point, then the iterations L*(L*<sup>t</sup>* , S*t* , K*<sup>t</sup>* ,R*<sup>t</sup>* ,Y*<sup>t</sup>* 1,Y*<sup>t</sup>* <sup>2</sup>) *in Algorithm 1 satisfy the residual convergence, objective convergence and dual variable convergence of Problem (29) as t* → ∞*.*

The proof of Theorem 2 is given in the Appendix A.

In a single iteration of Algorithm 1, the main cost comes from updating L*<sup>t</sup>* which involves computing FFT, IFFT and *n*<sup>3</sup> SVDs of *n*<sup>1</sup> × *n*<sup>2</sup> matrices [47]. Hence Algorithm 1 has per-iteration complexity of order *O n*1*n*2*n*3(*n*<sup>1</sup> ∧ *n*<sup>2</sup> + log *n*3) . Thus, if the total iteration number is *T*, then the total computational complexity is:

$$O\left(T n\_1 n\_2 n\_3 \left(\min\{n\_1, n\_2\} + \log n\_3\right)\right). \tag{38}$$

### *4.2. A Faster Algorithm*

To reduce the cost of computing TNN which is a main cost of Algorithm 1, we propose the following lemma which indicates that TNN is orthogonal invariant.

**Lemma 4.** *Given a tensor* <sup>X</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> *, let* <sup>Q</sup> <sup>∈</sup> <sup>R</sup>*n*1×*r*×*n*<sup>3</sup> *a two semi-orthogonal tensors, i.e.,* <sup>Q</sup> <sup>∗</sup> <sup>Q</sup> <sup>=</sup> <sup>I</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> *and r* <sup>≤</sup> min{*n*1, *<sup>n</sup>*2}*. Then, we have the following relationship:*

$$\|\underline{\mathbf{Q}} \* \underline{\mathbf{X}}\|\_{\text{TNN}} = \|\underline{\mathbf{X}}\|\_{\text{TNN}}.$$

The proof of Lemma 4 can be found in the appendix. Equipped with Lemma 4, we decompose the low-rank component in Problem (28) as follows:

$$
\underline{\mathbf{I}} = \underline{\mathbf{Q}} \* \underline{\mathbf{X}} \quad \text{s.t.} \ \underline{\mathbf{Q}} \ \ \* \underline{\mathbf{Q}} = \mathbf{I}\_{\mathbf{u}}.
$$

where <sup>I</sup>*<sup>r</sup>* <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> is an identity tensor. The similar strategy has been used in low-rank matrix recovery from gross corruptions by [50]. Furthermore, we propose the following model for Problem (28):

$$\begin{aligned} \min\_{\underline{\mathbf{Q}}, \underline{\mathbf{X}}, \underline{\mathbf{S}}} & \frac{1}{2} \| \underline{\mathbf{B}} \odot (\underline{\mathbf{Q}} \ast \underline{\mathbf{X}} + \underline{\mathbf{S}} - \underline{\mathbf{M}}) \|\_{\overline{\mathbf{F}}}^2 + \gamma (\| \underline{\mathbf{X}} \|\_{\text{TNN}} + \lambda \| \underline{\mathbf{S}} \|\_{1}) \\ \text{s.t.} & \underline{\mathbf{Q}}^\top \ast \underline{\mathbf{Q}} = \mathbf{I}, \end{aligned} \tag{39}$$

where *r* is an upper estimation of tubal rank of the underlying tensor *r*<sup>∗</sup> = *r*tubal(L0).

In contrast to Model (28), the proposed Model (39) is a non-convex optimization problem. That means Model (39) may have many local minima. We establish a connection between the proposed Model (39) with Model (28) in the following theorem.

**Theorem 3** (Connection between Model (39) and Model (28))**.** *Let* (Q∗, <sup>X</sup>∗, <sup>S</sup>∗) *be a global optimal solution to Problem (39). Furthermore, let* (L-, S-) *be the solution to Problem (28), and rtubal*(L-) ≤ *r, where r is the initialized tubal rank. Then* (Q<sup>∗</sup> <sup>∗</sup> <sup>X</sup>∗, S∗) *is also the optimal solution to Problem (28).*

The proof of Theorem 3 can be found in the appendix. Theorem 3 states that the global optimal point of the (non-convex) Model (39) coincides with solution of the (convex) Model (28). It further indicates that the accuracy of Model (39) cannot exceed Model (28), which can be validated numerically in the experiment section.

To solve Model (39), we also use the ADMM framework.

First, by adding auxiliary variables, we have the following problem:

$$\begin{split} \min\_{\mathbf{L}\in\underline{\mathbf{S}},\underline{\mathbf{Q}},\underline{\mathbf{Q}}} & \stackrel{\mathbf{1}}{\mathbf{2}} \|\underline{\mathbf{B}}\odot(\underline{\mathbf{L}}+\underline{\mathbf{S}}-\underline{\mathbf{M}})\|\_{\mathrm{F}}^{2} + \gamma(\|\underline{\mathbf{Q}}\|\_{\mathrm{TNN}} + \lambda\|\underline{\mathbf{R}}\|\_{1}) \\ \text{s.t.} \ \underline{\mathbf{Q}}\*\underline{\mathbf{A}} = \underline{\mathbf{L}}; \quad \underline{\mathbf{R}} = \underline{\mathbf{S}}; \quad \underline{\mathbf{Q}}^{\top}\*\underline{\mathbf{Q}} = \underline{\mathbf{I}}\_{\mathrm{r}}. \end{split} \tag{40}$$

The augmented Lagrangian of Problem (40) is:

$$\begin{split} L\_{2}'(\underline{\mathbf{L}}, \underline{\mathbf{S}}, \underline{\mathbf{B}}, \underline{\mathbf{Q}}, \underline{\mathbf{Q}}) &= \frac{1}{2} ||\underline{\mathbf{B}} \odot (\underline{\mathbf{L}} + \underline{\mathbf{S}} - \underline{\mathbf{M}})||\_{\mathbb{F}}^{2} + \gamma (||\underline{\mathbf{X}}||\_{\text{TNN}} + \lambda ||\underline{\mathbf{R}}||\_{1}) \\ &+ \langle \underline{\mathbf{X}} \rangle\_{\bullet} \underline{\mathbf{Q}} \* \underline{\mathbf{X}} - \underline{\mathbf{L}} \rangle + \frac{\rho}{2} ||\underline{\mathbf{Q}} \* \underline{\mathbf{X}} - \underline{\mathbf{L}}||\_{\mathbb{F}}^{2} + \langle \underline{\mathbf{Y}} \rangle\_{\bullet} \underline{\mathbf{R}} - \underline{\mathbf{S}} \rangle + \frac{\rho}{2} ||\underline{\mathbf{R}} - \underline{\mathbf{S}}||\_{\mathbb{F}}^{2} \\ \text{s.t.} &\underline{\mathbf{Q}}^{\top} \* \underline{\mathbf{Q}} = \underline{\mathbf{L}} \cdot \underline{\mathbf{U}} \end{split} \tag{41}$$

According the strategy of ADMM, we update prime variables (L, S) and (Q, X, R) by alternative minimization of AL in Problem (41) as follows

• Update (L, S): We update (L, S) by minimizing *L <sup>ρ</sup>* with other variables fixed as follows

$$\begin{split} & (\underline{\mathbf{L}}^{t+1}, \underline{\mathbf{S}}^{t+1}) \\ &= \operatorname{argmin}\_{\left(\underline{\mathbf{L}}, \underline{\mathbf{S}}\right)} \boldsymbol{\mathsf{L}}\_{\boldsymbol{\rho}}^{t} (\underline{\mathbf{L}}, \underline{\mathbf{S}}, \underline{\mathbf{Q}}^{t}, \underline{\mathbf{X}}^{t}, \underline{\mathbf{R}}^{t}, \underline{\mathbf{X}}^{t}, \underline{\mathbf{X}}^{t}) \\ &= \operatorname{argmin}\_{\left(\underline{\mathbf{L}}, \underline{\mathbf{S}}\right)} \frac{1}{2} \|\underline{\mathbf{B}}\odot\left(\underline{\mathbf{L}} + \underline{\mathbf{S}} - \underline{\mathbf{M}}\right)\|\_{F}^{2} + \langle \underline{\mathbf{X}}\_{1}^{t}, \underline{\mathbf{Q}}^{t} \* \underline{\mathbf{X}}^{t} - \underline{\mathbf{L}} \rangle + \frac{\rho}{2} \|\underline{\mathbf{Q}}^{t} \* \underline{\mathbf{A}}^{t} - \underline{\mathbf{L}}\|\_{F}^{2} + \langle \underline{\mathbf{V}}\_{2}^{t}, \underline{\mathbf{R}}^{t} - \underline{\mathbf{S}} \rangle + \frac{\rho}{2} \|\underline{\mathbf{R}}^{t} - \underline{\mathbf{S}}\|\_{F}^{2}. \end{split} \tag{42}$$

Taking derivatives of the right-hand side with respect to L and S respectively, and setting the results zero, we obtain:

$$\begin{aligned} \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{t+1} + \underline{\mathbf{S}}^{t+1}) - \underline{\mathbf{B}} \odot \underline{\mathbf{M}} - \underline{\mathbf{Y}}\_{1}^{t} + \rho (\underline{\mathbf{L}}^{t+1} - \underline{\mathbf{Q}}^{t} \* \underline{\mathbf{X}}^{t}) &= \underline{\mathbf{0}} \\ \underline{\mathbf{B}} \odot (\underline{\mathbf{L}}^{t+1} + \underline{\mathbf{S}}^{t+1}) - \underline{\mathbf{B}} \odot \underline{\mathbf{M}} - \underline{\mathbf{Y}}\_{2}^{t} + \rho (\underline{\mathbf{S}}^{t+1} - \underline{\mathbf{R}}^{t}) &= \underline{\mathbf{0}} \end{aligned} \tag{43}$$

Resolving the above equation group yields:

$$\begin{array}{l} \underline{\mathbf{L}}^{t+1} = \left( (1+\rho)\underline{\mathbf{Q}}^{t} \ast \underline{\mathbf{L}}^{t} + \underline{\mathbf{B}} \odot \underline{\mathbf{M}} + \underline{\mathbf{Y}}\_{1}^{t} - \underline{\mathbf{R}}^{t} \right) \odot (2\underline{\mathbf{B}} + \rho \underline{\mathbf{1}}),\\ \underline{\mathbf{S}}^{t+1} = \left( (1+\rho)\underline{\mathbf{R}}^{t} + \underline{\mathbf{B}} \odot \underline{\mathbf{M}} + \underline{\mathbf{Y}}\_{2}^{t} - \underline{\mathbf{Q}}^{t} \ast \underline{\mathbf{X}}^{t} \right) \odot (2\underline{\mathbf{B}} + \rho \underline{\mathbf{1}}). \end{array} \tag{44}$$

• Update Q. We update Q by minimizing *L <sup>ρ</sup>* with other variables fixed as follows:

$$\begin{split} & \min\_{\underline{\mathsf{Q}}^{\top} \prec \underline{\mathsf{Q}}^{t} = \mathsf{L}} L\_{\rho}(\underline{\mathsf{L}}^{t+1}, \underline{\mathsf{S}}^{t+1}, \underline{\mathsf{Q}} \cdot \underline{\mathsf{X}}^{t}, \underline{\mathsf{R}}^{t}, \underline{\mathsf{Y}}\_{1}^{t}, \underline{\mathsf{Y}}\_{2}^{t}) \\ & = \min\_{\underline{\mathsf{Q}}^{\top} \circ \underline{\mathsf{Q}} = \mathsf{L}} \left\langle \underline{\mathsf{X}}\_{1}^{t}, \underline{\mathsf{Q}} \ast \underline{\mathsf{X}}^{t} - \underline{\mathsf{L}}^{t+1} \right\rangle + \frac{\rho}{2} \| \underline{\mathsf{Q}} \ast \underline{\mathsf{X}}^{t} - \underline{\mathsf{L}}^{t+1} \|\_{\mathrm{F}}^{2} \cdot \\ & = \min\_{\underline{\mathsf{Q}}^{\top} \circ \underline{\mathsf{Q}} = \mathsf{L}} \frac{\rho}{2} \| \underline{\mathsf{Q}} \ast \underline{\mathsf{X}}^{t} - (\underline{\mathsf{L}}^{t+1} - \rho^{-1} \underline{\mathsf{X}}\_{1}^{t}) \|\_{\mathrm{F}}^{2} \\ & = \mathfrak{P} \left( (\underline{\mathsf{L}}^{t+1} - \rho^{-1} \underline{\mathsf{Y}}\_{1}^{t}) \* (\underline{\mathsf{X}}^{t})^{\top} \right), \end{split} \tag{45}$$

where operator P(·) is defined in Lemma 5 as follows.

**Lemma 5** ([51])**.** *Given any tensors* <sup>A</sup> <sup>∈</sup> <sup>R</sup>*r*×*n*2×*n*<sup>3</sup> , <sup>B</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> *, suppose tensor* <sup>B</sup> <sup>∗</sup> <sup>A</sup> *has t-SVD* <sup>B</sup> <sup>∗</sup> <sup>A</sup> <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> *and* <sup>V</sup> <sup>∈</sup> <sup>R</sup>*r*×*r*×*n*<sup>3</sup> *. Then, the problem:*

$$\min\_{\underline{\mathbf{Q}}^{\top} \ast \underline{\mathbf{Q}} = \underline{\mathbf{I}}} \| \underline{\mathbf{P}} \ast \underline{\mathbf{A}} - \underline{\mathbf{B}} \|\_{\mathrm{F}}^{2} \tag{46}$$

*has a closed-form solution as:*

$$
\underline{\mathbf{Q}} = \mathfrak{P}(\underline{\mathbf{B}} \* \underline{\mathbf{A}}^{\top}) := \underline{\mathbf{U}} \* \underline{\mathbf{V}}^{\top}. \tag{47}
$$

• Update (X, R):We update (X, S) by minimizing *L <sup>ρ</sup>* with other variables fixed as follows:

$$\begin{split} & \min\_{\left(\underline{\mathbf{X}},\underline{\mathbf{R}}\right)} L\_{\rho}(\underline{\mathbf{L}}^{t+1},\underline{\mathbf{S}}^{t+1},\underline{\mathbf{Q}}^{t+1},\underline{\mathbf{X}},\underline{\mathbf{R}},\underline{\mathbf{Y}}\_{1}^{t},\underline{\mathbf{Y}}\_{2}^{t}) \\ & = \min\_{\left(\underline{\mathbf{X}},\underline{\mathbf{R}}\right)} \gamma \|\underline{\mathbf{X}}\|\_{\mathrm{TNN}} + \gamma \lambda \|\underline{\mathbf{R}}\|\_{1} + \left\langle \underline{\mathbf{Y}}\_{1}^{t},\underline{\mathbf{Q}}^{t+1} \* \underline{\mathbf{X}} - \underline{\mathbf{L}}^{t+1} \right\rangle + \frac{\rho}{2} \|\underline{\mathbf{Q}}^{t+1} \* \underline{\mathbf{X}} - \underline{\mathbf{L}}^{t+1}\|\_{\mathrm{F}}^{2} \\ & \quad + \left\langle \underline{\mathbf{X}}\_{2}^{t},\underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1} \right\rangle + \frac{\rho}{2} \|\underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1}\|\_{\mathrm{F}}^{2} . \end{split} \tag{48}$$

Please note that Problem (48) can further be solved separately as follows:

$$\begin{split} \underline{\mathbf{K}}^{t+1} &= \operatorname\*{arg\,min}\_{\underline{\mathbf{x}}} \gamma \|\underline{\mathbf{X}}\|\_{\text{TNN}} + \left\langle \underline{\mathbf{x}}\_{1}^{t}, \underline{\mathbf{Q}}^{t+1} \* \underline{\mathbf{X}} - \underline{\mathbf{L}}^{t+1} \right\rangle + \frac{\rho}{2} \|\underline{\mathbf{Q}}^{\top} \* \underline{\mathbf{X}} - \underline{\mathbf{L}}^{t+1} \|\_{\text{F}}^{2} \\ &= \operatorname\*{arg\,min}\_{\underline{\mathbf{x}}} \gamma \|\underline{\mathbf{X}}\|\_{\text{TNN}} + \frac{\rho}{2} \|\underline{\mathbf{Q}}^{t+1} \* \underline{\mathbf{X}} - (\underline{\mathbf{L}}^{t+1} - \rho^{-1} \underline{\mathbf{L}}\_{1}^{t})\|\_{\text{F}}^{2} \\ &\overset{(i)}{=} \operatorname\*{arg\,min}\_{\underline{\mathbf{x}}} \gamma \|\underline{\mathbf{X}}\|\_{\text{TNN}} + \frac{\rho}{2} \|\underline{\mathbf{X}} - (\underline{\mathbf{Q}}^{t+1})^{\top} \* (\underline{\mathbf{L}}^{t+1} - \rho^{-1} \underline{\mathbf{Y}}\_{1}^{t})\|\_{\text{F}}^{2} \\ &= \boldsymbol{\Theta}^{\top}\_{\gamma p^{-1}} \left( (\underline{\mathbf{Q}}^{t+1})^{\top} \* (\underline{\mathbf{L}}^{t+1} - \rho^{-1} \underline{\mathbf{Y}}\_{1}^{t}) . \right) \end{split} \tag{49}$$

and

$$\begin{split} \underline{\mathbf{R}}^{t+1} &= \underset{\underline{\mathbf{R}}}{\operatorname{argmin}} \ \gamma \lambda || \underline{\mathbf{R}} ||\_1 + \left< \underline{\mathbf{Y}}\_1^t, \underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1} \right> + \frac{\rho}{2} || \underline{\mathbf{R}} - \underline{\mathbf{S}}^{t+1} ||\_{\mathbf{F}}^2 \\ &= \underline{\mathbf{S}}^{\parallel \cdot \parallel\_1}\_{\gamma \lambda \rho^{-1}} \left( \underline{\mathbf{K}}^{t+1} - \rho^{-1} \underline{\mathbf{Y}}\_2^t \right) . \end{split} \tag{50}$$

The equality (*i*) in Equation (49) holds because according to Q ∗ Q = I, we have:

$$\begin{split} \min\_{\widetilde{\mathbf{X}}} \|\underline{\mathbf{Q}} \ast \underline{\mathbf{X}} - \underline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} &= \min\_{\widetilde{\mathbf{X}}} \frac{1}{n\_{3}} \|\overline{\mathbf{Q}} \cdot \overline{\mathbf{X}} - \overline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} \\ &= \min\_{\widetilde{\mathbf{X}}} \frac{1}{n\_{3}} \|\overline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} - \frac{2}{n\_{3}} \left< \overline{\mathbf{Q}} \cdot \overline{\mathbf{X}}, \overline{\mathbf{Y}} \right> + \frac{1}{n\_{3}} \|\overline{\mathbf{Q}} \cdot \overline{\mathbf{X}}\|\_{\widetilde{\mathbf{F}}}^{2} \\ &= \min\_{\widetilde{\mathbf{X}}} \frac{1}{n\_{3}} \|\overline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} - \frac{2}{n\_{3}} \left< \overline{\mathbf{X}}, \overline{\mathbf{Q}}^{\mathsf{H}} \overline{\mathbf{Y}} \right> + \frac{1}{n\_{3}} \|\overline{\mathbf{X}}\|\_{\widetilde{\mathbf{F}}}^{2} \\ &= \min\_{\widetilde{\mathbf{X}}} \frac{1}{n\_{3}} \|\overline{\mathbf{X}} - \overline{\mathbf{Q}}^{\mathsf{H}} \overline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} \\ &= \min\_{\underline{\mathbf{X}}} \frac{\rho}{2} \|\underline{\mathbf{X}} - \underline{\mathbf{Q}}^{\mathsf{T}} \* \underline{\mathbf{Y}}\|\_{\widetilde{\mathbf{F}}}^{2} . \end{split} \tag{51}$$

• Update (Y1, Y2). The Lagrangian multipliers are updated by gradient ascent as follows:

$$\begin{aligned} \underline{\mathbf{Y}}\_{1}^{t+1} &= \underline{\mathbf{Y}}\_{1}^{t} + \rho(\underline{\mathbf{Q}}^{t+1} \* \underline{\mathbf{X}}^{t+1} - \underline{\mathbf{L}}^{t+1}), \\ \underline{\mathbf{Y}}\_{2}^{t+1} &= \underline{\mathbf{Y}}\_{2}^{t} + \rho(\underline{\mathbf{R}}^{t+1} - \underline{\mathbf{S}}^{t+1}). \end{aligned} \tag{52}$$

The algorithmic steps are summarized in Algorithm 2. The complexity analysis is given as follows.

In each iteration of Algorithm 2, the update of L requires FFT/IFFT, and *n*<sup>3</sup> multiplications of *n*1-by-*r* and *r*-by-*n*<sup>2</sup> matrices, which costs *O* (*n*1*n*<sup>2</sup> + *rn*<sup>1</sup> + *rn*2)*n*<sup>3</sup> log *n*<sup>3</sup> + *rn*1*n*2*n*<sup>3</sup> ; updating S costs *O n*1*n*2*n*<sup>3</sup> ; updating of Q involves FFT/IFFT and *n*<sup>3</sup> SVDs of *n*1-by-*r* matrices, which costs *O rn*1*n*<sup>3</sup> log *n*<sup>3</sup> + *r*2*n*1*n*<sup>3</sup> ; updating X involves FFT/IFFT and *n*<sup>3</sup> SVDs of *r*-by-*n*2, which costs *O rn*2*n*<sup>3</sup> log *n*<sup>3</sup> + *r*2*n*2*n*3) . Then, the per-iteration computational complexity of Algorithm 2 is dominated by:

$$O\left(\max\left\{n\_1n\_2n\_3\log n\_3, r^2(n\_1+n\_2)n\_3\right\}\right)\dots$$

Since the low-tubal-rank assumption *r* min{*n*1, *n*2} is adopted in this paper, the per-iteration of Algorithm 2 is much lower than Algorithm 1.

### **Algorithm 2** Solving Problem (40) using ADMM.

**Input:** The observed tensor M, an upper estimation *r* of *r*tubal(L0), the parameters *γ*, *λ*, *ρ*, *δ*. 1: Initialize *t* = 0, L0 = S<sup>0</sup> = R<sup>0</sup> = Y0 <sup>1</sup> = <sup>Y</sup><sup>0</sup> <sup>2</sup> <sup>=</sup> <sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*2×*n*<sup>3</sup> , Q0 <sup>=</sup> <sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*1×*r*×*n*<sup>3</sup> , X0 <sup>=</sup> <sup>0</sup> <sup>∈</sup>

R*r*×*n*2×*n*<sup>3</sup> .


(i) convergence of variables: A*t*+<sup>1</sup> <sup>−</sup> <sup>A</sup>*<sup>t</sup>* <sup>∞</sup> ≤ *δ*, ∀A ∈ {L, S, R, Q, X}

(ii) convergence of constraints: max{Q*t*+<sup>1</sup> <sup>∗</sup> <sup>X</sup>*t*+<sup>1</sup> <sup>−</sup> <sup>L</sup>*<sup>t</sup>* ∞, R*t*+<sup>1</sup> <sup>−</sup> <sup>S</sup>*t*+1∞} ≤ *<sup>δ</sup>*.

8: **end for**

**Output:** (Lˆ, Sˆ)=(L*t*<sup>+</sup>1, S*t*+1).

### **5. Experiments**
