Abstract
Matrix and tensor nuclear norms have been successfully used to promote the low-rankness of tensors in low-rank tensor completion. However, singular value decomposition (SVD), which is computationally expensive for large-scale matrices, frequently appears in solving those nuclear norm minimization models. Based on the tensor-tensor product (T-product), in this paper, we first establish the equivalence between the so-called transformed tubal nuclear norm for a third-order tensor and the minimum of the sum of two factor tensors’ squared Frobenius norms under a general invertible linear transform. Gainfully, we introduce a mode-unfolding (often named as “spatio-temporal” in the internet traffic data recovery literature) regularized tensor completion model that is able to efficiently exploit the hidden structures of tensors. Then, we propose an implementable alternating minimization algorithm to solve the underlying optimization model. It is remarkable that our approach does not require any SVDs and all subproblems of our algorithm enjoy closed-form solutions. A series of numerical experiments on traffic data recovery, color images and videos inpainting demonstrate that our SVD-free approach takes less computing time to achieve satisfactory accuracy than some state-of-the-art tensor nuclear norm minimization approaches.














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Notes
Matlab code: http://mp.cs.nthu.edu.tw/
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Acknowledgements
The authors are grateful to the two anonymous referees for their close reading and valuable comments, which helped us improve the quality of this paper greatly.
Funding
This work is supported in part by National Natural Science Foundation of China (Nos. 11771113 and 11971138) and Natural Science Foundation of Zhejiang Province (Nos. LY19A010019 and LD19A010002).
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A Appendix: Convergence Properties of Algorithm 1
A Appendix: Convergence Properties of Algorithm 1
In this appendix, we will show the convergence properties of Algorithm 1. We begin our analysis with introducing the following notations. Throughout this appendix, we use the notation \({\mathbb {E}}:={\mathbb {E}}_1\times {\mathbb {E}}_2\times \cdots \times {\mathbb {E}}_5\) with \({\mathbb {E}}_1={\mathbb {K}}_p^{m\times r}\) , \({\mathbb {E}}_2={\mathbb {K}}_p^{r\times n}\), and \({\mathbb {E}}_i={\mathbb {K}}_p^{m\times n}\) for \(i=3,4,5\), and let \({\mathcal {W}}:=({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {U}},{\mathcal {V}})\in {\mathbb {E}}\) and \(\varPsi ({\mathcal {W}})=\varPsi ({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {U}},{\mathcal {V}})\). In particular, we denote

where ‘\(+{\mathcal {X}}^{q+1}\)’ means using \({\mathcal {X}}^{q+1}\) in place of \({\mathcal {X}}^{q}\) in \(\varPsi \left( {\mathcal {W}}^{q}\right) \) and ‘\(-{\mathcal {X}}^{q}\)’ means taking \({\mathcal {X}}^{q}\) instead of \({\mathcal {X}}^{q+1}\) in \(\varPsi \left( {\mathcal {W}}^{q+1}\right) \).
Definition A.1
A point \({\mathcal {W}}^{\star }=({\mathcal {X}}^{\star },{\mathcal {Y}}^{\star },{\mathcal {Z}}^{\star },{\mathcal {U}}^{\star },{\mathcal {V}}^{\star })\in {\mathbb {E}}\) is a coordinate-wise minimum point of \(\varPsi \), if \({\mathscr {P}}_\varOmega ({\mathcal {Z}}^{\star })={\mathscr {P}}_\varOmega ({\mathcal {G}})\) and
where \({\mathcal {T}}_i({\mathcal {H}})=({\varvec{0}},\ldots ,\underbrace{{\mathcal {H}}}_{\mathrm{the}~i\mathrm{th~tensor}},\ldots ,{\varvec{0}})\) for \(i=1,2,\ldots ,5\), and in addition, \({\mathscr {P}}_\varOmega ({\mathcal {Z}}^{\star }+{\mathcal {H}})={\mathscr {P}}_\varOmega ({\mathcal {G}})\) when \(i=3\).
Consequently, problem (4.4) has at least one minimizer, and for any \(\bar{{\mathcal {W}}}\in {\mathbb {E}}\), all subproblems in (4.6)–(4.10) possess unique minimizer, which implies that Algorithm 1 is well-defined.
Proposition A.1
Let \(\left\{ {\mathcal {W}}^q\right\} \) be the sequence generated by Algorithm 1. Then \(\left\{ {\mathcal {W}}^q\right\} \) is bounded, and any limit point of \(\left\{ {\mathcal {W}}^q\right\} \) is a coordinate-wise minimum of (4.4).
Proof
From the special structure of the objective function \(\varPsi \), we see that \(\varPsi \) has bounded level set, i.e., \(\mathrm{Lev}(\varPsi , \alpha )=\{{\mathcal {W}}\in {\mathbb {E}}~|~\varPsi ({\mathcal {W}})\le \alpha \}\) is bounded for any \(\alpha \in {\mathbb {R}}_+\). Moreover, by Algorithm 1, the sequence of function values \(\{\varPsi \left( {\mathcal {W}}^q\right) \}\) is nonincreasing, which in particular implies that \(\left\{ {\mathcal {W}}^q\right\} \subseteq \mathrm{Lev}\left( \varPsi ,\varPsi ({\mathcal {W}}^0)\right) \). Hence, we know that the sequence \(\left\{ {\mathcal {W}}^q\right\} \) is bounded.
On the other hand, the strong convexity of every subproblem implies that each one has one unique minimizer. By [2, Theorem 14.3], it follows that any limit point of \(\left\{ {\mathcal {W}}^q\right\} \) is a coordinate-wise minimum point of (4.4). \(\square \)
Theorem A.1
Every coordinate-wise minimum point of (4.4) is its a stationary point.
Proof
Let \({\mathcal {W}}^{\star }\) be a coordinate-wise minimum point of (4.4). Then it follows that
where \({\bar{\varPsi }}_1({\mathcal {X}}):=\varPsi ({\mathcal {X}},{\mathcal {Y}}^{\star },{\mathcal {Z}}^{\star },{\mathcal {U}}^{\star },{\mathcal {V}}^{\star })\), which implies \(\nabla {\bar{\varPsi }}_1({\mathcal {X}}^{\star })=0\). Since \(\nabla _{{\mathcal {X}}}{\varPsi }({\mathcal {W}}^{\star })=\nabla {\bar{\varPsi }}_1({\mathcal {X}}^{\star })\), we obtain \(\nabla _{{\mathcal {X}}}{\varPsi }({\mathcal {W}}^{\star })=0\). Similarly, we can prove
Note that
with \(\delta _{{\mathbb {S}}}(\cdot )\) being an indicator function associated to \({\mathbb {S}}=\{{\mathcal {Z}}\in {\mathbb {K}}_p^{m\times n}~|~{\mathscr {P}}_\varOmega ({\mathcal {Z}})={\mathscr {P}}_\varOmega ({\mathcal {G}})\}\), which implies \(-\nabla _{{\mathcal {Z}}}\varPsi ({\mathcal {W}}^{\star })\in \partial \delta _{{\mathbb {S}}}({\mathcal {Z}}^{\star })\). From the definition of the subdifferential \(\partial \delta _{{\mathbb {S}}}(\cdot )\), we have \(\langle -\nabla _{{\mathcal {Z}}}\varPsi ({\mathcal {W}}^{\star }), {\mathcal {Z}}-{\mathcal {Z}}^{\star }\rangle \le 0\) for any \({\mathcal {Z}}\in S\). Consequently, it follows that \( (\nabla _{{\mathcal {Z}}}\varPsi ({\mathcal {W}}^{\star }))_{ijk}=0\) for any \((i,j,k)\not \in \varOmega \), and \({\mathcal {Z}}_{ijk}={\mathcal {G}}_{ijk}\) for any \((i,j,k)\in \varOmega \). Summarizing the above arguments, we know that \({\mathcal {W}}^{\star }\) is a stationary point of (4.4). \(\square \)
We now recall the well-known descent lemma for smooth functions, e.g., see [5].
Lemma A.1
(Descent lemma) Let \(\varphi : {\mathbb {R}}^d\rightarrow {\mathbb {R}}\) be a continuously differentiable function with gradient \(\nabla \varphi \) assumed to be \(\xi \)-Lipschitz continuous. Then,
Proposition A.2
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then, we have

where \({{{\bar{\tau }}}}_1^q={\min }_{1\le k\le p}\left\{ \tau _1^{(q,k)}\sigma ^2_{\min }(L)\right\} \) and \({{{\bar{\tau }}}}_2^q={\min }_{1\le k\le p}\left\{ \tau _2^{(q,k)}\sigma ^2_{\min }(L)\right\} \) with

Proof
We only prove (A.4), and inequality (A.5) can be proved similarly. It follows from the definition of \(\varPsi ({\mathcal {W}})\) in (4.5) and notation \((\varPhi _L({\mathcal {X}}^q))^{(k)}=(\widehat{{\mathcal {X}}}^q)^{(k)}\) that
where \(h^q_k(\cdot )\) is given by (4.13). It is trivial to see that the gradient \(\nabla h^q_k(\cdot )\) of \(h^q_k(\cdot )\) is \(\xi _{1}^{(q,k)}\)-Lipschitz continuous, where \(\xi _{1}^{(q,k)}:=\rho +\sigma ^2_{\max }((\widehat{{\mathcal {Y}}}^{q})^{(k)})\). Then, applying Lemma A.1 to \(h^q_k(\cdot )\) immediately yields
By invoking the updating scheme of \((\widehat{{\mathcal {X}}}^{q+1})^{(k)}\), we have
Consequently, it holds that
which, together with (A.7), implies that
We then conclude from (A.6) and (A.8) that
We obtain the desired results and complete the proof. \(\square \)
Proposition A.3
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then, we have
where \({{\bar{\tau }}}_{3}=(1+\beta _1+\beta _2)/2\).
Proof
Let us divide the tensor \({\mathcal {Z}}\) into two parts, denoted as \({\mathcal {Z}}_{\varOmega }\) and \({\mathcal {Z}}_{\varOmega ^c}\) respectively, where \(\varOmega ^c\) is the complement of \(\varOmega \), and the elements in \({\mathcal {Z}}_{\varOmega }\) are composed of elements in \({\mathcal {G}}_{\varOmega }\), i.e., \({\mathcal {Z}}_{\varOmega }={\mathcal {G}}_{\varOmega }\). Accordingly, denote
It is easy to see that
Moreover, it can be easily seen that \(\nabla f^q(\cdot )\) is Lipschitz continuous with constant \((1+\beta _1+\beta _2)\). By \(-\nabla f^q({\mathcal {Z}}^q_{\varOmega ^c})=(1+\beta _1+\beta _2)\left( {\mathcal {Z}}^{q+1}_{\varOmega ^c}-{\mathcal {Z}}^{q}_{\varOmega ^c}\right) \), an application of Lemma A.1 leads to
which implies that (A.9) holds, since \(\left\| {\mathcal {Z}}_{\varOmega ^c}^{q+1}-{\mathcal {Z}}_{\varOmega ^c}^q\right\| _F^2=\left\| {\mathcal {Z}}^{q+1}-{\mathcal {Z}}^q\right\| _F^2\). The proof is completed. \(\square \)
Proposition A.4
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then, we have

where

Proof
We only prove (A.10). The inequality (A.11) can be proved in a similar way. First, it is easy to see that
where the last equality comes from (4.2) and \(\Vert {\mathcal {A}}\Vert ^2_F=\sum _{k=1}^p\Vert {\mathcal {A}}^{(k)}\Vert _F^2\) for any \({\mathcal {A}}\in {\mathbb {K}}_p^{m\times n}\). Let
By the iterative scheme of \(({\mathcal {U}}^{q+1})^{(k)}\), it is obvious that the gradient of \(g_k^q(\cdot )\) at \(({\mathcal {U}}^q)^{(k)}\) reads as
Consequently, it holds that
By using the Lipschitz continuity of \(\nabla g_k^q(\cdot )\), we have
Applying Lemma A.1 to \(g_k^q(\cdot )\) and combining with (A.13) immediately yields
which, together with (A.12), immediately implies that
We obtain the desired results and complete the proof. \(\square \)
Theorem A.2
(Sufficient decrease property) Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then, the sequence \(\{\varPsi \left( {\mathcal {W}}^q\right) \}\) is nonincreasing. In particular, if there exists a real number \({{\bar{\kappa }}}>0\) such that
then
Proof
By Propositions A.2, A.4 and A.3, we have
which implies that the sequence \(\left\{ \varPsi \left( {\mathcal {W}}^q\right) \right\} \) is nonincreasing. As a consequence, combining the given condition on \({{\bar{\kappa }}}\) and (A.15) immediately leads to
which means the desired result follows.\(\square \)
Proposition A.5
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then there exists \(c>0\) such that for every q, we have
and
where \(c={\min }\left\{ \Vert {\mathcal {W}}^q\Vert _F~|~q=1,2,\ldots \right\} \) and \({\hat{c}}=\sigma ^2_{\max }(L)\max _{1\le k\le p}\{\vartheta _k\}\) is a constant.
Proof
We first prove (6.16). With notation \(\widehat{{\mathcal {A}}}:=\varPhi _L({\mathcal {A}})\) and (4.11), we let
Then we have
By a direct computation, we obtain
where throughout \(L_{k\cdot }\) represents the k-th row of matrix L and
Consequently, it follows from (6.19) that
By the invertible transform L, we have
Especially, when L is a unitary matrix, \(\sigma _{\max }(L)=1\) and the equality holds. Let \(\vartheta _k:=\big \Vert \mathrm{tube}(L_{k\cdot }^* )\big \Vert \) for \(k=1,2,\ldots , p\). Consequently, for \(i=1,2,\ldots ,m\) and \(l=1,2,\ldots ,r\), we have
which implies
Recall the fact that
which implies \(\nabla _{{\mathcal {X}}}\varPsi ({\mathcal {X}}^{q+1}, {\mathcal {Y}}^q,{\mathcal {Z}}^q,{\mathcal {U}}^q,{\mathcal {V}}^q)=0\). Consequently, by (6.18), (6.20), (6.21) and (6.22), it holds that
We obtain the desired result (6.16).
Now we prove (6.17). Similarly, we have
with
which implies
Similar to the optimal point \({\mathcal {X}}^{q+1}\), we also have \(\nabla _{{\mathcal {Y}}}\varPsi ({\mathcal {X}}^{q+1}, {\mathcal {Y}}^{q+1},{\mathcal {Z}}^q,{\mathcal {U}}^q,{\mathcal {V}}^q)=0\). Consequently, by (6.25), we have
which means (6.17) holds. \(\square \)
Proposition A.6
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then for every q, we have
Proof
By the definition of \(\varPsi ({\mathcal {W}})\) and \(f^q\) given in Proposition A.3, we have
Since \(\nabla _{{\mathcal {Z}}_{\varOmega ^c}}\varPsi ({\mathcal {X}}^{q+1},{\mathcal {Y}}^{q+1},{\mathcal {Z}}^{q+1},{\mathcal {U}}^{q},{\mathcal {V}}^{q})=0\), it holds that
We obtain the desired result and complete the proof. \(\square \)
Proposition A.7
Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then for every q, we have
Proof
From the structure of \(\varPsi \), we know that
which implies, together with the fact \(\nabla _{{\mathcal {U}}}\varPsi ({\mathcal {X}}^{q+1},{\mathcal {Y}}^{q+1},{\mathcal {Z}}^{q+1},{\mathcal {U}}^{q+1},{\mathcal {V}}^{q})=0\), that \(\nabla _{{\mathcal {U}}}\varPsi \left( {\mathcal {W}}^{q+1}\right) =0\), i.e., \(\big \Vert \nabla _{{\mathcal {U}}}\varPsi \left( {\mathcal {W}}^{q+1}\right) \big \Vert _F=0\). Clearly, \(\big \Vert \nabla _{{\mathcal {V}}}\varPsi \left( {\mathcal {W}}^{q+1}\right) \big \Vert _F=0\) can be proved similarly. \(\square \)
Theorem A.3
(Relative error property) Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then there exists \(\varrho >0\) such that
holds for every q.
Proof
It is obvious that
It follows from Propositions A.5, A.6, and A.7. \(\square \)
Theorem A.4
(Convergence to a stationary point) Let \(\left\{ {\mathcal {W}}^q\right\} \) be a sequence generated by Algorithm 1. Then the sequence \(\left\{ {\mathcal {W}}^q\right\} \) has a finite length, i.e.,
and hence \(\left\{ {\mathcal {W}}^q\right\} \) is a Cauchy sequence which converges to a stationary point of (4.4).
Proof
It is obvious that \(\varPsi \) satisfies the Kurdyka-Łojasiewicz inequality, since \(\varPsi (\cdot )\) is a semi-algebraic function. Moreover, by Theorems A.2 and A.3, we know that all conditions in [1, Theorem 2.9] are satisfied, and hence the desired result follows. \(\square \)
Theorem A.5
(Local convergence to global minima) For each \(\delta _0>0\), there exist \(\delta \in (0,\delta _0)\) and \(\eta >0\) such that for the starting point \({\mathcal {W}}^0\) satisfying \(\big \Vert {\mathcal {W}}^0-{\mathcal {W}}^{\star }\big \Vert _F<\delta \) and \({\min }\;\varPsi ({\mathcal {W}})<\varPsi ({\mathcal {W}}^0)<{\min }\;\varPsi ({\mathcal {W}})+\eta \), the sequence \(\left\{ {\mathcal {W}}^q\right\} \) generated by Algorithm 1 satisfies
-
(i).
\({\mathcal {W}}^q\in {\mathbb {N}}({\mathcal {W}}^{\star },\delta _0)\) for every q;
-
(ii).
\(\left\{ {\mathcal {W}}^q\right\} \) converges to some \({{\mathcal {W}}^{\infty }}\) and \(\sum _{q=1}^{+\infty }\big \Vert {\mathcal {W}}^{q+1}-{\mathcal {W}}^q\big \Vert _F<+\infty \);
-
(iii).
\(\varPsi ({{\mathcal {W}}^{\infty }})=\varPsi ({\mathcal {W}}^{\star })\).
Proof
It follows from [1, Theorem 2.12]. \(\square \)
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He, H., Ling, C. & Xie, W. Tensor Completion via A Generalized Transformed Tensor T-Product Decomposition Without t-SVD. J Sci Comput 93, 47 (2022). https://doi.org/10.1007/s10915-022-02006-3
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DOI: https://doi.org/10.1007/s10915-022-02006-3