Original: Tensor Wheel Decomposition and Its Tensor Completion Application
Revised: Tensor Wheel Decomposition and Its Tensor Completion Application

Original: Abstract
Revised: Abstract

Original: Recently, tensor network (TN) decompositions have gained prominence in computer vision and contributed promising results to high-order data recovery tasks.
Revised: Recently, tensor network (TN) decompositions have gained prominence in computer vision and contributed promising results to high-order data recovery tasks.

Original: However, current TN models are rather being developed towards more intricatestructures to pursue incremental improvements, which instead leads to a dramaticincrease in rank numbers, thus encountering laborious hyper-parameter selection,especially for higher-order cases.
Revised: However, current TN models are rather being developed towards more intricate structures to pursue incremental improvements, which instead leads to a dramatic increase in rank numbers, thus encountering laborious hyper-parameter selection, especially for higher-order cases.

Original: In this paper, we propose a novel TN decomposition, dubbed tensor wheel (TW) decomposition, in which a high-order tensoris represented by a set of latent factors mapped into a speciﬁc wheel topology.
Revised: In this paper, we propose a novel TN decomposition, dubbed tensor wheel (TW) decomposition, in which a high-order tensor is represented by a set of latent factors mapped into a speciﬁc wheel topology.

Original: Such decomposition is constructed starting from analyzing the graph structure,aiming to more accurately characterize the complex interactions inside objectiveswhile maintaining a lower hyper-parameter scale, theoretically alleviating the abovedeﬁciencies.
Revised: Such decomposition is constructed starting from analyzing the graph structure, aiming to more accurately characterize the complex interactions inside objectives while maintaining a lower hyper-parameter scale, theoretically alleviating the above deﬁciencies.

Original: Furthermore, to investigate the potentiality of TW decomposition, weprovide its one numerical application, i.e., tensor completion (TC), yet developan efﬁcient proximal alternating minimization-based solving algorithm with guaranteed convergence.
Revised: Furthermore, to investigate the potentiality of TW decomposition, we provide its one numerical application, i.e., tensor completion (TC), yet develop an efﬁcient proximal alternating minimization-based solving algorithm with guaranteed convergence.

Original: Experimental results elaborate that the proposed method issigniﬁcantly superior to other tensor decomposition-based state-of-the-art methodson synthetic and real-world data, implying the merits of TW decomposition. 
Revised: Experimental results elaborate that the proposed method is signiﬁcantly superior to other tensor decomposition-based state-of-the-art methods on synthetic and real-world data, implying the merits of TW decomposition. The code is available at: https://github.com/zhongchengwu/code_TWDec .

Original:  Introduction
Revised:  Introduction

Original: Tensors, as a higher-order generalization of matrices, can naturally preserve the complex interactionsinside high-order tensor data, e g., multispectral images (MSIs), color videos (CVs) and hyperspectralvideos (HSVs).
Revised: Tensors, as a higher-order generalization of matrices, can naturally preserve the complex interactions inside high-order tensor data, e.g., multispectral images (MSIs), color videos (CVs), and hyperspectral videos (HSVs).

Original: Tensor decompositions refer to the approximation of high-order tensor data by usinglatent factors, which can be considered as the principal components or features, leading to the reducedstorage cost and computational complexity in large-scale and intractable tensor data analysis.
Revised: Tensor decompositions refer to the approximation of high-order tensor data by using latent factors, which can be considered as the principal components or features, leading to the reduced storage cost and computational complexity in large-scale and intractable tensor data analysis.

Original: Previously, different tensor decompositions have been proposed and widely applied in various ﬁelds, e.g.,signal processing [6, 7, 19], machine learning [25, 29, 1] and medical imaging [31].
Revised: Previously, different tensor decompositions have been proposed and widely applied in various ﬁelds, e.g., signal processing [6, 7, 19], machine learning [25, 29, 1], and medical imaging [31].

Original: Among them,tensor singular value decomposition (t-SVD) [16], especially CANDECOMP/PARAFAC (CP) [13]and Tucker [26] decompositions, are the most classical and widely investigated [17, 34].
Revised: Among them, tensor singular value decomposition (t-SVD) [16], especially CANDECOMP/PARAFAC (CP) [13] and Tucker [26] decompositions, are the most classical and widely investigated [17, 34].

Original: More recently, tensor network (TN) decompositions [27, 9, 14, 11] have consecutively been involvedin computational mathematics and numerical analysis, and exhibit superior properties, such as supercompressing and efﬁcient computing [24, 15].
Revised: More recently, tensor network (TN) decompositions [27, 9, 14, 11] have consecutively been involved in computational mathematics and numerical analysis, and exhibit superior properties, such as super compressing and efﬁcient computing [24, 15].

Original: Particularly, some of TN models have been introducedto the computer vision community as powerful tools for high-order data recovery applications owingto their excellent characterization capabilities. The popular tensor train (TT) decomposition [22] isone such decomposition, in which an N th-order tensor is expressed as strictly ordered multi-linearproducts over one matrix, followed by N -2 third-order tensors and another matrix, as graphicallyillustrated in Figure 1(a). Afterwards, tensor ring (TR) decomposition [35, 23, 21], considered as ageneralized form of TT decomposition, instead expands the near-border two matrices in TT factors to
Revised: Particularly, some of TN models have been introduced

Original: [Figure]
Revised: [Figure]

Original: X
Revised: X

Original: = 

Revised: =

Original: I
Revised: I

Original: N
Revised: N

Original: G
Revised: G

Original: G N
Revised: G N

Original: R
Revised: R

Original: R N G k k
Revised: R N G k k

Original: I
Revised: I

Original: I I I

Revised: I I I

Original: I 

Revised: I

Original: I 4 I k I
Revised: I 4 I k I

Original: N I
Revised: N I

Original: I
 I 

Revised: I
 I

Original: I 
 I
Revised: I 
 I

Original: R R
Revised: R R

Original: R NR
Revised: R NR

Original: R
Revised: R

Original: R
Revised: R

Original: Rk I G
Revised: Rk I G

Original: G 1 G
Revised: G 1 G

Original: G
Revised: G

Original: G N
Revised: G N

Original: G
Revised: G

Original: G k R k
Revised: G k R k

Original: R k+
Revised: R k+

Original: N I
Revised: N I

Original: C
Revised: C

Original: L L
Revised: L L

Original: L L
Revised: L L

Original: L k
Revised: L k

Original: L N
Revised: L N

Original: R k R kX
Revised: R k R kX

Original: = 2 I
Revised: = 2 I

Original: N
Revised: N

Original: G k k
Revised: G k k

Original: I
Revised: I

Original: I I I

Revised: I I I

Original: I 

Revised: I

Original: I 4 I k I
Revised: I 4 I k I

Original: N I
Revised: N I

Original: R R
Revised: R R

Original: R N-1 G
Revised: R N-1 G

Original: R k R kR N
Revised: R k R kR N

Original: X =
Revised: X =

Original: I

Revised: I

Original: I 

Revised: I

Original: I 4 I k I
Revised: I 4 I k I

Original: N I
Revised: N I

Original: G
Revised: G

Original: G N  two third-order tensors and then constructs the cyclically sequential multi-linear products over Nthird-order tensor factors, see Figure 1(b).
Revised: G N to the computer vision community as powerful tools for high-order data recovery applications owing to their excellent characterization capabilities. The popular tensor train (TT) decomposition [22] is one such decomposition, in which an N th-order tensor is expressed as strictly ordered multi-linear products over one matrix, followed by N -2 third-order tensors and another matrix, as graphically illustrated in Figure 1(a). Afterwards, tensor ring (TR) decomposition [35, 23, 21], considered as a generalized form of TT decomposition, instead expands the near-border two matrices in TT factors to two third-order tensors and then constructs the cyclically sequential multi-linear products over N third-order tensor factors, see Figure 1(b).

Original: Currently, a more sophisticated decomposition, referredfully-connected tensor network (FCTN) decomposition [37], has also been explored by establishing astructure evolved from the complete graph in graph theory, where all decomposed N th-order factorsare connected to each other.
Revised: Currently, a more sophisticated decomposition, referred fully-connected tensor network (FCTN) decomposition [37], has also been explored by establishing a structure evolved from the complete graph in graph theory, where all decomposed N th-order factors are connected to each other.

Original: As of now, TT, TR, and FCTN formats have been properly evaluated fortheir effectiveness and successfully applied to a variety of ﬁelds, e.g., classiﬁcation [30], compressivesensing [5], and image/video recovery [8, 10, 36, 3, 28, 37, 12].
Revised: As of now, TT, TR, and FCTN formats have been properly evaluated for their effectiveness and successfully applied to a variety of ﬁelds, e.g., classiﬁcation [30], compressive sensing [5], and image/video recovery [8, 10, 36, 3, 28, 37, 12].

Original: Despite the great success achieved by TT, TR, and FCTN decompositions, three intrinsic limitationsare rather reﬂected in their TN structures.
Revised: Despite the great success achieved by TT, TR, and FCTN decompositions, three intrinsic limitations are rather reﬂected in their TN structures.

Original: More speciﬁcally, ( i ) TT, even TR, models only establishthe connection over adjacent latent factors while ignoring other possible interactions between nonadjacent dimensions, leading to an inadequate relation construction.
Revised: More speciﬁcally, ( i ) TT, even TR, models only establish the connection over adjacent latent factors while ignoring other possible interactions between nonadjacent dimensions, leading to an inadequate relation construction.

Original: ( ii )
Revised: ( ii )

Original: From the perspective oftensor subspace, the intricate features inside an N th-order tensor can be well-preserved in the samedimensional tensor space without destroying the structures.
Revised: From the perspective of tensor subspace, the intricate features inside an N th-order tensor can be well-preserved in the same dimensional tensor space without destroying the structures.

Original: However, when N ≥ 4 , TT and TRmodels express an N th-order tensor employing only third even lower-order factors, which deﬁnitelycannot form the desired space.
Revised: However, when N ≥ 4 , TT and TR models express an N th-order tensor employing only third even lower-order factors, which deﬁnitely cannot form the desired space.

Original: ( iii ) From FCTN-ranks, FCTN model generates ( N 2 − N ) / 2 hyperparameters, which scale quadratically with tensor order N .
Revised: ( iii ) From FCTN-ranks, FCTN model generates ( N 2 − N ) / 2 hyperparameters, which scale quadratically with tensor order N .

Original: Thus, numerically specifying an optimalcollection of FCTN-ranks can be challenging for higher order, making it a costly and tedious task.
Revised: Thus, numerically specifying an optimal collection of FCTN-ranks can be challenging for higher order, making it a costly and tedious task.

Original: Given the above limitations, we argue that a more rational TN model should include three aspects.
Revised: Given the above limitations, we argue that a more rational TN model should include three aspects.

Original: Namely, ( i ) all factors with dimension modes (i.e., I k , k = 1 , 2 , · · · , N ) are certainly interconnectedwith their adjacent ones, while remaining as equivalently close as possible to the others.
Revised: Namely, ( i ) all factors with dimension modes (i.e., I k , k = 1 , 2 , · · · , N ) are certainly interconnected with their adjacent ones, while remaining as equivalently close as possible to the others.

Original: ( ii ) At leastone N th-order factor is required to physically inherit the complex interactions from an N th-ordertensor, thus expecting superior characterization for high-order tensors without sacriﬁcing parameterstorage.
Revised: ( ii ) At least one N th-order factor is required to physically inherit the complex interactions from an N th-order tensor, thus expecting superior characterization for high-order tensors without sacriﬁcing parameter storage.

Original: ( iii )
Revised: ( iii )

Original: The TN structure keeps the scale of hyper-parameters lower, i.e., the smaller collectionof ranks.
Revised: The TN structure keeps the scale of hyper-parameters lower, i.e., the smaller collection of ranks.

Original: Remarkably, the ﬁrst two folds correspond to the characterization capabilities, which arecrucial for TN models in high-order data recovery.
Revised: Remarkably, the ﬁrst two folds correspond to the characterization capabilities, which are crucial for TN models in high-order data recovery.

Original: Consequently, we propose the tensor wheel (TW)decomposition, which decomposes an N th-order tensor into both N fourth-order ring factors and an
Revised: Consequently, we propose the tensor wheel (TW) decomposition, which decomposes an N th-order tensor into both N fourth-order ring factors and an

Original: N th-order core factor, and establishes the multi-linear products by wheel topology (see Figure 1(c)),aiming to agree above superior properties.
Revised: N th-order core factor, and establishes the multi-linear products by wheel topology (see Figure 1(c)), aiming to agree above superior properties.

Original: The main contributions of this paper are as follows:
Revised: The main contributions of this paper are as follows:

Original: • We analytically propose a novel TW decomposition, which allows more expressive characterizationfor correlation than TT and TR formats, especially preserving the linear scaling for the number ofhyper-parameters (TW-ranks) with increased tensor dimension (Section 2.3).
Revised: • We analytically propose a novel TW decomposition, which allows more expressive characterization for correlation than TT and TR formats, especially preserving the linear scaling for the number of hyper-parameters (TW-ranks) with increased tensor dimension (Section 2.3).

Original: Also, we present analternating least squares (ALS)-based algorithm for learning TW decomposition (Section 2.4).
Revised: Also, we present an alternating least squares (ALS)-based algorithm for learning TW decomposition (Section 2.4).

Original: • We further provide one numerical application of TW decomposition, i e., tensor completion(TC), and then formulate a TW decomposition-based TC model, abbreviated TW-TC.
Revised: • We further provide one numerical application of TW decomposition, i.e., tensor completion (TC), and then formulate a TW decomposition-based TC model, abbreviated TW-TC.

Original: Moreover,we develop an efﬁcient proximal alternating minimization (PAM)-based algorithm to solve theproposed TW-TC model and establish the theoretical proof of algorithm convergence (Section 3)
Revised: Moreover, we develop an efﬁcient proximal alternating minimization (PAM)-based algorithm to solve the proposed TW-TC model and establish the theoretical proof of algorithm convergence (Section 3).

Original: Tensor Wheel Decomposition2.1 Notations and Preliminaries

Original: In general, scalars, vectors, matrices, and tensors are denoted by lowercase letters, e.g., x , lowercasebold letters, e.g., x , uppercase bold letters, e.g., X , and calligraphic letters, e.g., X , respectively.

Original: For an N th-order tensor X ∈ R I 1 ×

Original: I 2 ×···× I N , its element on index ( i 1 , i 2 , · · · , i N ) is expressed by

Original: X
Revised: X

Original: I

Revised: I

Original: I k
Revised: I k

Original: I
Revised: I

Original: M I m -based k
Revised: M I m -based k

Original: -unfolding then
Revised: -unfolding then

Original: J2 J k J
Revised: J2 J k J

Original: N J Y = ×
Revised: N J Y = ×

Original: I k m I 1 m
Revised: I k m I 1 m

Original: I M m
Revised: I M m

Original: I k + 1 m transposing n -based k -unfolding
Revised: I k + 1 m transposing n -based k -unfolding

Original: J N n J k +1 n
Revised: J N n J k +1 n

Original: J k n
Revised: J k n

Original: J 1 n
Revised: J 1 n

Original: I M m
Revised: I M m

Original: I k + 1 m
Revised: I k + 1 m

Original: J N n J k +1 n (1,2 M + N - 2k )-based ( M-k )-folding
Revised: J N n J k +1 n (1,2 M + N - 2k )-based ( M-k )-folding

Original: I M m
Revised: I M m

Original: I k +1 m
Revised: I k +1 m

Original: I k m
Revised: I k m

Original: I 1 m
Revised: I 1 m

Original: J N n
Revised: J N n

Original: J k +1 n J k n
Revised: J k +1 n J k n

Original: J 1 n = =
Revised: J 1 n = =

Original: X m →
Revised: X m →

Original: Y n →
Revised: Y n →

Original: X
Revised: X

Original: Y × 1 m k m , ,… 1 n k n , ,… , ,…
Revised: Y × 1 m k m , ,… 1 n k n , ,… , ,…

Original:  X ( i
Revised: In general, scalars, vectors, matrices, and tensors are denoted by lowercase letters, e.g., x , lowercase bold letters, e.g., x , uppercase bold letters, e.g., X , and calligraphic letters, e.g., X , respectively.

Original: [Equation]
Revised: For an N th-order tensor X ∈ R I 1 × I 2 ×···× I N , its element on index ( i 1 , i 2 , · · · , i N ) is expressed by

Original:  , i [Equation] · , i N ) .
Revised: X ( i 1 , i 2 , · · · , i N ) .

Original: The Frobenius norm is deﬁned as [Equation]
Revised: The Frobenius norm is deﬁned as (cid:107)X(cid:107) F = (cid:113)(cid:80) i 1 ,i 2 , ··· ,i N X ( i 1 , i 2 , · · · , i N ) 2 .

Original: For p, q ∈ N + and p ≤ q , we especially denote by X p : q the sequence ( X p , · · · , X q ) for brevity.
Revised: For p, q ∈ N + and p ≤ q , we especially denote by X p : q the sequence ( X p , · · · , X q ) for brevity.

Original: 2.2 Basic Theory
Revised: 2.2 Basic Theory

Original: To numerically construct the TW decomposition, we ﬁrstly develop several new tensor deﬁnitions.
Revised: To numerically construct the TW decomposition, we ﬁrstly develop several new tensor deﬁnitions.

Original: Deﬁnition 1 ( Tensor Permutation ) .
Revised: Deﬁnition 1 ( Tensor Permutation ) .

Original: Given an N th-order tensor X ∈ R I 

Revised: Given an N th-order tensor X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I N .
Revised: I 2 ×···× I N .

Original: Assumingthat vector n = ( n 1 , n 2 , · · · , n N ) is a speciﬁc reordering of vector (1 , 2 , · · · , N ) , then we deﬁne thevector n -based permutation of tensor X as a tensor (cid:126) X n ∈ R I n 1 × I n 2 ×···× I nN , whose elements obey
Revised: Assuming that vector n = ( n 1 , n 2 , · · · , n N ) is a speciﬁc reordering of vector (1 , 2 , · · · , N ) , then we deﬁne the vector n -based permutation of tensor X as a tensor (cid:126) X n ∈ R I n 1 × I n 2 ×···× I nN , whose elements obey

Original: [Equation]
Revised: [Equation]

Original: Deﬁnition 2 ( Generalized Tensor k -Unfolding ) .
Revised: Deﬁnition 2 ( Generalized Tensor k -Unfolding ) .

Original: Let n = ( n 1 , n 2 , · · · , n N ) be a reordering ofvector (1 , 2 , · · · , N ) , then for an N th-order tensor X ∈ R
Revised: Let n = ( n 1 , n 2 , · · · , n N ) be a reordering of vector (1 , 2 , · · · , N ) , then for an N th-order tensor X ∈ R

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 2 ×···× I N , the generalized k-unfolding(0 ≤ k ≤ N, k ∈ Z ) of X is deﬁned as a matrix X [ n ; k ] ∈ R (cid:81)
Revised: I 2 ×···× I N , the generalized k-unfolding (0 ≤ k ≤ N, k ∈ Z ) of X is deﬁned as a matrix X [ n ; k ] ∈ R (cid:81)

Original: ki
Revised: ki

Original: [-=
-] I ni × (cid:81)
Revised: [-=
-][Equation] 1 I ni × [Equation]

Original: Nj =
Revised: Nj =

Original: k +

Revised: k +

Original: I nj , which requires
Revised: I nj , which requires

Original: [Equation]
Revised: [Equation]

Original: where (cid:126) X n is the tensor permutation of X , the multi-indices i n 1 i n 2 · · · i n k and i n k +1 i n k +2 · · · i n N aredeﬁned by 1 + (cid:80) ki
Revised: where (cid:126) X n is the tensor permutation of X , the multi-indices i n 1 i n 2 · · · i n k and i n k +1 i n k +2 · · · i n N are deﬁned by 1 + (cid:80) ki

Original: =1 ( i n i − 1) (cid:81)
Revised: =1 ( i n i − 1) (cid:81)

Original: i − 1 j =

Revised: i − 1 j =

Original: I n j and 1 + (cid:80) Ni
Revised: I n j and 1 + (cid:80) Ni

Original: = k +

Revised: = k +

Original: ( i n i − 1) (cid:81)
Revised: ( i n i − 1) (cid:81)

Original: i − 1 j = k
Revised: i − 1 j = k

Original: +1 I n j , respectively.
Revised: +1 I n j , respectively.

Original: Whenk is 0 and N , x [ n ;0] ∈ R 1 × (cid:81)
Revised: When k is 0 and N , x [ n ;0] ∈ R 1 × (cid:81)

Original: Nj =1 I nj and x [ n ; N ] ∈ R (cid:81)
Revised: Nj =1 I nj and x [ n ; N ] ∈ R (cid:81)

Original: Ni
Revised: Ni

Original: =

Revised: =

Original: I ni × 1 imply two generalized vectorizations.
Revised: I ni × 1 imply two generalized vectorizations.

Original: Conversely, its inverse operator yields X
Revised: Conversely, its inverse operator yields X

Original: = Fold [ n ; k ] ( X [ n ; k ] ) or X = Fold [ n ; k ] ( x [ n ; k ] ) for k = 0 , N .
Revised: = Fold [ n ; k ] ( X [ n ; k ] ) or X = Fold [ n ; k ] ( x [ n ; k ] ) for k = 0 , N .

Original: Following Deﬁnition 2 with associated multi-indices, X [( k, 1 , 2 , ··· ,k − 1 ,k +1 , ··· ,N );1] can be specialized92 as X ( k ) , which is the classical mode-
Revised: Following Deﬁnition 2 with associated multi-indices, X [( k, 1 , 2 , ··· ,k − 1 ,k +1 , ··· ,N );1] can be specialized as X ( k ) , which is the classical mode-

Original: k unfolding in [17].
Revised: k unfolding in [17].

Original: Also, X [( k,k +1 , ··· ,N, 1 , 2 , ··· ,k − 1);1] can be93 specialized as X <k> used in TR decomposition [35].
Revised: Also, X [( k,k +1 , ··· ,N, 1 , 2 , ··· ,k − 1);1] can be specialized as X <k> used in TR decomposition [35].

Original: Correspondingly, the mode- k folding operatorsof these two special unfoldings can be expressed as Fold ( k ) ( · ) and Fold <k> ( · ) , respectively.
Revised: Correspondingly, the mode- k folding operators of these two special unfoldings can be expressed as Fold ( k ) ( · ) and Fold <k> ( · ) , respectively.

Original: Deﬁnition 3 ( Generalized Tensor k -Contraction ) .
Revised: Deﬁnition 3 ( Generalized Tensor k -Contraction ) .

Original: Given an M th-order tensor X ∈ R I 

Revised: Given an M th-order tensor X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I Mand an N th-order tensor Y ∈ R J 1 × J 2 ×···× J N with k common modes ( 1 ≤ k ≤ min ( M, N ) , k ∈ Z ).
Revised: I 2 ×···× I M and an N th-order tensor Y ∈ R J 1 × J 2 ×···× J N with k common modes ( 1 ≤ k ≤ min ( M, N ) , k ∈ Z ).

Original: Assume that two vectors m = ( m 1 , m 2 , · · · , m M ) and n = ( n 1 , n 2 , · · · , n N ) respectively indicatethe reordering of vectors (1 , 2 , · · · , M ) and (1 , 2 , · · · , N ) , satisfying I m i =
Revised: Assume that two vectors m = ( m 1 , m 2 , · · · , m M ) and n = ( n 1 , n 2 , · · · , n N ) respectively indicate the reordering of vectors (1 , 2 , · · · , M ) and (1 , 2 , · · · , N ) , satisfying I m i =

Original: J n i for i = 1 , 2 , · · · , k ,m k
Revised: J n i for i = 1 , 2 , · · · , k , m k

Original: +1 < m k +2 < · · · < m M and n k +1 < n k +2 < · · · < n N .
Revised: +1 < m k +2 < · · · < m M and n k +1 < n k +2 < · · · < n N .

Original: Then the generalized k-contractionbetween X and Y along the k modes speciﬁes an ( M + N − 2 k ) th-order tensor X × m 1 ,m 2 ··· ,m k n 1 ,n 2 , ··· ,n k Y ∈
Revised: Then the generalized k-contraction between X and Y along the k modes speciﬁes an ( M + N − 2 k ) th-order tensor X × m 1 ,m 2 ··· ,m k n 1 ,n 2 , ··· ,n k Y ∈

Original: RI mk
Revised: R

Original: [Equation]
Revised: [Table]

Original: ×···× I mM 
Revised: I mM ×

Original: × J nk +1 ×···× J nN , which is given (also intuitively shown in Figure 2) as follows,
Revised: J nk +1 ×···× J nN , which is given (also intuitively shown in Figure 2) as follows,

Original: [Equation]
Revised: [Equation]

Original: where X [ m ; k ] and Y [ n ; k ] are the m -based and n -based k-unfoldings of tensors X and Y , respectively.
Revised: where X [ m ; k ] and Y [ n ; k ] are the m -based and n -based k-unfoldings of tensors X and Y , respectively.

Original: Signiﬁcantly, the generalized tensor k-contraction operator requires a left-side precedence andnon-commutative property.
Revised: Signiﬁcantly, the generalized tensor k-contraction operator requires a left-side precedence and non-commutative property.

Original: That is, neither the associative nor commutative properties are satisﬁed.
Revised: That is, neither the associative nor commutative properties are satisﬁed.

Original: 2.3 Tensor Wheel Model
Revised: 2.3 Tensor Wheel Model

Original: As illustrated in Figure 1(c), let X ∈ R
Revised: As illustrated in Figure 1(c), let X ∈ R

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 2 ×···× I N be an N th-order tensor, then TW decompositionaims to parameterize it by both N fourth-order ring factors G k ∈ R R k
Revised: I 2 ×···× I N be an N th-order tensor, then TW decomposition aims to parameterize it by both N fourth-order ring factors G k ∈ R R k

Original: ×
Revised: ×

Original: I k
Revised: I k

Original: × L k ×
Revised: × L k ×

Original: R k +1 , k = 1 , 2 , · · · , N ,and an N th-order core factor C ∈ R L 1 × L 2 ×···× L k ×···× L N .
Revised: R k +1 , k = 1 , 2 , · · · , N , and an N th-order core factor C ∈ R L 1 × L 2 ×···× L k ×···× L N .

Original: When k = N , R N +1 stands for R 1 .
Revised: When k = N , R N +1 stands for R 1 .

Original: Mathematically, TW model establishes the element-wise relation as follows,
Revised: Mathematically, TW model establishes the element-wise relation as follows,

Original: [Equation]
Revised: [Equation]

Original: According to the wheel topology, when the tensor contractions are executed in the simplest manner,
Revised: According to the wheel topology, when the tensor contractions are executed in the simplest manner,

Original: TW model (4) can be naturally expressed in the tensor form by X =
Revised: TW model (4) can be naturally expressed in the tensor form by X =

Original: G 1 × 41 G 2 × 61 · · · × 2 k 1 G k × 2 k +2 1· · · × 2 N, 1 1 , 4 G N × 2 , 4 , ··· , 2 N 1 , 2 , ··· ,N C , which provides the actual numerical computation for TW decomposition.
Revised: G 1 × 41 G 2 × 61 · · · × [Equation] G k × 2 k +2· · · × 2 N, 1 1 , 4 G N × 2 , 4 , ··· , 2 N 1 , 2 , ··· ,N C , which provides the actual numerical computation for TW decomposition.

Original: Symbolically, we employ TW (cid:74) G 1 , G 2 , · · · , G N ; C (cid:75) or, more compactly, TW (cid:74) {G k } Nk =1 ; C (cid:75) to denote114 TW decomposition.
Revised: Symbolically, we employ TW (cid:74) G 1 , G 2 , · · · , G N ; C (cid:75) or, more compactly, TW (cid:74) {G k } Nk =1 ; C (cid:75) to denote TW decomposition.

Original: Moreover, the ring factors G k , k = 1 , 2 , · · · , N , and core factor C are collectively called TW factors .
Revised: Moreover, the ring factors G k , k = 1 , 2 , · · · , N , and core factor C are collectively called TW factors .

Original: Apart from the above basic form, TW decomposition also has numerousgeneralized tensor forms, but enumerating all the cases is laborious and pointless.
Revised: Apart from the above basic form, TW decomposition also has numerous generalized tensor forms, but enumerating all the cases is laborious and pointless.

Original: In what follows,we exhibit two special tensor forms by Theorems 1 and 2, aiming to illustrate its invariance property.
Revised: In what follows, we exhibit two special tensor forms by Theorems 1 and 2, aiming to illustrate its invariance property.

Original: Theorem 1 ( Core-Centered Circular Invariance ) .
Revised: Theorem 1 ( Core-Centered Circular Invariance ) .

Original: Given an N th-order tensor X ∈ R I 

Revised: Given an N th-order tensor X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I Nand its TW decomposition TW (cid:74) {G k } Nk =1 ; C (cid:75) .
Revised: I 2 ×···× I N and its TW decomposition TW (cid:74) {G k } Nk =1 ; C (cid:75) .

Original: Assume that n = ( n 1 , n 2 , · · · , n N ) is the circular reordering of vector (1 , 2 , · · · , N ) , then the core-centered invariance gives (cid:126) X n = TW (cid:74) {G n k } Nk =1 ; (cid:126) C n (cid:75) .
Revised: Assume that n = ( n 1 , n 2 , · · · , n N ) is the circular reordering of vector (1 , 2 , · · · , N ) , then the core-centered invariance gives (cid:126) X n = TW (cid:74) {G n k } Nk =1 ; (cid:126) C n (cid:75) .

Original: Theorem 2 ( Core-Connected Invariance ) .
Revised: Theorem 2 ( Core-Connected Invariance ) .

Original: Assume that the TW decomposition of X ∈ R I 1 ×···×I Nis (cid:126) X n = TW (cid:74) {G n k } Nk =
Revised: Assume that the TW decomposition of X ∈ R I 1 ×···×

Original: 1 ; (cid:126) C n (cid:75) , where n = ( n 1 , n 2 , · · · , n N ) is any vector that circularly shifts vector123 (1 , 2 , · · · , N ) .
Revised: I N is (cid:126) X n = TW (cid:74) {G n k } Nk =1 ; (cid:126) C n (cid:75) , where n = ( n 1 , n 2 , · · · , n N ) is any vector that circularly shifts vector (1 , 2 , · · · , N ) .

Original: Let vector e
Revised: Let vector e

Original: =
Revised: =

Original: ( n 1 , n k , n 2 , · · · , n k − 1 , n k +1 , · · · , n N ) (3 ≤ k < N, k ∈ Z ) , then(cid:126)
Revised: ( n 1 , n k , n 2 , · · · , n k − 1 , n k +1 , · · · , n N ) (3 ≤ k < N, k ∈ Z ) , then (cid:126)

Original: X e =
Revised: X e =

Original: ( G n 1 × 
Revised: ( G n 1 × 31 (cid:126) C e × 43 G n k ) × 3 , 4 , ··· ,k + [Equation] ,N +

Original: [Equation]
Revised: [Equation]

Original: e × 43 G n k ) × 3 , 4 , ··· ,k +1 ,N +2 1 , 3 , ··· , 2 k − 3 , 2 k − 2 U n 2 , ··· ,n k − 1 × 1 , 3 , 4 , ··· ,N − k +2 ,N − k +4 2( N − k )+2 , 3 , 5 , ··· , 2( N − k )+1 , 1 V n k +1 , ··· ,n N , where U n 2 , ··· ,n k − 1 =
Revised: [Equation] [Table] ··· , 2 k − 3 , [Table]

Original: G n 2 × 4 1 · · · × 2( k − 2)
Revised: k − 2 U n [Table] ,n [Table]

Original: G n k − 1 and V n k +1 , ··· ,n N = G n k +1 × 4 1 · · · × 
Revised: × [Table] N

Original: [Equation]
Revised: [Table]

Original: ( N − k )
Revised: − k )+1 , 1 V n k +1 , ··· ,n N , where U n 2 , ··· ,n k − 1 = G n 2 × 41 · · · × 2( k − 2) G n k − 1 and V n k +1 , ··· ,n N = G n k +1 × 41 · · · × 2( N − k )

Original: G n N .
Revised: G n N .

Original: Theorem 2 reveals any two non-adjacent ring factors with dimension modes (i.e., I k , k = 1 , 2 , · · · , N )can be connected by the core factor.
Revised: Theorem 2 reveals any two non-adjacent ring factors with dimension modes (i.e., I k , k = 1 , 2 , · · · , N ) can be connected by the core factor.

Original: Accordingly, TW topology can comprehensively establish allpossible mode interactions of a high-order tensor without being impeded by other ring factors, whichendows TW decomposition with adequate relation construction over TT and TR decompositions.
Revised: Accordingly, TW topology can comprehensively establish all possible mode interactions of a high-order tensor without being impeded by other ring factors, which endows TW decomposition with adequate relation construction over TT and TR decompositions.

Original: Theorem 3 ( Tensor Subwheel Equation ) .
Revised: Theorem 3 ( Tensor Subwheel Equation ) .

Original: Assume that the TW decomposition of X ∈ R I 

Revised: Assume that the TW decomposition of X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I Nis (cid:126) X n = TW (cid:74) {G n k } Nk =1 ; (cid:126) C n (cid:75) , where n = ( n 1 , n 2 , · · · , n N ) is any vector that circularly shifts vector131 (1 , 2 , · · · , N ) .
Revised: I 2 ×···× I N is (cid:126) X n = TW (cid:74) {G n k } Nk =1 ; (cid:126) C n (cid:75) , where n = ( n 1 , n 2 , · · · , n N ) is any vector that circularly shifts vector (1 , 2 , · · · , N ) .

Original: Let m =
Revised: Let m =

Original: ( N + 1 , N + 2 , 1 , 2 , · · · , N ) and v = (2 , 4 , · · · , 2 N, 1 , 3 , · · · , 2 N − 1) ,then there inherently exists the following two tensor subwheel equations,
Revised: ( N + 1 , N + 2 , 1 , 2 , · · · , N ) and v = (2 , 4 , · · · , 2 N, 1 , 3 , · · · , 2 N − 1) , then there inherently exists the following two tensor subwheel equations,

Original: [Equation]
Revised: [Equation]

Original: and(cid:126)
Revised: and

Original: [Equation]
Revised: [Equation]

Original: [1: N, [Equation] (cid:126) [Equation]
Revised: where

Original: n [1: N, 0] ( N (cid:54) = C ) [ v ; N ] , (5) where M (cid:54) = n N ∈ R R n 1 × I n 1 ×···× I nN − 1 × R nN ×
Revised: M (cid:54) = n N ∈ R R n 1 × I n 1 ×···× I nN − 1 × R nN ×

Original: L nN is an ( N + 2) th-order subwheel tensor, whichmerges all TW factors but G n N , i.e., M (cid:54) =n N [Equation] G
Revised: L nN is an ( N + 2) th-order subwheel tensor, which merges all TW factors but G n N , i.e., M (cid:54) =

Original: [Equation] × 3 , 5 , [Equation] [Equation]{+41 · · · × 2 k 1 G n k × 2 k +2· · · × 2 N − 
+}
Revised: n N = G n 1 × {+41 · · · × 2 k 1 G n k × 2 k +2· · · × 2 N − 
+}

Original:  N − [Equation] [Equation] ,N − 1 (cid:126) C n , and N (cid:54) =
Revised: G n N − 1 × 3 , 5 , ··· , 2 N − 1 1 , 2 , ··· ,N − 1 (cid:126) C n , and N (cid:54) =

Original: C ∈ R I n 1 × L n [Equation] I nN ×
Revised: C ∈ R I n 1 × L n 1 ×···× I nN ×

Original: L nN is another 2 N th-order subwheeltensor obtained by only merging {G n k } Nk =1 , i.e.
Revised: L nN is another 2 N th-order subwheel tensor obtained by only merging {G n k } Nk =1 , i.e.

Original: , N (cid:54) =
Revised: , N (cid:54) =

Original: C =
Revised: C =

Original: G n 1 × 41 · [Equation] × [Equation]
Revised: G n 1 × 41 · · · × 2 k 1 G n k × 2 k +2· · · × 2 N, 1 1 , 4 G n N .

Original: G

Original: [Equation]

Original: k ×

Original: [Equation]

Original: +

Original: [Equation]

Original: [Equation]

Original: N [Equation] 1 1 , 4 G n N . Theorem 3 clariﬁes that all TW factors, i.e., {G k } Nk =1 and C , can be individually separated from
Revised: Theorem 3 clariﬁes that all TW factors, i.e., {G k } Nk =1 and C , can be individually separated from

Original: TW (cid:74) {G k } Nk =1 ; C (cid:75) , which contributes to those alternating direction-based algorithms, e.g., ALS and139 PAM ones in Sections 2.4 and 3.1, respectively.
Revised: TW (cid:74) {G k } Nk =1 ; C (cid:75) , which contributes to those alternating direction-based algorithms, e.g., ALS and PAM ones in Sections 2.4 and 3.1, respectively.

Original: Furthermore, (cid:126) x n [
Revised: Furthermore, (cid:126) x n [1: N, 0]

Original: [Equation]
Revised: = (cid:126) c

Original:  0] ( N (cid:54) =
Revised: n [1: N, 0] ( N (cid:54) =

Original: C ) [ v ; N ] informula (5) is essentially (cid:126)
Revised: C ) [ v ; N ] in formula (5) is essentially (cid:126)

Original: X n = (cid:126) C n × 1 , 2 , ··· ,N 2 , 4 , ··· , 2 N N (cid:54) =
Revised: X n = (cid:126) C n × 1 , 2 , ··· ,N 2 , 4 , ··· , 2 N N (cid:54) =

Original: C , appearing as a tensor subspace representation.
Revised: C , appearing as a tensor subspace representation.

Original: Since all modes of (cid:126) X n are contained in N (cid:54) =
Revised: Since all modes of (cid:126) X n are contained in N (cid:54) =

Original: C , (cid:126) C n is purely a coefﬁcient tensor, and N (cid:54) =
Revised: C , (cid:126) C n is purely a coefﬁcient tensor, and N (cid:54) =

Original: C can beregarded as a basis tensor.
Revised: C can be regarded as a basis tensor.

Original: From such a perspective, the intrinsic structure of a high-order tensor can bewell-preserved by the core factor C , leading to a high characterization ability of TW decomposition.
Revised: From such a perspective, the intrinsic structure of a high-order tensor can be well-preserved by the core factor C , leading to a high characterization ability of TW decomposition.

Original: Theorem 4.
Revised: Theorem 4.

Original: Assume that X = TW (cid:74) {G k } Nk =1 ; C (cid:75) with N ring factors G k ∈ R R k
Revised: Assume that X = TW (cid:74) {G k } Nk =1 ; C (cid:75) with N ring factors G k ∈ R R k

Original: ×
Revised: ×

Original: I k
Revised: I k

Original: × L k ×
Revised: × L k ×

Original: R k +1 , then
Revised: R k +1 , then

Revised: Rank ( X ( k ) )

Revised: =

Revised: Rank

Revised: [Equation]

Revised: X

Original: [Equation]
Revised: [Equation]

Revised: k

Revised: [Equation]

Revised: Obtain the subwheel tensor M (cid:54) and vector m by Theorem 3.

Original:  Theorem 4 justiﬁes that scalars R k and L k , k = 1 , 2 , · · · , N , can upper bound the rank of alldimension-mode tensor unfolding and control the low-rankness behavior to a certain extent.
Revised: Obtain subwheel Check the convergence criterion: 10: end while Theorem 4 justiﬁes that scalars R k and L k , k = 1 , 2 , · · · , N , can upper bound the rank of all dimension-mode tensor unfolding and control the low-rankness behavior to a certain extent.

Original: Thus, theyare assigned as the TW-ranks , usually collected by a vector r =
Revised: Thus, they are assigned as the TW-ranks , usually collected by a vector r =

Original: ( R 1 , · · · , R N , L 1 , · · · , L N ) ∈ R 2 N .
Revised: ( R 1 , · · · , R N , L 1 , · · · , L N ) ∈ R 2 N .

Original: The TW-ranks essentially determines the actual TN structure, e.g., Tucker topology when R k = 1and TR topology when L k = 1 with k = 1 , 2 , · · · , N , graphically proving the superiority of TWdecomposition over both Tucker and TR decompositions.
Revised: The TW-ranks essentially determines the actual TN structure, e.g., Tucker topology when R k =and TR topology when L k = 1 with k = 1 , 2 , · · · , N , graphically proving the superiority of TW decomposition over both Tucker and TR decompositions.

Original: Along such a perspective, TT, TR, and
Revised: Along such a perspective, TT, TR, and

Original: FCTN decompositions can be viewed as a special case of the adaptive TN (ATN) decomposition [11],which adaptively identiﬁes TN structures by eliminating the relevant edges from the complete graphtopology.
Revised: FCTN decompositions can be viewed as a special case of the adaptive TN (ATN) decomposition [11], which adaptively identiﬁes TN structures by eliminating the relevant edges from the complete graph topology.

Original: Nevertheless, provided that decomposing the same N th-order tensor, TW decomposition is exactly excluded from the setting of ATN decomposition since the latter cannot ﬁts the structureswith an internal core tensor, e.g., Tucker [26], hierarchical Tucker [9], and projected entangled statepairs (PEPS) [27] topologies.
Revised: Nevertheless, provided that decomposing the same N th-order tensor, TW decomposition is exactly excluded from the setting of ATN decomposition since the latter cannot ﬁt the structures with an internal core tensor, e.g., Tucker [26], hierarchical Tucker [9], and projected entangled state pairs (PEPS) [27] topologies.

Original: Compared with FCTN decomposition with ( N 2 − N ) / 2 ranks, TWdecomposition forms the collection of 2 N ranks, which grows only linearly rather than quadraticallywith the tensor order N , thereby improving its ﬂexibility for higher-order applications.
Revised: Compared with FCTN decomposition with ( N 2 − N ) / 2 ranks, TW decomposition forms the collection of 2 N ranks, which grows only linearly rather than quadratically with the tensor order N , thereby improving its ﬂexibility for higher-order applications.

Original: Limitation.
Revised: Limitation.

Original: The proposed TW topology allows for more expressive characterization capability usingthe linear number of hyper-parameters.
Revised: The proposed TW topology allows for more expressive characterization capability using the linear number of hyper-parameters.

Original: Nonetheless, such a structure inevitably leads to a limitationon the storage costs.
Revised: Nonetheless, such a structure inevitably leads to a limitation on the storage costs.

Original: For a given N th-order tensor X ∈ R I ×···× I , whose TW-ranks is assumed as thesame value R , i.e., r =
Revised: For a given N th-order tensor X ∈ R I ×···× I , whose TW-ranks is assumed as the same value R , i.e., r =

Original: ( R, · · · , R ) , then TW decomposition requires O ( NIR 3 + R N ) parametercosts for storage.
Revised: ( R, · · · , R ) , then TW decomposition requires O ( NIR 3 + R N ) parameter costs for storage.

Original: It is clear that the O ( NIR 3 + R N ) scales exponentially with the tensor order N ,resulting in a greater storage burden for larger-scale and higher-order tensor recovery problems.
Revised: It is clear that the O ( NIR 3 + R N ) scales exponentially with the tensor order N , resulting in a greater storage burden for larger-scale and higher-order tensor recovery problems.

Original: Unlike Tucker decomposition, TW one considers the potential relationship between adjacent factorsand establishes a connection for a higher characterization capacity.
Revised: Unlike Tucker decomposition, TW one considers the potential relationship between adjacent factors and establishes a connection for a higher characterization capacity.

Original: Actually, such a strategy alsoreduces the loadings of core factor, which contributes a smaller { L i } Ni =1 in TW-ranks than Tuckerdecomposition, thus alleviating the limitations of high storage and computational complexity.
Revised: Actually, such a strategy also reduces the loadings of core factor, which contributes a smaller { L i } Ni =1 in TW-ranks than Tucker decomposition, thus alleviating the limitations of high storage and computational complexity.

Original: 2.4 Alternating Least Squares (ALS) Algorithm
Revised: 2.4 Alternating Least Squares (ALS) Algorithm

Original: In this section, we develop an ALS-based algorithm to perform TW decomposition, called TW-ALS.
Revised: In this section, we develop an ALS-based algorithm to perform TW decomposition, called TW-ALS.

Original: Given X ∈ R I 1 ×
Revised: Given X ∈ R I 1 ×

Original: I 2 ×···× I N and its predeﬁned TW-ranks, then the TW-ALS algorithm mainly aims toseek a set of TW factors such that a relative error between X and TW (cid:74) {G k } Nk =1 ; C (cid:75) is minimum, i.e.,
Revised: I 2 ×···× I N and its predeﬁned TW-ranks, then the TW-ALS algorithm mainly aims to seek a set of TW factors such that a relative error between X and TW (cid:74) {G k } Nk =1 ; C (cid:75) is minimum, i.e.,

Original: [Equation]
Revised: [Equation]

Original:  Based on Theorem 3, minimization problem (7) can be optimized under the ALS framework by alternatively solving N + 1 least-squares subproblems, i.e., min ( G k ) (2) (cid:107) X <k> − ( G k ) (2) ( M (cid:54) = k ) [ m ;3] (cid:107) F ,k = 1 , 2 , · · · , N , and min c [1: N, 0] (cid:107) x [1: N, 0] − c [1: N, 0] ( N (cid:54) = C )
Revised: G N [Equation] Based on Theorem 3, minimization problem (7) can be optimized under the ALS framework by alternatively solving N + 1 least-squares subproblems, i.e., min ( G k ) (2) (cid:107) X <k> − ( G k ) (2) ( M (cid:54) = k ) [ m ;3] (cid:107) F , k = 1 , 2 , · · · , N , and min c [1: N, 0] (cid:107) x [1: N, 0] − c [1: N, 0] ( N (cid:54) = C )

Original: [ v ; N ] (cid:107) F .
Revised: [ v ; N ] (cid:107) F .

Original: Compared to formula (5), sincewe utilize the simplest case without permutation for updating C , x [1: N, 0] and c [1: N, 0] imply (cid:126) x n [1: N, 0]and (cid:126) c n [1: N, 0] with n = (1 , 2 , · · · , N ) , respectively.
Revised: Compared to formula (5), since we utilize the simplest case without permutation for updating C , x [1: N, 0] and c [1: N, 0] imply (cid:126) x n [1: N, 0] and (cid:126) c n [1: N, 0] with n = (1 , 2 , · · · , N ) , respectively.

Original: The numerical procedure is detailed in Algorithm 1. 
Revised: The numerical procedure is detailed in Algorithm 1. [Equation] [Equation] [Equation]

Original: Numerical Application to Tensor Completion
Revised: [Equation] [Equation] [Equation] [Equation]

Original: To validate the rationality and superiority of TW decomposition, we employ it to one classical tensorrecovery problem, i.e., tensor completion (TC), whose objective is recovering the missing entriesfrom a partially observed tensor, then formulating a TW decomposition-based TC (TW-TC) model.
Revised: [Equation] [Equation] [Equation] [Equation]

Original: The proposed TW-TC method is committed to exploring latent TW factors from the partially observedtensor, then predicting the missing entries using these constructed factors.
Revised: [Equation] [Equation] [Equation] [Equation]

Original: This essentially evaluatesthe potential capability of TW decomposition to characterize the high-order tensors.
Revised: [Equation] [Equation] [Equation] [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: 9: end if 10:
Revised: 9: end if 10:

Original: [Equation]
Revised: [Equation]

Original: Check the convergence criterion: (cid:107)X ( t +1) − X (
Revised: Check the convergence criterion: (cid:107)X ( t +

Original: [Equation]
Revised: [Equation]

Original: F
Revised: − X ( [Equation] (cid:107)

Original: [Equation]
Revised: [Equation]

Original: F <
Revised: / (cid:107)X ( t

Original: [Equation]
Revised: [Equation]

Revised: F < (cid:15) .

Original: 12: t ← t + 1 .
Revised: 12: t ← t + 1 .

Original: 13: end while
Revised: 13: end while

Original: Output: The recovered tensor X ∈ R I 

Revised: Output: The recovered tensor X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I N .
Revised: I 2 ×···× I N .

Revised:  Numerical Application to Tensor Completion

Revised: To validate the rationality and superiority of TW decomposition, we employ it to one classical tensor recovery problem, i.e., tensor completion (TC), whose objective is recovering the missing entries from a partially observed tensor, then formulating a TW decomposition-based TC (TW-TC) model.

Revised: The proposed TW-TC method is committed to exploring latent TW factors from the partially observed tensor, then predicting the missing entries using these constructed factors.

Revised: This essentially evaluates the potential capability of TW decomposition to characterize the high-order tensors.

Original: 3.1 Model and Algorithm
Revised: 3.1 Model and Algorithm

Original: Given a partially observed tensor F ∈ R I 

Revised: Given a partially observed tensor F ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I N at location set Ω , then the proposed TW-TCmodel aims to identify an optimum-TW-ranks approximation X ∈ R I 

Revised: I 2 ×···× I N at location set Ω , then the proposed TW-TC model aims to identify an optimum-TW-ranks approximation X ∈ R I

Original: ×
Revised: ×

Original: I 2 ×···× I N of the tensor F .
Revised: I 2 ×···× I N of the tensor F .

Original: Mathematically, the TW-TC model can be formulated as the unconstrained problem as follows,min X , G 1: N , C
Revised: Mathematically, the TW-TC model can be formulated as the unconstrained problem as follows, [Equation] ι

Original: [Equation]
Revised: [Equation]

Original: where P Ω ( · ) is a projection function keeping the entries in Ω while forcing all the others to zeros.
Revised: where P Ω ( · ) is a projection function keeping the entries in Ω while forcing all the others to zeros.

Original: Obviously, the object function in formula (8) is not jointly but independently convex for all variables.
Revised: Obviously, the object function in formula (8) is not jointly but independently convex for all variables.

Original: Thus, we develop a proximal alternating minimization (PAM) [2] based algorithm to alternately andrecursively optimize each variable, leading to the following procedure for k = 1 , 2 , · · · , N ,
Revised: Thus [Equation] we develop a proximal alternating minimization (PAM) [2] based algorithm to alternately and recursively optimize each variable, leading to the following procedure for k = 1 , 2 , · · · , N ,

Original: [Equation]
Revised: [Equation]

Original: where ρ > 0 is a proximal parameter.
Revised: where ρ > 0 is a proximal parameter.

Original: Following Theorem 3 again, each univariate minimizationproblem in formula (9) can certainly be reduced to a linear least-squares problem, which is expressedby matrix form in the supplementary material, as in Section 2.4.
Revised: Following Theorem 3 again, each univariate minimization problem in formula (9) can certainly be reduced to a linear least-squares problem, which is expressed by matrix form in the supplementary material, as in Section 2.4.

Original: Then, the corresponding closed-formsolutions can easily be given.
Revised: Then, the corresponding closed-form solutions can easily be given.

Original: In most numerical TC experiments, the variable C typically tends to bestable when it exceeds 200 iterations.
Revised: In most numerical TC experiments, the variable C typically tends to be [Figure] [Figure]

Original:  Therefore, we utilize the modulo operation (marked mod ( · , · ) )with a step size to intermittently update C after 200 iterations, aiming to save computing resources.
Revised: [Figure] stable when it exceeds 200 iterations. Therefore, we utilize the modulo operation (marked mod ( · , · ) ) with a step size to intermittently update C after 200 iterations, aiming to save computing resources.

Original: The PAM-based solving procedure for the developed TW-TC model is summarized in Algorithm 2.
Revised: The PAM-based solving procedure for the developed TW-TC model is summarized in Algorithm 2.

Original: 3.2 Computational Complexity Analysis
Revised: 3.2 Computational Complexity Analysis

Original: For simplicity, we assume that all dimensions of an N th-order tensor X ∈ R I 1 ×···× I k ×···× I N are Iand all values of TW-ranks are R , i.e.,
Revised: For simplicity, we assume that all dimensions of an N th-order tensor X ∈ R I 1 ×···× I k ×···× I N are I and all values of TW-ranks are R , i.e.,

Original: I k
Revised: I k

Original: = I and R k
Revised: = I and R k

Original: =
Revised: =

Original: L k
Revised: L k

Original: = R for k
Revised: = R for k

Original: = 1 , 2 , · · · , N .
Revised: = 1 , 2 , · · · , N .

Original: Then, the computational complexities involved in updating factors {G k } Nk =1 , C , and X are O
Revised: Then, the computational complexities involved in updating factors {G k } Nk =1 , C , and X are O

Original: ( N (cid:80) N − 1 l =2 I l R l +3 +
Revised: ( N (cid:80) N − 1 l =2 I l R l +3 +

Original: NI N R 3 + NR 9 ) , O
Revised: NI N R 3 + NR 9 ) , O

Original: ( I N R 2 N ) , and O ( I N R N ) , respectively.
Revised: ( I N R 2 N ) , and O ( I N R N ) , respectively.

Original: Therefore, the computational complexity for each nonintermittent iteration (i.e., t ∈ { t : 0 ≤ t ≤ 200 or mod ( t, s )
Revised: Therefore, the computational complexity for each nonintermittent iteration (i.e., t ∈ { t : 0 ≤ t ≤ 200 or mod ( t, s )

Original: = 0 } ) in [-Algorithm 
 [Figure] [Figure] [Figure]-]
Revised: = 0 } ) in [-Algorithm 
 [Figure] [Figure] [Figure]-]Algorithmis O

Original: The smaller Log 10 (RSE) value implies higher recovery accuracy.

Original: is O

Original: ( N (cid:80) N − 1 l =

Revised: ( N (cid:80) N − 1 l =

Original: I l R l +3 + NI N R 

Revised: I l R l +3 + NI N R

Original: + I N R 2 N ) .
Revised: + I N R 2 N ) .

Original: Akin to the storage parameters in the Limitationof Section 2.3, the computational complexity also increases exponentially with dimensionality N ,implying that more computing resources are required for higher-order tensor recovery tasks.
Revised: Akin to the storage parameters in the Limitation of Section 2.3, the computational complexity also increases exponentially with dimensionality N , implying that more computing resources are required for higher-order tensor recovery tasks.

Original: 3.3 Convergence Analysis
Revised: 3.3 Convergence Analysis

Original: This section consists in theoretically proving the convergence of Algorithm 2.
Revised: This section consists in theoretically proving the convergence of Algorithm 2.

Original: Before that, we employthe Z =
Revised: Before that, we employ the Z =

Original: ( X , G 1: N , C ) and Φ( Z ) to denote the variables and objective function in (8), respectively.
Revised: ( X , G 1: N , C ) and Φ( Z ) to denote the variables and objective function in (8), respectively.

Original: Theorem 5.
Revised: Theorem 5.

Original: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2, then it globally converges toa critical point (i.e., local minimum point)
Revised: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2, then it globally converges to a critical point (i.e., local minimum point)

Original: Z ∗ =
Revised: Z ∗ =

Original: ( X ∗ , G ∗ 1: N , C ∗ ) of the objective function Φ( Z ) .
Revised: ( X ∗ , G ∗ 1: N , C ∗ ) of the objective function Φ( Z ) .

Original: Proof.
Revised: Proof.

Original: According to the ﬁnite length property (see [4, Theorem 1, pag 479]), the sequence {Z ( t ) } t ∈ Ncan converge to the critical point Z ∗ of Φ( Z ) , since the following Lemmas 1-4 are satisﬁed.
Revised: According to the ﬁnite length property (see [4, Theorem 1, pag 479]), the sequence {Z ( t ) } t ∈ N can converge to the critical point Z ∗ of Φ( Z ) , since the following Lemmas 1-4 are satisﬁed.

Original: Lemma 1.
Revised: Lemma 1.

Original: The objective function Φ( Z ) in formula (8) is a Kurdyka-Łojasiewicz (KŁ) function.
Revised: The objective function Φ( Z ) in formula (8) is a Kurdyka-Łojasiewicz (KŁ) function.

Original: Lemma 2 ( Sufﬁcient Decrease Condition ) .
Revised: Lemma 2 ( Sufﬁcient Decrease Condition ) .

Original: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2.
Revised: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2.

Original: Then, the sequence { Φ( Z ( t ) ) } t ∈ N explicitly satisﬁes Φ( Z ( t ) ) − Φ( Z ( t +1) ) ≥ ρ/ 2 (cid:107)Z ( t +1) − Z ( t ) (cid:107) 2 F ,where (cid:107)Z ( t +1) − Z ( t ) (cid:107) 2 F
Revised: Then, the sequence { Φ( Z ( t ) ) } t ∈ N explicitly satisﬁes Φ( Z ( t ) ) − Φ( Z ( t +1) ) ≥ ρ/ 2 (cid:107)Z ( t +1) − Z ( t ) (cid:107) 2 F , where (cid:107)Z ( t +1) − Z ( t ) (cid:107) 2 F

Original: = (cid:107)X ( t +1) − X ( t ) (cid:107) 2 F + (cid:80) Nk
Revised: = (cid:107)X ( t +1) − X ( t ) (cid:107) 2 F + (cid:80) Nk

Original: =1 (cid:107)G ( t +1)
Revised: =1 (cid:107)G ( t +1)

Original: k − G ( t ) k
Revised: k − G ( t ) k

Original: (cid:107) 2 F + (cid:107)C ( t +1) − C
Revised: (cid:107) 2 F + (cid:107)C ( t +1) − C

Original: ( t ) (cid:107) 2 F .
Revised: ( t ) (cid:107) 2 F .

Original: Lemma 3 ( Relative Error Condition ) .
Revised: Lemma 3 ( Relative Error Condition ) .

Original: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2.
Revised: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2.

Original: Then, there exists (cid:107) ∂
Revised: Then, there exists (cid:107) ∂

Original: Φ( Z ( t +1) )
Revised: Φ( Z ( t +1) )

Original: (cid:107) F ≤ { L Φ + ( N + 2) ρ }(cid:107)Z ( t +1) − Z ( t ) (cid:107) F , where L Φ sums the
Revised: (cid:107) F ≤ { L Φ + ( N + 2) ρ }(cid:107)Z ( t +1) − Z ( t ) (cid:107) F , where L Φ sums the

Original: Lipschitz constants of { ∂ G k Φ( Z ) } Nk =1 and ∂ C Φ( Z ) , i.e., L Φ =
Revised: Lipschitz constants of { ∂ G k Φ( Z ) } Nk =1 and ∂ C Φ( Z ) , i.e., L Φ =

Original: (cid:80)
Revised: (cid:80)

Original: Nk
Revised: Nk

Original: =1 L G k + L C .
Revised: =1 L G k + L C .

Original: Lemma 4.
Revised: Lemma 4.

Original: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2, then it is bounded.
Revised: Let {Z ( t ) } t ∈ N be the sequence generated by Algorithm 2, then it is bounded.

Original: Considering the page limitations, the proof of Lemmas 1-4 is detailed in the supplementary material.
Revised: Considering the page limitations, the proof of Lemmas 1-4 is detailed in the supplementary material.

Original: 3.4 Numerical Experiments
Revised: 3.4 Numerical Experiments

Original: In this section, we design substantial numerical experiments on synthetic and real-world data to verifythe superiority of the proposed TW-TC method compared with others, which are constructed basedon several commonly used tensor decompositions.
Revised: In this section, we design substantial numerical experiments on synthetic and real-world data to verify the superiority of the proposed TW-TC method compared with others, which are constructed based on several commonly used tensor decompositions.

Original: All the experiments are implemented in MATLAB(R2021a) on a computer of 64Gb RAM and Intel(R) Core(TM) i9-10900KF CPU: @3.70 GHz.
Revised: All the experiments are implemented in MATLAB (R2021a) on a computer of 64Gb RAM and Intel(R) Core(TM) i9-10900KF CPU: @3.70 GHz.

Original: Also,the MATLAB code is provided in the supplementary material, see Appendix E for more details .

Original: 3.4.1 Synthetic Data Completion
Revised: 3.4.1 Synthetic Data Completion

Original: To testify the superiority of TW decomposition over four related decompositions, i.e., TT [22],
Revised: To testify the superiority of TW decomposition over four related decompositions, i.e., TT [22],

Original: TR [35], FCTN [37], and Tucker [26] ones, we ﬁrstly conduct the numerical experiments on synthetictensor data by horizontally comparing the TW-TC method with their PAM-based TC algorithms,i.e., TT-TC (PAM), TR-TC (PAM), FCTN-TC (PAM), and Tucker-TC (PAM), respectively.
Revised: TR [35], FCTN [37], and Tucker [26] ones, we ﬁrstly conduct the numerical experiments on synthetic tensor data by horizontally comparing the TW-TC method with their PAM-based TC algorithms, i.e., TT-TC (PAM), TR-TC (PAM), FCTN-TC (PAM), and Tucker-TC (PAM), respectively.

Original: Sincethe Tucker factors comprise an underlying high-order structure, which may be closer to reality, weconstruct the synthetic tensors by Tucker decomposition using Tucker factors sampled from theuniform distribution U (0 , 1) .
Revised: Since the Tucker factors comprise an underlying high-order structure, which may be closer to reality, we construct the synthetic tensors by Tucker decomposition using Tucker factors sampled from the uniform distribution U (0 , 1) .

Original: More speciﬁcally, the synthetic data consists of 64 third-order, 81fourth-order, and 32 ﬁfth-order tensors, whose sizes are { I 

Revised: More speciﬁcally, the synthetic data consists of 64 third-order,fourth-order, and 32 ﬁfth-order tensors, whose sizes are { I

Original: ×
Revised: ×

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 3 : I 1 , I 2 , I 3 ∈ { 45 , 50 , 55 , 60 }} ,{ I 1 ×
Revised: I 3 : I 1 , I 2 , I 3 ∈ { 45 , 50 , 55 , 60 }} , { I 1 ×

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 4 : I 1 , I 2 , I 3 , I 4 ∈ { 18 , 20 , 22 }} , and { I 1 ×
Revised: I 4 : I 1 , I 2 , I 3 , I 4 ∈ { 18 , 20 , 22 }} , and { I 1 ×

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 

Revised: I

Original: ×
Revised: ×

Original: I 4 ×
Revised: I 4 ×

Original: I 5 : I 1 , I 2 , I 3 , I 4 , I 5 ∈{ 7 , 8 }} , and Tucker-ranks are (6 , 6 , 6) , (5 , 5 , 5 , 5) , and (3 , 3 , 3 , 3 , 3) , respectively.
Revised: I 5 : I 1 , I 2 , I 3 , I 4 , I 5 ∈ { 7 , 8 }} , and Tucker-ranks are (6 , 6 , 6) , (5 , 5 , 5 , 5) , and (3 , 3 , 3 , 3 , 3) , respectively.

Original: All synthetic dataare numerically renormalized into [0, 1].
Revised: All synthetic data are numerically renormalized into [0, 1].

Original: Subsequently, the partially observed tensors are generatedby random sampling with two sampling rates (SRs): 20% , 40% .
Revised: Subsequently, the partially observed tensors are generated by random sampling with two sampling rates (SRs): 20% , 40% .

Original: For each case, the hyper-parameters,i.e., TT-ranks, TR-ranks, FCTN-ranks, Tucker-ranks, and TW-ranks, also the storage complexitiesare provided in Appendix C .
Revised: For each case, the hyper-parameters, i.e., TT-ranks, TR-ranks, FCTN-ranks, Tucker-ranks, and TW-ranks, also the storage complexities are provided in Appendix C.

Original: The performance is assessed by the residual standard error (RSE), i.e.,(cid:107)X − X real (cid:107) F / (cid:107)X real (cid:107) F , where X is the recovered tensor.
Revised: The performance is assessed by the residual standard error (RSE), i.e., (cid:107)X − X real (cid:107) F / (cid:107)X real (cid:107) F , where X is the recovered tensor.

Original: Figure 3 depicts the recovery result versus recovery time for ﬁve PAM-based TC models under sixdifferent cases.
Revised: Figure 3 depicts the recovery result versus recovery time for ﬁve PAM-based TC models under six different cases.

Original: Compared with TT-TC (PAM), TR-TC (PAM), and FCTN-TC (PAM), the TW-TCmethod achieves the optimal RSE values for the majority of samples using both competitive computingefﬁciency and lower storage complexity (see Appendix C), which experimentally demonstrates thesuperior characterization ability of TW decomposition and strongly supports the theoretical analysispresented in Sections 1 and 2.3.
Revised: Compared with TT-TC (PAM), TR-TC (PAM), and FCTN-TC (PAM), the TW-TC method achieves the optimal RSE values for the majority of samples using both competitive computing efﬁciency and lower storage complexity (see Appendix C), which experimentally demonstrates the superior characterization ability of TW decomposition and strongly supports the theoretical analysis presented in Sections 1 and 2.3.

Original: Moreover, the faster computational efﬁciency of TW-TC over the
Revised: Moreover, the faster computational efﬁciency of TW-TC over the

Original: Tucker-TC model implies that TW decomposition can alleviate the curse of high-order core factor oncomputational burden, achieving greater applicability than Tucker decomposition.
Revised: Tucker-TC model implies that TW decomposition can alleviate the curse of high-order core factor on computational burden, achieving greater applicability than Tucker decomposition.

Original: 3.4.2 Real-World Data Completion
Revised: 3.4.2 Real-World Data Completion

Original: Furthermore, we investigate the TW-TC method on real-world data experiments by comparing it withseveral tensor decomposition-based state-of-the-art methods, including HaLRTC [18], t-SVD [33],
Revised: Furthermore, we investigate the TW-TC method on real-world data experiments by comparing it with several tensor decomposition-based state-of-the-art methods, including HaLRTC [18], t-SVD [33],

Original: TMacTT [3], TRLRF [32], and FCTN-TC [37].
Revised: TMacTT [3], TRLRF [32], and FCTN-TC [37].

Original: All hyper-parameters adopted in these competitorsare ﬁne-tuned within a speciﬁc range suggested by their authors to achieve better performance.
Revised: All hyper-parameters adopted in these competitors are ﬁne-tuned within a speciﬁc range suggested by their authors to achieve better performance.

Original: Thereal-world data mainly comprises two types, i.e., the multispectral image (MSI) and the video data.
Revised: The real-world data mainly comprises two types, i.e., the multispectral image (MSI) and the video data.

Original: Similarly, all experimental data are numerically pre-normalized into [0, 1].
Revised: Similarly, all experimental data are numerically pre-normalized into [0, 1].

Original: Afterwards, the partiallyobserved tensors are created by random sampling with three SRs: 5% , 10% , 20% .
Revised: Afterwards, the partially observed tensors are created by random sampling with three SRs: 5% , 10% , 20% .

Original: A popular metric,i.e., the peak signal-to-noise ratio (PSNR), is employed for quantitative evaluation.
Revised: A popular metric, i.e., the peak signal-to-noise ratio (PSNR), is employed for quantitative evaluation.

Original: When applied tothe MSI and video data, the mean of PSNRs across all channels is measured, termed MPSNR.
Revised: When applied to the MSI and video data, the mean of PSNRs across all channels is measured, termed MPSNR.

Original: MSI Data.
Revised: MSI Data.

Original: The tested MSI data sizes 200 × 200 × 31 (i.e., height × width × spectral), called Toy ,which is cropped from the CAVE dataset 1 .
Revised: The tested MSI data sizes 200 × 200 × 31 (i.e., height × width × spectral), called Toy , which is cropped from the CAVE dataset 2 .

Original: Regarding TW-ranks, we empirically assign R 1 =
Revised: Regarding TW-ranks, we empirically assign R 1 =

Original: L 3 and
Revised: L 3 and

Original: R 

Revised: R

Original: =
Revised: =

Original: L 1 =
Revised: L 1 =

Original: L 2 based on observations of multiple third-order real-world data experiments, and thenselect ranks R 1 , R 2 and R 3 from the candidate sets { 3 , 4 , 5 } , { 10 , 15 , 20 , 25 } and { 2 , 3 } , respectively.
Revised: L 2 based on observations of multiple third-order real-world data experiments, and then select ranks R 1 , R 2 and R 3 from the candidate sets { 3 , 4 , 5 } , { 10 , 15 , 20 , 25 } and { 2 , 3 } , respectively.

Original: Numerically, Table 1 (Data Index: Toy ) reports the MPSNR values under three different SRs and theaverage computing costs of all compared methods.
Revised: Numerically, Table 1 (Data Index: Toy ) reports the MPSNR values under three different SRs and the average computing costs of all compared methods.

Original: Also, Figure 4 presents their visual performancesand the corresponding error maps when SR= 20% .
Revised: Also, Figure 4 presents their visual performances and the corresponding error maps when SR= 20% .

Original: From the quantitative and qualitative results, weobserve that our TW-TC model achieves higher MPSNR values and better visual reconstruction thanthe other methods, conﬁrming its effectiveness for real-world third-order data. 
Revised: From the quantitative and qualitative results, we observe that our TW-TC model achieves higher MPSNR values and better visual reconstruction than the other methods, conﬁrming its effectiveness for real-world third-order data. Observed HaLRTC [18] t-SVD [33] TMacTT [3]

Revised: TRLRF [32]

Original: Video Data.
Revised: FCTN-TC [37] TW-TC Ground-truth [Figure]

Original:  The tested video data contains two color videos 2 (CVs): News and Container , and onehyperspectral video 3 (HSV) [20].
Revised: [Table] [Table] Video Data. The tested video data contains two color videos 3 (CVs): News and Container , and one hyperspectral video 4 (HSV) [20].

Original: The former two are of size 144 × 176 × 3 × 20 (i.e., height × width ×channel × frame), and the latter is of size 40 × 40 ×
Revised: The former two are of size 144 × 176 × 3 × 20 (i.e., height × width × channel × frame), and the latter is of size 40 × 40 ×

Original: 20 × 20 (i.e., height × width × spectral × frame).
Revised: 20 × 20 (i.e., height × width × spectral × frame).

Original: When tested on two CVs, we also empirically set R 3 =
Revised: When tested on two CVs, we also empirically set R 3 =

Original: R 4 and L 1 =
Revised: R 4 and L 1 =

Original: L 2 =
Revised: L 2 =

Original: L 3 =
Revised: L 3 =

Original: L 4 , and thenspecify ranks R 1 , R 2 , R 3 and L 1 from the candidate sets { 2 , 3 , 4 , 5 } , { 10 , 15 , 20 , 25 } , { 3 , 4 , 5 } and{ 2 , 3 } , respectively.
Revised: L 4 , and then specify ranks R 1 , R 2 , R 3 and L 1 from the candidate sets { 2 , 3 , 4 , 5 } , { 10 , 15 , 20 , 25 } , { 3 , 4 , 5 } and { 2 , 3 } , respectively.

Original: More directly, we designate R 1 =
Revised: More directly, we designate R 1 =

Original: R 2 = R 3 =
Revised: R 2 = R 3 =

Original: R 4 = 6 and L 1 =
Revised: R 4 = 6 and L 1 =

Original: L 2 =
Revised: L 2 =

Original: L 3 =
Revised: L 3 =

Original: L 4 = 4 to the HSV data experiment for all three SRs, aiming to alleviate the challenge of determining
Revised: L 4 = 4 to the HSV data experiment for all three SRs, aiming to alleviate the challenge of determining

Original: TW-ranks for these data with more balanced dimensions.
Revised: TW-ranks for these data with more balanced dimensions.

Original: The numerical and graphical results of all

Original: Observed

Original: HaLRTC [18] t-SVD [33]

Original: TMac-TT [3]

Original: TRLRF [32]

Original: FTCN-TC [37]

Original: TW-TC

Original: Ground-truth [Figure] [Table] compared methods are exhibited in Table 1 and Figure 
Revised: The numerical and graphical results of all compared methods are exhibited in Table 1 and Figure 4, respectively.

Original: [Table]

Original: respectively. From the results, we detect thatthe TW-TC method invariably contributes the best performances and further enhanced advantages [Table] which jointly illustrates the superiority of the TW topology for higher-order tensor characterization.
Revised: From the results, we detect that the TW-TC method invariably contributes the best performances and further enhanced advantages, which jointly illustrates the superiority of the TW topology for higher-order tensor characterization.

Original: Besides, further studies, e.g., the effect investigations of TW-ranks, are conducted in Appendix D.
Revised: Besides, further studies, e.g., the effect investigations of TW-ranks, are conducted in Appendix D.

Original:  Conclusion
Revised:  Conclusion

Original: This paper suggested a novel TN decomposition,

Original: i

Original: [Table] e., TW decomposition [Table] and presented an ALS-basedalgorithm to learn such a decomposition iteratively.
Revised: This paper suggested a novel TN decomposition, i.e., TW decomposition, and presented an ALS-based algorithm to learn such a decomposition iteratively.

Original: The TW decomposition veriﬁed theoretically andnumerically potential for more accurately characterizing the complex interactions inside higher-ordertensors using only linear hyper-parameter scaling, signiﬁcantly contributing to the advancement of
Revised: The TW decomposition veriﬁed theoretically and numerically potential for more accurately characterizing the complex interactions inside higher-order tensors using only linear hyper-parameter scaling, signiﬁcantly contributing to the advancement of

Original: TNs in machine learning.
Revised: TNs in machine learning.

Original: Moreover, we provided a TW decomposition-based TC model, i.e., TW-TC,for high-order tensor data recovery and developed an efﬁcient PAM-based solving algorithm, whichenjoys a convergence guarantee.
Revised: Moreover, we provided a TW decomposition-based TC model, i.e., TW-TC, for high-order tensor data recovery and developed an efﬁcient PAM-based solving algorithm, which enjoys a convergence guarantee.

Original: Substantial experiments on synthetic and real-world data conﬁrmedthat the TW-TC model markedly outperforms other decomposition-based state-of-the-art methods.
Revised: Substantial experiments on synthetic and real-world data conﬁrmed that the TW-TC model markedly outperforms other decomposition-based state-of-the-art methods.

Original: Checklist
Revised: Acknowledgments and Disclosure of Funding

Original: For all authors...
Revised: The authors would like to thank the area chair and anonymous reviewers for their insightful comments and constructive suggestions. This research is supported by NSFC (Grant Nos.

Original: (a) Do
Revised: Natural Science Foundation of Sichuan Province (Grant No. 2022NSFSC0501), Key Projects of Applied Basic Research in Sichuan Province (Grant No. 2020YJ0216), National Key Research and

Original: the main claims made
Revised: Development Program of China (Grant No. 2020YFA0714001), Major Key Project of PCL (Grant No. PCL2021A12), and China NSFC projects under contract (Grant No. U1811461).

Original: in the
Revised: Checklist

Original:  abstract and introduction accurately reﬂect the paper’scontributions and scope
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

Original: [Yes] Our abstract and introduction clearly describe ourcontribution, the algorithm, the theoretical results, and the experimental results.
Revised: [Yes] Our abstract and introduction clearly describe our contribution, the algorithm, the theoretical results, and the experimental results.

Original: (b) Did you describe the limitations of 
Revised: (b) Did you describe the limitations of your work?

Original: your work?

Original: [Yes] The limitations are clearly elaborated in the last paragraph of Section 3, also in Section 3.2.
Revised: [Yes] The limitations are clearly elaborated in a later paragraph of Section 2.3, also in Section 3.2.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No] This work is a theoretical research and numerical application for tensor learning.
Revised: [No] This work is a theoretical research and numerical application for tensor learning.

Original: For the time being, it does not present any foreseeable negative societal impact.
Revised: For the time being, it does not present any foreseeable negative societal impact.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] All assumptions are clearly stated and referenced in Section 2.2.
Revised: [Yes] All assumptions are clearly stated and referenced in Section 2.2.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] The proofs of all Theorems and Lemmas are provided in Appendix A of the supplementary material.
Revised: [Yes] The proofs of all Theorems and Lemmas are provided in Appendix A of the supplementary material.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] The code for TW-TC model is provided in Appendix E of the supplementary material.
Revised: [Yes] The code for TW-TC model is provided in Appendix E of the supplementary material.

Original: (b) Did you specify all the training details (e g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See Section 3.
Revised: [Yes] See Section 3.

Original: (c) Did you report error bars (e g., with respect to the random seed after running ex- periments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [No] Testing our model is time-consuming and we lack sufﬁcient resources/time for multiple repetitions of our experiments.
Revised: [No] Testing our model is time-consuming and we lack sufﬁcient resources/time for multiple repetitions of our experiments.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] The computational resources is shown in the ﬁrst paragraph of Section 3. 
Revised: [Yes] The computational resources is shown in the ﬁrst paragraph of Section 3. 4.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] See Section 3.4.(b)
Revised: [Yes] See Section 3.4.

Original:  Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [No] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [No] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes] The code for TW-TC model is provided in Appendix E of the supplementary material.
Revised: [Yes] The code for TW-TC model is provided in Appendix E of the supplementary material.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [No] All datasets used in our work are publicly available.
Revised: [No] All datasets used in our work are publicly available.

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]