Tucker-O-Minus Decomposition for Multi-view Tensor Subspace Clustering Yingcong Lu Yipeng Liu Zhen Long Zhangxin Chen Ce Zhu

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Tucker-O-Minus Decomposition for Multi-view

Tensor Subspace Clustering

Yingcong Lu, Yipeng Liu, Zhen Long, Zhangxin Chen, Ce Zhu ∗

October 25, 2022

Abstract

With powerful ability to exploit latent structure of self-representation

information, diﬀerent tensor decompositions have been employed into low

rank multi-view clustering (LRMVC) models for achieving signiﬁcant per-

formance. However, current approaches suﬀer from a series of problems re-

lated to those tensor decomposition, such as the unbalanced matricization

scheme, rotation sensitivity, deﬁcient correlations capture and so forth.

All these will lead to LRMVC having insuﬃcient access to global informa-

tion, which is contrary to the target of multi-view clustering. To alleviate

these problems, we propose a new tensor decomposition called Tucker-

O-Minus Decomposition (TOMD) for multi-view clustering. Speciﬁcally,

based on the Tucker format, we additionally employ the O-minus struc-

ture, which consists of a circle with an eﬃcient bridge linking two weekly

correlated factors. In this way, the core tensor in Tucker format is re-

placed by the O-minus architecture with a more balanced structure, and

the enhanced capacity of capturing the global low rank information will

be achieved. The proposed TOMD also provides more compact and pow-

erful representation abilities for the self-representation tensor, simultane-

ously. The alternating direction method of multipliers is used to solve the

proposed model TOMD-MVC. Numerical experiments on six benchmark

data sets demonstrate the superiority of our proposed method in terms of

F-score, precision, recall, normalized mutual information, adjusted rand

index, and accuracy.

1 Introduction

Tensor, a higher order generation of vector and matrix, provides a natural rep-

resentation for high order data. For example, ORL multi-view data set [54] is

a 3rd-order tensor XI×Cv×V(v= 1,· · · , V ), where Iis the number of samples,

Cvis the feature size of the v-th view, and Vis the number of views. Tensor de-

composition allows us to simultaneously explore all dimensions to obtain more

∗All the authors are with the School of Information and Communication Engineering,

University of Electronic Science and Technology of China (UESTC), Chengdu, 611731, China.

E-mail: yipengliu@uestc.edu.cn

arXiv:2210.12638v1 [cs.LG] 23 Oct 2022

latent information, which has attracted attention in a series of ﬁelds, e.g., image

processing [8,32], machine learning [15,16] and signal processing [1,5]. Tensor-

based multi-view clustering (MVC) is one of them, which separates multi-view

data into clusters by exploiting their latent information.

Most tensor MVC methods are based on the assumption that their self-

representation tensors are low rank [53]. For example, Chen et al. [7] combine

the low-rank tensor graph and the subspace clustering into a uniﬁed model in

multi-view clustering, which applies the Tucker decomposition [41,42] to ex-

plore the low-rank information of representation tensor. With the emergence of

a new type of decomposition for 3rd-order tensors [20,21], multiple multi-view

subspace clustering methods based on the tensor singular value decomposition

(t-SVD) have been proposed [45,46,58]. In addition, tensor networks provide a

more compact and ﬂexible representation for higher order tensor than traditional

tensor decomposition [27,30,31,33]. In this way, Yu et al. [51] have proposed

a novel non-negative tensor ring (NTR) decomposition and graph-regularized

NTR (GNTR) for the non-negative multi-way representation learning with sat-

isfying performance in clustering tasks.

Although the aforementioned methods have achieved promising clustering

performance, there are still several problems. Since the Tucker rank is related to

the unbalanced mode-nunfolding matrices [2], capturing the global information

of the self-representation tensor by simply processing Tucker decomposition may

be diﬃcult in LRMVC. Besides, the t-SVD suﬀers from rotation sensitivity and

the low rank information can not be fully discovered in the 3rd mode [10].

Thus, the t-SVD based methods always need to utilize the rotation operation of

the self-representation tensor to explore the correlations across diﬀerent views.

Naturally, the acquisition of the correlations among samples will be inadequate.

Furthermore, tensor ring (TR) has shown better performance in exploring the

low-rank information with the well-balanced matricization scheme. However,

the interaction of neighboring modes in TR is stronger than that of two modes

separated by a large distance, whose correlations are also have been ignored.

Considering the above problems, we propose the Tucker-O-Minus decompo-

sition for multi-view subspace clustering. For employing the additional factors

and correlations to construct a more compact and balanced tensor network, a

simple way seems to replace the core factor of the Tucker structure with the

TR format. However, as we mentioned above, the TR architecture suﬀers from

a deﬁciency in exploring the correlations between the weakly-connected modes,

which will result in the loss of essential clustering information in LRMVC. To

this end, we propose the O-minus structure for LRMVC in the following three

considerations. Firstly, from the perspective of two-point correlations in high-

energy physics [3], appending moderate lines with eﬀective vertexes between

two slightly correlated factors will strengthen their relationships. Similarly, we

try to add the ”bridge” with a tensor factor based on the TR structure to better

capture the low rank information. Nonetheless, more links will generate more

loops, which will cause huge diﬃculties in tensor contraction and the computa-

tional complexity burden of the tensor networks [9]. Simultaneously, since the

number of samples IVin LRMVC, compared with higher-order information

across diﬀerent views, the information related to the correlations of instances

requires more connections to be fully discovered. Accordingly, to eﬃciently ex-

plore the low rank information in LRMVC, the correlations related to samples

(i.e., I1, and I3in Fig. 1) are further strengthened by the special ”bridge”. This

unique internal architecture is similar to ””, thus namely O-Minus. And the

whole architecture called Tucker-O-Minus decomposition is illustrated in Fig. 1.

In this way, the low rank based multi-view clustering problem can be success-

fully solved with satisfying results. The main contributions of this paper are

summarized as follows:

1. We propose the Tucker-O-Minus decomposition. Diﬀerent from the exist-

ing tensor networks, it allows more valid interaction among nonadjacent

factors and obtains a better low rank representation for high-dimensional

data. Numerical experimental results on gray image reconstruction show

that the performance of TOMD in exploring low rank information is su-

perior to others.

2. We apply TOMD to a uniﬁed multi-view clustering framework. The pro-

posed model, namely TOMD-MVC, utilizes the TOMD to capture the

low-rank properties of self-representation tensor from multi-view data.

The alternating direction method of multipliers (ADMM) is applied to

solve the optimization model.

3. We conduct extensive experiments on six real-world multi-view data sets

to demonstrate the performance of our method. Compare with state-of-

the-art models, TOMD-MVC achieves highly competitive or even better

performance in terms of six evaluation metrics.

The remainder of this paper is organized as follows. Section 2gives the used

notations, mathematical backgrounds, and review the related works. The pro-

posed tensor network is presented in details in Section 3. Section 4gives the

multi-view clustering method based on the newly proposed tensor network. The

experimental results are demonstrated and analyzed in Section 5. Section 6

draws the conclusion ﬁnally.

2 Notations, Preliminaries, and Related Works

2.1 Notations

We give a brief introduction about notations and preliminaries in this section.

Throughout this paper, we use lower case letters, bold lower case letters, bold

upper case letters, and calligraphic letters to denote scalars, vectors, matrices

and tensors, respectively, e.g., a,a,Aand A. Some other frequently used

notations are listed in Table 1, where Ais a 3rd-order tensor and Ais a matrix.

Figure 1: The graphical illustration of TOMD for a 4th-order tensor X ∈

RI1×I2×I3×I4, where U(i)∈RIi×Ri(i= 1,· · · ,4) are factor matrices, G(n),

n= 1,· · · ,5, are the sub-tensors.

Table 1: Summary of notations in this paper.

Denotation Deﬁnition

AiAi=A(:,:, i)

ai1,i2,i3The (i1, i2, i3)-th entry of tensor A

tr(A) tr(A) = Piai,i

kAkFkAkF=qPi1i2a2

i1,i2

kAk∞kAk∞= maxi1i2|Ai1,i2|

kAk2,1kAk2,1=PjkA(:, j)k2

kAkFkAkF=qPi1i2i3|ai1,i2,i3|2

2.2 Preliminaries

Deﬁnition 1. (Mode-nunfolding) [22] For tensor A ∈ RI1×···×IN, its ma-

tricization along the n-th mode is denoted as A(n)∈RIn×I1I2···In−1In+1···IN.

Deﬁnition 2. (n-unfolding) [9] Given an N-way tensor A ∈ RI1×···×IN, the

deﬁnition of its n-unfolding is expressed as Ahni∈RI1···In×In+1···IN.

Deﬁnition 3. (Mode-nproduct) [22] The mode-nproduct of A ∈ RI1×···×IN

and matrix B∈RJ×Inis deﬁned as

X=A ×nB∈RI1×···×In−1×J×In+1×···×IN.(1)

Deﬁnition 4. (Tensor Network Contraction) [38] Given a tensor network

composed of Nsub-tensors {G(n)}(n= 1,· · · , N), then Gis denoted as the

tensor network contraction result of {G(n)}(n= 1,· · · , N), whose general math-

ematical equation can be written as

n=1

1,Rn

2,···

1,rn

2,···

{

D1,D2,···

d1,d2···

n=1

G(n)({rn

1, rn

2· · · , d1, d2· · · })}

= TC({G(n)}N

n=1),

(2)

where {D1, D2,· · · } are geometrical indexes, each of which is shared normally

two tensors and will be contracted, R={R1

1, R1

2,· · · , RN

1, RN

2,· · · } are open

bounds, each of which only belongs to one tensor. After contracting all the geo-

metrical indexes, the Grepresents a P-th order tensor with Pthe total number

of the open indexes R.

2.3 Related Works

This section reviews some closely related works, including low-rank approxima-

tion based multi-view subspace clustering, and multi-view graph clustering.

2.3.1 Low-rank Approximation Based Multi-view Subspace Cluster-

ing

Subspace learning based multi-view clustering methods assume that each sample

can be represented as a linear combination of all the samples [40].

Chen et al. [7] utilize the Tucker decomposition to obtain the ”clean” repre-

sentation tensor from view speciﬁc matrices. He et al. [12] introduce a tensor-

based multi-linear multi-view clustering (MMC) method. Under the consid-

eration of Canonical Polyadic Decomposition (CPD) [19], the MMC is able

to explore the higher-order interaction information among the multiple views.

Recently, tensor singular value decomposition (t-SVD) based tensor nuclear

norm [20,21] shows satisfying performance in capturing low rank information

for 3rd-order tensor. Thus, many t-SVD based works [45,46,48] are proposed

for better capturing the high-order correlation hidden in 3rd-order tensor from

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Tucker-O-MinusDecompositionforMulti-viewTensorSubspaceClusteringYingcongLu,YipengLiu,ZhenLong,ZhangxinChen,CeZhu*October25,2022AbstractWithpowerfulabilitytoexploitlatentstructureofself-representationinformation,dierenttensordecompositionshavebeenemployedintolowrankmulti-viewclustering(LRMVC)modelsf...

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Tucker-O-Minus Decomposition for Multi-view Tensor Subspace Clustering Yingcong Lu Yipeng Liu Zhen Long Zhangxin Chen Ce Zhu

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