# slrmvtc_smooth_lowrank_multiview_tensor_clustering__0a43e566.pdf

SLR-MVTC: Smooth Low-Rank Multi-View Tensor Clustering

Zhen Long, Yipeng Liu, Yazhou Ren, Ce Zhu

University of Electronic Science and Technology of China in Chengdu, 611731, China {zhen.long,yipengliu,yazhou.ren,eczhu}@uestc.edu.cn

Multi-view tensor clustering (MVTC) has gained much attention for its effectiveness in capturing global high-order correlations across views. However, current MVTC methods suffer from two limitations: 1) adopting a two-stage process to learn the latent features for clustering, and 2) either ignoring local similarities within views or treating local similarities and global high-order correlations equally. In this paper, we propose a smooth low-rank MVTC (SLRMVTC) method, which aims to extract latent features that are smooth within each view and low-rank across views, enhancing clustering performance. Specifically, we first learn latent features from each view using orthogonal projection and then construct the latent feature tensor by concatenation and rotation. Then, we introduce a new smooth tensor nuclear norm to depict the low-rank components of the lowfrequency parts in the feature tensor. Benefiting from the fast Fourier transform along the sample dimension, the obtained low-frequency components effectively capture local smoothness within views, while their low-rank parts further explore global correlations across views. Experimental results on six multi-view datasets demonstrate that SLR-MVTC outperforms state-of-the-art algorithms in terms of clustering performance and CPU time.

Code https://github.com/longzhen520/SLR MVTC

Introduction Multi-view data collected from multiple feature extractors or sensors are ubiquitous in numerous real-world scenarios, with each view providing a distinct feature description (Zhang et al. 2019; Cui et al. 2023; Huang et al. 2021, 2023; Xu et al. 2024). For example, in assessing breast cancer risk, multi-view data include various types of ultrasound (US) images, such as US B-mode, US color Doppler, and US elastography images (Qian et al. 2021). Multi-view data can provide both consistency and complementary information, improving the performance of related data analysis tasks and leading to the development of multi-view learning (Zhang et al. 2024; Xu et al. 2023a; Yu et al. 2023). Among them, multi-view clustering (MVC) aims to divide data into several clusters by fully leveraging information from different views. It has garnered much attention in medical image

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v v v X = P U

Multi-view data

Multi-view data

graph learning

affinity matrices

feature learning

(a) Tensor based methods (b) MF based methods

shared features

shared features

Figure 1: Two ways for exploring inter-view correlations.

segmentation (Zhou, Ruan, and Canu 2019), brain network analysis (Liu et al. 2018), and single-cell multi-omics integration (Huizing et al. 2023). Existing MVC methods are mainly categorized into similarity learning-based and feature learning-based approaches based on the spaces in which they explore inter-view correlations. Similarity learning-based methods focus on capturing global high-order inter-view correlations by constructing affinity matrices. These matrices are typically derived through subspace learning or graph learning, which map the original data into sample spaces where the elements of the affinity matrix represent the similarity between pairs of samples. The final shared affinity matrix is then transformed into embedding features by performing eigenvalue decomposition on its Laplacian matrix, which is then used for K-means clustering (Khan and Maji 2019; Xia et al. 2022; Long et al. 2023), as shown in Fig. 1 (a). It can be observed this group employs a two-stage process to learn the latent features. Besides, one goal of clustering is to minimize the distance between samples within the same cluster, which means that samples should exhibit local similarities. Therefore, both high-order correlations across views and local similarities within each view are crucial. However, this group either ignores local similarities or treats local and global high-order correlations equally. Another group focuses on directly learning consistent latent features from multi-view data for clustering, typically using deep leaning (Xu et al. 2023b; Ren et al. 2024; Yan et al. 2023) and matrix factorization (MF) (Liu et al. 2013; Wan et al. 2023; He et al. 2023; Liu et al. 2021). However, deep MVC requires substantial data and a meticulously designed network architecture. MF-based MVC maps the orig-

The Thirty-Ninth AAAI Conference on Artificial Intelligence (AAAI-25)

inter-view information

intra-view information

Smooth low-rank tensor approximation

Multi-view data

V X v v v X = P U

Clustering results K-means

low frequency components

low rank approximation

Figure 2: The framework of SLR-MVTC. This process involves first learning latent features Uv from the given data Xv using orthogonal projection Pv. These latent features {Uv}V v=1 are then formed into a tensor U, where a newly defined smooth low-rank tensor approximation operator is applied to explore inter-view and intra-view correlations respectively.

inal features from different views into the consensus latent feature space for clustering, as shown in Fig. 1 (b). For instance, (Wang et al. 2017) proposed a diverse nonnegative MF to enhance the diversity between latent features. (Wan et al. 2023) fused the latent features by mapping them into a consensus low-dimensional space using rotation matrices. However, current MF-based MVC methods explore pairwise correlations between views, ignoring higher-order correlations across views and local smoothness within views. To address these issues, we propose a smooth lowrank multi-view tensor clustering (SLR-MVTC) method, designed to efficiently explore high-order inter-view and local smooth intra-view correlations of latent features respectively, as shown in Fig. 2. In particular, given multiview data {Xv RDv N}V v=1, we first obtain latent features {Uv RC N}V v=1 using orthogonal projection matrices {Pv RDv C}V v=1. Next, to respectively capture intra-view local similarities and inter-view global correlations of latent features, we introduce a new norm U STNN on the feature tensor U RC V N. Here U is formed by concatenating Uv along the third dimension and then rotating it. The norm U STNN is defined based on the work (Long et al. 2024), which introduced tensor low-frequency operators to obtain smooth representations of samples. The newly defined norm advances the approach by further depicting the low-rank component of the low-frequency parts. In this way, the low-frequency component provides a smooth representation within views, and its low-rank parts further explore the global correlations across views. Finally, the shared features U = 1

V PV v=1 Uv are fed into the K-means algorithm for clustering. Experimental results on six multi-view datasets show that SLR-MVTC outperforms state-of-the-art algorithms in both clustering performance and computational efficiency. Our contributions beyond existing MVC methods are:

We develop the norm U STNN, which captures both global correlations across views and local smoothness within views, with a computational complexity of O(N log N). We integrate the newly defined norm with latent feature learning into a unified framework, allowing the smooth low-rank prior to guiding the learning of latent features

from multi-view data during each iteration, resulting in enhanced clustering performance. Experimental results on six multi-view datasets show that SLR-MVTC outperforms state-of-the-art algorithms in both clustering performance and CPU time.

Notations and Problem Formulation Notations For clarity, we present frequently used notations in Table 1.

Symbols Descriptions

a, a, A, A scalar, vector, matrix, tensor

q PI i=1 PJ j=1 a2 i,j A *

PI i=1 σi, σi is the i-th singular value of A A 2,1

PI i=1 a 2 trace(A) PI i=1 ai,i a floor function N, V,C number of samples, views, clusters v = 1, , V index ranges from 1 to its capital version Xv RDv N multi-view data in the v-th view Pv RDv C orthogonal projection matrix in the v-th view Uv RC N latent features in the v-th view U RC V N latent feature tensor

Table 1: Frequently used notations in this paper.

Preliminaries Definition 1. (t-SVD) (Kilmer et al. 2013; Lu et al. 2020; Braman 2010) Given a tensor U RC V N, its t-SVD can be expressed as:

U = S V DT, (1)

where S RC C N and D RV V N are orthogonal tensors, and V RC V N is an f-diagonal tensor.

The t-SVD can be obtained by Algorithm 1, where b U = fft(U, [ ], 3) denotes applying the fast Fourier transform (FFT) operator along the 3-rd dimension, and ifft denotes the inverse FFT.

Algorithm 1: t-SVD

Input: U RC V N Output: S RC C N, V RC V N, D RV V N; b U fft(U, [ ], 3); for n = 1 to N do [S, V, D] = svd( b U(:, :, n)); b S(:, :, n) = S; b V(:, :, n) = V; b D(:, :, n) = D. end for S = ifft(b S, [ ], 3); V = ifft(b V, [ ], 3); D = ifft(b D, [ ], 3).

Definition 2. (Tensor nuclear norm (TNN)) (Kilmer and Martin 2011; Xie et al. 2021) Given a tensor U RC V N, its t-SVD-based TNN is defined as

n=1 b U(:, :, n) =

j=1 σ j( b U(:, :, n)), (2)

where σ j( ) is the j-th largest singular value of b U(:, :, n), (n = 1, 2, . . . , N) and b U(:, :, n) is the n-th frontal slice of b U. Definition 3. (Tensor low-frequency component (TLFC)) (Long et al. 2024) The TLFC of a tensor U RC V N is defined as the Frobenius norm of the frontal slices in the low-frequency domain. Mathematically, it can be expressed as follows:

U TLFC = b U(:, :, 1) +

n=2 ( b U(:, :, n) + b U(:, :, N + 2 n)) F,

(3) where L is the number of low-frequency bands, b U(:, :, 1) is the 0-frequency component and b U(:, :, N +2 n) = conj( b U(: , :, n)) means conjugate symmetry according to the properties of the FFT.

Related Works MF-based MVC MF-based MVC methods seek to discover the common features U that reveal the consensus structure from the given multi-view data {Xv RDv N}V v=1. The general framework is described as follows:

min Pv 0,Uv 0

v=1 Xv Pv Uv F + λ

v=1 R(U, Uv), (4)

where R(U, Uv) is the regularization terms. For instance, (Liu et al. 2013) used U Uv F to determine the common features. Furthermore, (Wang et al. 2017) introduced a diversity constraint, trace(Uv UT w), to promote orthogonality between Uv and Uw. Besides, some approaches have removed the non-negativity constraint on Pv and Uv, and instead introduce orthogonality regularization for Pv to get more distinctive embedding features. The approach is as follows:

v=1 Xv Pv Uv F, s. t. Pv PT v = IC. (5)

According to this framework, (Liu et al. 2021) further decomposes the features Uv into a consensus indicator matrix

and centroid matrix. Besides, (Wan et al. 2023) maps the original feature matrix of each view into several latent features and dynamically adjusts their weights based on their corresponding contributions. The correlations in multi-view data (V >2) are of a higher order. However, this group only considers pairwise correlations between views and do not explore the higher-order correlations across views .

Tensor-based MVC Tensor-based MVC methods aim to identify the common affinity matrix by capturing high-order correlations of memberships between different views. Given multi-view data {Xv RDv N}V v=1, the general model can be formulated as follows:

min Ev,Zv L(Z) + λR(E), s. t. Xv = f(Zv) + Ev, (6)

where Zv refers to the relationship matrix, and Z = Ω(Z1, Z2, . . . , ZV) means the combination of these relationship matrices into a third-order tensor. E = [E1; E2; . . . ; EV] is the noise. L(Z) and R(E) are the regularization terms. When f(Zv) = Xv Zv, the problem (6) can be transformed into the optimization problem for subspace learning. For instance, (Xie et al. 2018) first considered t-SVD-based TNN on the rotated self-representation tensor to explore highorder correlations across views. Furthermore, some lowrank tensor network approximations are considered on the self-representation tensor to characterize the intra-view and inter-view relationships (Lu et al. 2023; Liu et al. 2024). When f(Zv) = Av Zv, the problem (6) is the optimization problem of anchor learning. For example, (Long et al. 2023) considered low-rank MERA approximation on the anchor graph tensor to capture the inter/intra-view correlations. (Ji and Feng 2023) split the anchor graph tensor into two parts: the common and the specific. To investigate the common and specific, respectively, enhanced tensor rank and tensorial exclusive regularization are taken into consideration. Furthermore, (Long et al. 2024) investigated the learning of embedding features from the provided anchor graph and introduced a low-frequency operator to obtain a smooth representation of the samples. (Li et al. 2023) introduced a method to directly learn the common indicator feature using orthogonal non-negative tensor factorization on the anchor graph tensor. This group employs a two-stage processing approach to acquire the final embedding features. Besides, the aforementioned methods either ignore local similarities or treat local similarity and global high-order correlations equally, failing to effectively explore multi-view data for clustering.

Proposed Method

In this paper, we focus on directly learning the latent features from multi-view data and introduce a smooth low-rank tensor approximation method on the feature tensor to capture the local similarity within views and the global high-order correlations across views.

Model Development Given {Xv RDv N}V v=1, the general optimization problem of SLR-MVTC is expressed as:

min {Ev,Uv,Pv}V v=1 γ U STNN + λ

v=1 ||Ev||2,1

s. t. Xv = Pv Uv + Ev, PT v Pv = IC, v = 1, , V.

Here, the orthogonal projection matrix Pv RDv C is used to obtain more distinctive latent features Uv RC N, where C is the number of clusters. In addition, Ev denotes the sparse noise, which is typically removed using convex surrogates such as ℓ1 or ℓ2,1 norms. U = Ω(U1, , UV) RC V N is the latent feature tensor, where Ωis an operator that stacks all latent features into a 3-rd tensor and subsequently rotates it. The inverse operator is denoted as Uv = Ω 1 v (U). U STNN is the newly defined norm, which aims to explore the global high-order inter-view correlations and local intra-view similarities. According to definitions of TNN and TLFC, we introduce the definition of STNN as follows: Definition 4. (Smooth tensor nuclear norm (STNN)) Given a tensor U RC V N, its STNN is defined as the nuclear norm of the frontal slices in the low-frequency part. Mathematically, it can be expressed as follows:

U STNN = b U(:, :, 1) +

n=2 b U(:, :, n) + b U(:, :, N+2 n) ,

(8) where b U(:, :, 1) is the nuclear norm of 0-frequency component and L represents the number of low-frequency bands. Note that the low-frequency components obtained using the FFT along the sample dimension will result in a local smooth representation. Furthermore, the low-rank approximation in the low-frequency part will capture the global high-order correlations. To make the optimization problem (7) separable, we introduce an auxiliary variable Y, leading to the following reformulation:

min {Ev,Uv,Pv}V v=1,Y γ Y STNN + λ

v=1 ||Ev||2,1

s. t. Xv = Pv Uv + Ev, PT v Pv = IC, v = 1, , V, U = Y.

Solutions To solve the problem with the constraints mentioned above, the Alternating Direction Method of Multipliers (ADMM) (Boyd et al. 2011) framework can be used. The corresponding augmented Lagrangian function can be defined as follows:

L {{Uv, Ev, Pv, Qv}V v=1, Y, F

v=1 ( Qv, Xv Pv Uv Ev + µ1

2 Xv Pv Uv Ev 2 F

+ λ||Ev||2,1) + γ Y STNN + F , U Y + µ2

2 U Y 2 F, (10)

Algorithm 2: Updating Y

Input: H, L, γ Initialize: H = (U + F

µ2 ), τ = γ

µ2 . Output: Y; b H fft(H, [ ], 3); b Y(:, :, 1)) = SVTτ( b H(:, :, 1)); for n = 2 to L do b Y(:, :, n) = SVTτ( b H(:, :, n)); b Y(:, :, N + 2 n) = conj(b Y(:, :, n)). end for Y = ifft(b Y, [ ], 3).

under constraints PT v Pv = IC, v = 1, , V, where {Qv}V v=1, F are Lagrange multipliers and µ1, µ2 are penalty factors. Under the ADMM framework, problem (10) can be divided into several subproblems, where each subproblem alternately updates one variable while keeping the others fixed. Update {Pv}V v=1: Fixing other variables, the subproblem of Pv can be rewritten as:

max Pv:PTv Pv=IC trace(Pv(Uv(Qv + µ1Xv µ1Ev)T)). (11)

This subproblem is a well-known orthogonal Procrustes problem. Letting H = Uv(Qv+µ1Xv µ1Ev)T, and [S, V, D] = svd(H) is the singular value decomposition of H, the solution Pv = DST. The computational complexity of updating Pv is O(C2Dv), where Dv is the original feature dimension. Update {Uv}V v=1: The subproblem of Uv can be written as:

min Uv Qv, Xv Pv Uv Ev + µ1

2 Xv Pv Uv Ev 2 F

+ F , U Y + µ2

2 U Y 2 F. (12)

By taking the derivative and setting it to zero, the solution of Uv can be obtained as follows:

Uv = PT v (Qv + µ1Xv µ1Ev) + Ω 1 v (µ2Y F ) µ1 + µ2 , (13)

where Ω 1 v (Y) = Yv. The computational complexity of solving Uv is O(NCDv). Update {Ev}V v=1: The solution of Ev can be updated by

Ev = sth(Xv Pv Uv + (1/µ1)Qv, λ/µ1), (14)

where sth(x, τ) is the well-known soft thresholding operator, defined as follows: sth(x, τ) = sgn(x) max(|x| τ, 0). The computational complexity of solving Ev is O(NCDv). Update Y: The subproblem of Y is rewritten as:

min Y γ µ2 Y STNN + 1

µ2 ) 2 F, (15)

which can be solved by Algorithm 2. Here SVTτ(H) is the well-known singular value thresholding operator, defined as follows:

SVTτ(H) = S sth(V, τ)DT,

Algorithm 3: SLR-MVTC

Input:Multi-view data {Xv}V v=1, low frequency parameter L,low rank parameter γ, regularization parameter λ Initialize: µ1 = 5 10 4, µ2 = 10 3 while not converged do for v = 1 to V do Update Pv via Eq.11; Update Uv via Eq.13 Update Ev via Eq.14; Update Qv via Eq.16; end for Update Y via Alg.2; Update F via Eq. 17; µ1 = min(µ1 1.5, 1010), µ2 = min(µ2 1.5, 1010); L = min(L 1.5, N/2) ; end while Apply K-means on U = 1

V PV v=1 Uv Output: Clustering result.

where [S, V, D] = svd(H). The computational complexity of updating Y is O(max(N log N, V2C)), where O(N log N) is from the FFT and O(V2C) comes from the SVD operator. Update Lagrangian multipliers:

Qv = Qv + µ1(Xv Pv Uv Ev), v = 1, , V, (16) F = F + µ2(U Y). (17)

Overall, the solution of SLR-MVTC is summarized in Algorithm 3, with convergence achieved when RE 10 5, where RE=max(maxv( Xv Pv Uv Ev ), U Y ). Besides, the main computational complexity of SLR-MVTC is O(max(N log N, NC PV v=1 Dv)).

Experiments Experimental Settings Multi-View Datasets Description Six well-known multiview datasets were selected to evaluate the performance of SLR-MVTC: ORL (Samaria and Harter 1994), CCV (Jiang et al. 2011), ALOI 100 (Geusebroek, Burghouts, and Smeulders 2005), Reuters (Lewis et al. 2004), Aw A (Lampert, Nickisch, and Harmeling 2009), and CIFAR100 (Krizhevsky, Hinton et al. 2009). Detailed statistical information about these datasets is provided in Table 2.

Datasets N V C (D1, , DV)

ORL 400 3 40 (4096, 3304, 6750) CCV 6773 3 20 (20, 20, 20) ALOI 100 10800 4 100 (77, 13, 64, 125)

Reuters 18758 5 6 (21531, 24892, 34251, 15506, 11547) Aw A 30475 6 50 (2688, 2000, 252, 2000, 2000) CIFAR100 50000 3 100 (512,2048,1024)

Table 2: The statistical information of multi-view datasets, where N, V, C are the number of samples, views, and clusters, respectively. (D1, , DV) are original feature sizes.

Compared Clustering Algorithms To evaluate the clustering performance, we compare the proposed approach against eight state-of-the-art clustering methods. These include four pairwise inter-view correlation-based methods: Binary Multi-View Clustering (BMVC) (Zhang et al. 2018), Fast Parameter-free Multi-view Subspace Clustering with Consensus Anchor Guidance (FPMSC-CAG) (Wang et al. 2022), and Auto-Weighted Multi-View Clustering for Large-Scale Data (AWMVC) (Wan et al. 2023), A Simple yet Efficient Scalable Multi-View Tensor Clustering (S2MVTC) (Long et al. 2024). Additionally, we compare three high-order inter-view correlation-based multiview clustering methods: scalable MERA based multi-view clustering (s MERA-MVC) (Long et al. 2023), Orthogonal Non-negative Tensor Factorization-based Multi-view Clustering (Orth-NTF) (Li et al. 2023), and High-order Complementarity Induced Fast Multi-View Clustering with Enhanced Tensor Rank Minimization (CFMVC-ETR) (Ji and Feng 2023). We also add one updated multi-view clustering method: Efficient Balanced Multi-view Graph Clustering via Good Neighbor Fusion (EBMGC-GNF) (Wu et al. 2024). All tests are tuned best and accomplished on a desktop computer with a 2.10 GHz 13th Gen Intel(R) Core(TM) i7 Processor, 64 GB RAM, and MATLAB 2020b.

Evaluation Metrics The performance of the clustering methods is evaluated using seven standard metrics: Accuracy (ACC), Normalized Mutual Information (NMI), Fscore, Precision, Recall, Adjusted Rand Index (ARI), and CPU time. For all metrics except CPU time, higher values indicate better clustering performance.

Parameter Settings SLR-MVC has three parameters λ, γ, and L which control the weight on noise, the low-rank component, and the low-frequency component, respectively. We select these parameters by a brute-force search, with λ and γ ranging from {10 7, 10 6, 10 5, 10 4, 10 3, 10 2, 10 1} and L ranging from {2:2:40}. Fig. 3 shows partial results with some parameters fixed. For example, with L = 22, the ranges of λ and γ are shown in Fig. 3 (a), where ACC performs better and remains stable with smaller γ and λ = 10 4. Additionally, Fig. 3 (b) demonstrates that when L > 16, the ACC shows little variation and remains above 70%.

0 10 20 30 40 L

(b) γ = 10 5, λ = 10 4

Figure 3: The change of clustering performance on CCV dataset. (a) γ and λ changed with L = 22; (b) L changed with γ = 10 5, λ = 10 4.

Datasets Methods F-score Precision Recall NMI ARI ACC CPU time (s)

BMVC (TPAMI 18) 59.61(0.00) 55.12(0.00) 64.89(0.00) 84.32(0.00) 58.60(0.00) 71.25(0.00) 0.08 AWMVC (AAAI 23) 66.61(2.87) 62.39(3.32) 71.48(2.71) 87.80(1.16) 65.78(2.95) 73.88(2.24) 6.41 FPMVS-CAG (TIP 22) 57.74(2.24) 50.91(2.95) 66.78(1.10) 83.84(0.70) 56.63(2.32) 67.63(2.12) 26.33 S2MVTC (CVPR 24) 76.40(0.00) 67.43(0.00) 88.11(0.00) 92.49(0.00) 75.78(0.00) 81.50(0.00) 0.08 EBMGC-GNF (TPAMI 24) 80.64(0.00) 79.85(0.00) 81.44(0.00) 92.76(0.00) 80.19(0.00) 85.50(0.00) 1.26 s MERA-MVC (TMM 24) 85.92(3.56) 81.52(4.58) 90.88(2.39) 95.67(1.09) 85.58(3.65) 88.38(3.06) 4.22 CFMVC-ETR (ACMMM 23) 90.10(0.00) 84.80(0.00) 96.11(0.00) 97.74(0.00) 89.86(0.00) 90.00(0.00) 5.47 Orth-NTF (ICCV 24) 59.04(0.00) 57.29(0.00) 60.89(0.00) 84.57(0.00) 58.06(0.00) 72.50(0.00) 2.09 SLR-MVC 91.37(1.66) 88.62(2.09) 94.29(1.33) 97.30(0.56) 91.16(1.70) 92.90(1.49) 2.10

BMVC (TPAMI 18) 9.48(0.00) 9.55(0.00) 9.68(0.00) 9.80(0.00) 4.02(0.00) 15.44(0.00) 1.25 AWMVC (AAAI 23) 11.29(0.09) 11.86(0.11) 10.77(0.08) 14.59(0.18) 6.17(0.10) 18.85(0.17) 0.40 FPMVS-CAG (TIP 22) 12.99(0.18) 13.62(0.24) 12.42(0.15) 18.37(0.26) 7.96(0.21) 22.13(0.34) 7.36 S2MVTC (CVPR 24) 48.58(0.00) 40.54(0.00) 60.60(0.00) 60.87(0.00) 44.79(0.00) 58.39(0.00) 0.97 EBMGC-GNF (TPAMI 24) 10.45(0.00) 11.23(0.00) 9.78(0.00) 14.06(0.00) 5.41(0.00) 17.27(0.00) 9.51 s MERA-MVC (TMM 24) 59.85(3.21) 59.54(4.99) 60.29(1.85) 71.24(1.07) 57.37(3.49) 67.84(3.52) 2.09 CFMVC-ETR (ACMMM 23) 60.19(0.00) 60.29(0.00) 60.10(0.00) 75.49(0.00) 57.76(0.00) 68.96(0.00) 3.42 Orth-NTF (ICCV 24) 26.37(0.00) 27.74(0.00) 25.13(0.00) 46.25(0.00) 22.10(0.00) 35.43(0.00) 54.07 SLR-MVC 65.53(1.70) 65.28(1.92) 65.80(1.78) 74.54(0.53) 63.42(1.81) 72.84(2.07) 0.72

BMVC (TPAMI 18) 1.96(0.00) 0.99(0.00) 100.00(0.00) 0.00(0.00) 0.00(0.00) 1.00(0.00) 3.29 AWMVC (AAAI 23) 50.24(1.25) 46.92(1.79) 54.08(0.64) 75.27(0.45) 49.71(1.27) 61.73(1.31) 1.63 FPMVS-CAG (TIP 22) 60.88(1.45) 56.27(1.96) 66.32(0.95) 83.22(0.41) 60.45(1.47) 69.31(1.51) 16.25 S2MVTC (CVPR 24) 52.92(0.00) 38.09(0.00) 86.66(0.00) 86.59(0.00) 52.26(0.00) 50.81(0.00) 1.74 EBMGC-GNF (TPAMI 24) 71.52(0.00) 71.16(0.00) 71.87(0.00) 86.38(0.00) 71.23(0.00) 81.95(0.00) 17.00 s MERA-MVC (TMM 24) 71.96(1.34) 69.96(1.45) 74.08(1.25) 90.81(0.39) 71.67(1.35) 78.42(1.46) 5.73 CFMVC-ETR (ACMMM 23) 68.18(0.00) 61.01(0.00) 77.25(0.00) 89.80(0.00) 67.82(0.00) 74.33(0.00) 31.02 Orth-NTF (ICCV 24) 33.29(0.00) 30.13(0.00) 37.19(0.00) 73.68(0.00) 32.55(0.00) 45.29(0.00) 1229.19 SLR-MVC 81.89(1.19) 79.09(1.60) 84.90(0.80) 93.89(0.26) 81.70(1.20) 85.49(1.51) 7.14

BMVC (TPAMI 18) 47.28(0.00) 51.33(0.00) 43.83(0.00) 33.12(0.00) 34.34(0.00) 56.69(0.00) 6.03 AWMVC (AAAI 23) 38.33(0.00) 33.90(0.00) 44.11(0.00) 25.59(0.00) 18.65(0.00) 45.86(0.00) 6082.89 FPMVS-CAG (TIP 22) - - - - - - - S2MVTC (CVPR 24) 95.86(0.00) 95.91(0.00) 95.81(0.00) 92.23(0.00) 94.73(0.00) 96.50(0.00) 5.59 EBMGC-GNF (TPAMI 24) 38.01(0.00) 41.64(0.00) 34.96(0.00) 30.22(0.00) 22.96(0.00) 47.60(0.00) 9973.23 s MERA-MVC (TMM 24) 90.25(1.36) 91.61(0.41) 88.95(2.26) 89.47(0.87) 87.64(1.69) 88.13(1.42) 2513.41 CFMVC-ETR (ACMMM 23) 97.39(0.00) 98.14(0.00) 96.65(0.00) 94.78(0.00) 96.68(0.00) 97.90(0.00) 6119.14 Orth-NTF (ICCV 24) 51.26(0.00) 57.81(0.00) 46.04(0.00) 59.94(0.00) 39.84(0.00) 55.40(0.00) 363.98 SLR-MVC 99.38(0.00) 99.35(0.00) 99.41(0.00) 98.48(0.00) 99.21(0.00) 99.60(0.00) 780.51

BMVC (TPAMI 18) 5.96(0.00) 4.96(0.00) 7.47(0.00) 12.89(0.00) 3.25(0.00) 9.98(0.00) 10.21 AWMVC (AAAI 23) 4.44(0.06) 4.74(0.07) 4.17(0.05) 10.34(0.15) 2.30(0.06) 9.09(0.12) 249.99 FPMVS-CAG (TIP 22) 5.12(0.10) 4.76(0.07) 5.54(0.16) 11.13(0.13) 2.68(0.09) 8.55(0.13) 645.32 S2MVTC (CVPR 24) 59.32(0.00) 53.44(0.00) 66.66(0.00) 81.93(0.00) 58.24(0.00) 66.96(0.00) 10.80 EBMGC-GNF (TPAMI 24) 4.56(0.00) 2.34(0.00) 99.67(0.00) 0.31(0.00) -0.00(0.00) 3.84(0.00) 2579.66 s MERA-MVC (TMM 24) 84.67(1.63) 87.59(1.64) 81.96(1.91) 91.56(0.46) 84.32(1.67) 86.23(1.79) 344.93 CFMVC-ETR (ACMMM 23) 73.06(0.00) 77.02(0.00) 69.49(0.00) 89.18(0.00) 72.45(0.00) 76.70(0.00) 511.94 Orth-NTF (ICCV 24) OM OM OM OM OM OM OM SLR-MVC 88.01(2.70) 86.54(4.23) 89.61(1.80) 94.00(0.61) 87.72(2.77) 88.62(2.64) 158.13

BMVC (TPAMI 18) 89.57(0.00) 81.76(0.00) 99.03(0.00) 98.42(0.00) 89.46(0.00) 91.27(0.00) 42.22 AWMVC (AAAI 23) 93.12(0.99) 89.17(1.46) 97.45(0.64) 98.69(0.20) 93.05(1.00) 91.92(1.26) 377.81 FPMVS-CAG (TIP 22) 90.64(1.96) 85.24(3.09) 96.81(0.60) 98.27(0.35) 90.54(1.98) 89.98(1.93) 1386.99 S2MVTC (CVPR 24) 85.89(0.00) 78.34(0.00) 95.06(0.00) 96.64(0.00) 85.73(0.00) 86.10(0.00) 6.87 EBMGC-GNF (TPAMI 24) 99.92(0.00) 99.92(0.00) 99.92(0.00) 99.94(0.00) 99.92(0.00) 99.96(0.00) 1404.59 s MERA-MVC (TMM 24) 94.22(0.89) 91.26(1.40) 97.39(0.32) 98.16(0.22) 94.16(0.90) 94.77(0.96) 163.26 CFMVC-ETR (ACMMM 23) 87.81(0.00) 87.14(0.00) 88.48(0.00) 91.60(0.00) 87.68(0.00) 92.48(0.00) 778.61 Orth-NTF (ICCV 24) 36.86(0.00) 35.33(0.00) 38.53(0.00) 70.03(0.00) 36.20(0.00) 46.67(0.00) 12418.21 SLR-MVC 98.36(0.54) 97.04(1.00) 99.72(0.12) 99.65(0.12) 98.34(0.55) 98.36(0.56) 128.02

Table 3: The comparison of clustering results, including mean values (standard deviation), using different methods on six multiview datasets. ( OM indicates out of memory, and - signifies that the algorithm took more than four hours.)

Clustering Performance Analysis Table 3 presents the clustering performance of various methods across six multi-view datasets, evaluated using F-score, Precision, Recall, ARI, NMI, ACC, and CPU time. The best and second-best results for each metric are highlighted in bold and underlined, respectively. Firstly, among pairwise inter-view correlation-based methods, S2MVTC, which considers local smoothness within each view, achieves the best performance on the ORL, CCV, Reuters, and AWA fea datasets. It implies that these datasets exhibit strong local consistency within views. Furthermore, tensor-based MVC methods, such as s MERAMVC, CFMVC-ETR, and SLR-MVC, generally outperform pairwise inter-view methods like BMVC, AWMVC, and FPMVS-CAG, particularly on datasets such as ORL, CCV, and Reuters. This shows the benefits of leveraging highorder inter-view correlations across different views, leading to significantly improved clustering performance. Specifically, compared to CFMVC-ETR, which focuses on enhanced tensor rank and tensorial exclusive regularization to effectively capture inter-view high-order correlations, SLR-MVTC achieves higher ACC performance, with improvements of 2.90%, 3.88%, 11.16%, 11.29%, 1.70%, and 5.88% on the ORL, CCV, ALOI 100, Reuters, AWA fea, and CIFAR100 datasets, respectively. This suggests the significant role of intra-view correlations. When compared to s MERA-MVC, which treats the learning of both interand intra-view correlations equally, SLR-MVTC demonstrates a higher F-score performance with increases of 5.44%, 5.68%, 9.93%, 9.13%, 3.34%, and 4.14% on the same datasets. This indicates that local similarities within views and global highorder correlations across views should be weighted differently. Additionally, while EBMGC-GNF achieves the best performance on CIFAR-100, our method performs similarly but is approximately ten times faster in terms of CPU time. In conclusion, the proposed SLR-MVTC method not only outperforms state-of-the-art methods in clustering performance but also demonstrates significantly better computational efficiency.

Model Discussion Convergence Analysis Fig. 4 illustrates the convergence performance of the proposed SLR-MVTC across six multiview datasets, where our method converges when RE 10 5. It can be observed that RE decreases rapidly during the initial iterations on all datasets, stabilizing after approximately 10 iterations, and eventually approaches zero. Similarly, the ACC increases quickly in the first few iterations and shows minimal changes after 10 iterations. This suggests that within the first 10 iterations, both intra-view local similarity and inter-view global high-order correlations are effectively captured from the multi-view data, enhancing clustering performance. Moreover, the method converges quickly, especially on ORL, CCV, ALOI 100, and Reuters, typically reaching convergence around the 20th iteration.

Ablation Study The newly defined norm U STNN, which calculates the low-rank (LR) approximation of the lowfrequency (LF) components, is crucial in the SLR-MVTC

0 10 20 30 number of iteration

ORL CCV ALOI_100 Reuters Aw A_fea CIFAR100

0 10 20 30 number of iteration

Figure 4: The convergence performance of SLR-MVTC.

model. To further investigate its impact, we conducted two modifications: 1) replacing U STNN with U TNN (resulting in the w/o LF model), and 2) replacing U STNN with U TLFC (resulting in the w/o LR model). The results, shown in Table 4, reveal that on the CCV, ALOI100, Reuters, and AWA-fea datasets, removing the LR component led to a slight decline in clustering performance, while removing the LF component significantly degraded performance. This suggests that inter-view and intra-view correlations have different impacts on clustering, with local smoothness exploration within views being more effective in enhancing clustering outcomes. Furthermore, compared to the S2MVTC model in Table 3, which also considers local smoothness within views, w/o LR performs better. This improvement is likely due to S2MVTC s two-stage learning process, whereas w/o LR directly utilizes learn latent features from multi-view data, making it more efficient.

Datasets w/o LF w/o LR SLR-MVTC

ORL 90.80(2.58) 91.12(1.20) 92.90(1.49) CCV 43.92(0.24) 72.53(3.70) 72.84(2.07) ALOI 100 67.95(2.05) 84.16(1.05) 85.49(1.51) Reuters 43.90(0.02) 96.44(0.00) 99.60(0.00) Aw A fea 74.40(1.69) 88.49(1.13) 88.62(2.64) CIFAR100 89.09(1.45) 91.25(2.67) 98.36(0.56)

Table 4: The accuracy of ablation study on six datasets.

Conclusion In this paper, we focus on directly learning the shared latent features from multi-view data for clustering, where the learned features exhibit local smoothness within views and global low-rank structure across views. By minimizing the STNN norm, which optimizes the low-rank approximation of the low-frequency component on the feature tensor, SLRMVTC effectively captures local similarities within views and global high-order correlations across views. Numerical experiments on six multi-view datasets demonstrate that SLR-MVTC outperforms all state-of-the-art methods.

Acknowledgements This research is supported by the National Natural Science Foundation of China (NSFC) (No.6240011126,

No.62020106011, No.62171088), the Sichuan Science and Technology Program (No.2024NSFSC1473), the Central Guidance for Local Science and Technology Development Fund Project (No.2024ZYD0268) and the China Postdoctoral Science Foundation (No.2024M750356).

References Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; and Eckstein, J. 2011. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine learning, 3(1): 1 122. Braman, K. 2010. Third-order tensors as linear operators on a space of matrices. Linear Algebra and its Applications, 433(7): 1241 1253. Cui, C.; Ren, Y.; Pu, J.; Pu, X.; and He, L. 2023. Deep multi-view subspace clustering with anchor graph. In IJCAI, IJCAI 23. ISBN 978-1-956792-03-4. Geusebroek, J.-M.; Burghouts, G. J.; and Smeulders, A. W. 2005. The Amsterdam library of object images. IJCV, 61: 103 112. He, Z.; Wan, S.; Zappatore, M.; and Lu, H. 2023. A similarity matrix low-rank approximation and inconsistency separation fusion approach for multiview clustering. TAI, 5(2): 868 881. Huang, Z.; Ren, Y.; Pu, X.; and He, L. 2021. Non-linear fusion for self-paced multi-view clustering. In ACMMM, 3211 3219. Huang, Z.; Ren, Y.; Pu, X.; Huang, S.; Xu, Z.; and He, L. 2023. Self-supervised graph attention networks for deep weighted multi-view clustering. In AAAI, volume 37, 7936 7943. Huizing, G.-J.; Deutschmann, I. M.; Peyr e, G.; and Cantini, L. 2023. Paired single-cell multi-omics data integration with Mowgli. Nature Communications, 14(1): 7711. Ji, J.; and Feng, S. 2023. High-order complementarity induced fast multi-view clustering with enhanced tensor rank minimization. In ACMMM, 328 336. Jiang, Y.; Ye, G.; Chang, S.; Ellis, D.; and Loui, A. 2011. Consumer video understanding: A benchmark database and an evaluation of human and machine performance. In Proc. ACM Con. Multimedia Retrieval, 1 8. Khan, A.; and Maji, P. 2019. Approximate graph Laplacians for multimodal data clustering. TPAMI, 43(3): 798 813. Kilmer, M.; Braman, K.; Hao, N.; and Hoover, R. 2013. Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl., 34(1): 148 172. Kilmer, M. E.; and Martin, C. D. 2011. Factorization strategies for third-order tensors. Linear Algebra and its Applications, 435(3): 641 658. Krizhevsky, A.; Hinton, G.; et al. 2009. Learning multiple layers of features from tiny images. Lampert, C. H.; Nickisch, H.; and Harmeling, S. 2009. Learning to detect unseen object classes by between-class attribute transfer. In CVPR, 951 958. IEEE.

Lewis, D. D.; Yang, Y.; Russell-Rose, T.; and Li, F. 2004. Rcv1: A new benchmark collection for text categorization research. JMLR, 5(Apr): 361 397. Li, J.; Gao, Q.; WANG, Q.; Yang, M.; and Xia, W. 2023. Orthogonal Non-negative Tensor Factorization based Multiview Clustering. In Neur IPS, volume 36, 18186 18202. Liu, J.; Liu, X.; Yang, Y.; Liu, L.; Wang, S.; Liang, W.; and Shi, J. 2021. One-pass multi-view clustering for large-scale data. In ICCV, 12344 12353. Liu, J.; Wang, C.; Gao, J.; and Han, J. 2013. Multi-view clustering via joint nonnegative matrix factorization. In Proceedings of the 2013 SIAM international conference on data mining, 252 260. SIAM. Liu, Y.; Chen, J.; Lu, Y.; Ou, W.; Long, Z.; and Zhu, C. 2024. Adaptively topological tensor network for multi-view subspace clustering. TKDE. Liu, Y.; He, L.; Cao, B.; Yu, P.; Ragin, A.; and Leow, A. 2018. Multi-view multi-graph embedding for brain network clustering analysis. In AAAI, volume 32. Long, Z.; Wang, Q.; Ren, Y.; Liu, Y.; and Zhu, C. 2024. S2MVTC: a Simple yet Efficient Scalable Multi-View Tensor Clustering. In CVPR, 26213 26222. Long, Z.; Zhu, C.; Chen, J.; Li, Z.; Ren, Y.; and Liu, Y. 2023. Multi-view MERA Subspace Clustering. TMM, 1 11. Lu, C.; Feng, J.; Chen, Y.; Liu, W.; Lin, Z.; and Yan, S. 2020. Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm. TPAMI, 42(4): 925 938. Lu, Y.; Liu, Y.; Long, Z.; Chen, Z.; and Zhu, C. 2023. Ominus decomposition for multi-view tensor subspace clustering. TAI. Qian, X.; Pei, J.; Zheng, H.; Xie, X.; Yan, L.; Zhang, H.; Han, C.; Gao, X.; Zhang, H.; Zheng, W.; et al. 2021. Prospective assessment of breast cancer risk from multimodal multiview ultrasound images via clinically applicable deep learning. Nature biomedical engineering, 5(6): 522 532. Ren, Y.; Pu, J.; Yang, Z.; Xu, J.; Li, G.; Pu, X.; Philip, S. Y.; and He, L. 2024. Deep clustering: A comprehensive survey. TNNLS. Samaria, F. S.; and Harter, A. C. 1994. Parameterisation of a stochastic model for human face identification. In Proceedings of 1994 IEEE workshop on applications of computer vision, 138 142. IEEE. Wan, X.; Liu, X.; Liu, J.; Wang, S.; Wen, Y.; Liang, W.; Zhu, E.; Liu, Z.; and Zhou, L. 2023. Auto-weighted multi-view clustering for large-scale data. In AAAI, volume 37, 10078 10086. Wang, J.; Tian, F.; Yu, H.; Liu, C. H.; Zhan, K.; and Wang, X. 2017. Diverse non-negative matrix factorization for multiview data representation. TCYB, 48(9): 2620 2632. Wang, S.; Liu, X.; Zhu, X.; Zhang, P.; Zhang, Y.; Gao, F.; and Zhu, E. 2022. Fast Parameter-Free Multi-View Subspace Clustering With Consensus Anchor Guidance. TIP, 31: 556 568.

Wu, D.; Yang, Z.; Lu, J.; Xu, J.; Xu, X.; and Nie, F. 2024. EBMGC-GNF: Efficient Balanced Multi-view Graph Clustering via Good Neighbor Fusion. TPAMI. Xia, W.; Gao, Q.; Wang, Q.; Gao, X.; Ding, C.; and Tao, D. 2022. Tensorized bipartite graph learning for multi-view clustering. TPAMI. Xie, D.; Gao, Q.; Deng, S.; Yang, X.; and Gao, X. 2021. Multiple graphs learning with a new weighted tensor nuclear norm. Neural Networks, 133: 57 68. Xie, Y.; Tao, D.; Zhang, W.; Liu, Y.; Zhang, L.; and Qu, Y. 2018. On unifying multi-view self-representations for clustering by tensor multi-rank minimization. IJCV, 126(11): 1157 1179. Xu, C.; Si, J.; Guan, Z.; Zhao, W.; Wu, Y.; and Gao, X. 2024. Reliable conflictive multi-view learning. In AAAI, volume 38, 16129 16137. Xu, C.; Zhao, W.; Zhao, J.; Guan, Z.; Yang, Y.; Chen, L.; and Song, X. 2023a. Progressive deep multi-view comprehensive representation learning. In AAAI, volume 37, 10557 10565. Xu, J.; Ren, Y.; Tang, H.; Yang, Z.; Pan, L.; Yang, Y.; Pu, X.; Yu, P. S.; and He, L. 2023b. Self-Supervised Discriminative Feature Learning for Deep Multi-View Clustering. TKDE, 35(7): 7470 7482. Yan, W.; Zhang, Y.; Lv, C.; Tang, C.; Yue, G.; Liao, L.; and Lin, W. 2023. Gcfagg: Global and cross-view feature aggregation for multi-view clustering. In CVPR, 19863 19872. Yu, X.; Xu, M.; Zhang, Y.; Liu, H.; Ye, C.; Wu, Y.; Yan, Z.; Zhu, C.; Xiong, Z.; Liang, T.; et al. 2023. Mvimgnet: A large-scale dataset of multi-view images. In CVPR, 9150 9161. Zhang, C.; Jia, X.; Li, Z.; Chen, C.; and Li, H. 2024. Learning Cluster-Wise Anchors for Multi-View Clustering. In AAAI, volume 38, 16696 16704. Zhang, R.; Nie, F.; Li, X.; and Wei, X. 2019. Feature selection with multi-view data: A survey. Information Fusion, 50: 158 167. Zhang, Z.; Liu, L.; Shen, F.; Shen, H. T.; and Shao, L. 2018. Binary multi-view clustering. TPAMI, 41(7): 1774 1782. Zhou, T.; Ruan, S.; and Canu, S. 2019. A review: Deep learning for medical image segmentation using multimodality fusion. Array, 3: 100004.