Tensorized Multi-View Subspace Clustering via Tensor Nuclear Norm and Block Diagonal Representation

Abstract

Recently, a growing number of researchers have focused on multi-view subspace clustering (MSC) due to its potential for integrating heterogeneous data. However, current MSC methods remain challenged by limited robustness and insufficient exploitation of cross-view high-order latent information for clustering advancement. To address these challenges, we develop a novel MSC framework termed TMSC-TNNBDR, a tensorized MSC framework that leverages t-SVD based tensor nuclear norm (TNN) regularization and block diagonal representation (BDR) learning to unify view consistency and structural sparsity. Specifically, each subspace representation matrix is constrained by a block diagonal regularizer to enforce cluster structure, while all matrices are aggregated into a tensor to capture high-order interactions. To efficiently optimize the model, we developed an optimization algorithm based on the inexact augmented Lagrange multiplier (ALM). The TMSC-TNNBDR exhibits both optimized block-diagonal structure and low-rank properties, thereby enabling enhanced mining of latent higher-order inter-view correlations while demonstrating greater resilience to noise. To investigate the capability of TMSC-TNNBDR, we conducted several experiments on certain datasets. Benchmarking on circumscribed datasets demonstrates our method’s superior clustering performance over comparative algorithms while maintaining competitive computational overhead.

1. Introduction

In the field of pattern recognition, subspace clustering (SC) is a very important research topic [1,2,3,4]. In recent decades, scholars have created many subspace clustering algorithms, among which spectral-type methods have shown good performance.

Sparse subspace clustering (SSC) [5] and low-rank representation (LRR) [6], which have achieved significant success, are two typical self-representation subspace clustering methods. Block diagonal structure is a matrix form where non-zero elements are confined to square blocks along the main diagonal, with all elements outside these blocks being zero. Recent studies [7,8] have shown that the block diagonal structure within the learned low-dimensional subspace projection serves the purpose of obtaining the correct clustering results. However, SSC and LRR pursue the block diagonal representation (BDR) matrix indirectly since they only impose nuclear-norm and L1-norm on the subspace representation, respectively. Furthermore, Feng et al. [8] imposed a block diagonal prior on the subspace representation matrices obtained using SSC and LRR, and their clustering performance was improved. However, it is difficult to optimize Feng’s method since the rank constraint is NP-hard. To tackle this problem, Lu et al. [7] developed a simple BDR that relaxes the rank constraint. Compared with Feng’s method, BDR is more easily optimized since it is smooth. Xu et al. [9] developed a learning projective model for BDR to deal with the large-scale subspace clustering problem. Xing et al. [10] proposed an enhanced version of DBSCAN, which is a highly prevalent algorithm in data mining, to improve the clustering process using the block diagonal property of affinity matrices. Meanwhile, Guo et al. [11] put forward a spectral clustering algorithm with BDR for large-scale datasets.

The above-mentioned approaches are single-view-based since they assume that there is only one data source. However, in fact, data are generally sourced from various origins. For instance, one event can be represented by images, text, videos. As such, multi-view clustering (MVC) methods, which often demonstrate a better clustering performance than single-view methods [12,13,14,15,16,17,18,19], are becoming increasingly popular.

The authors of [20] proposed co-training learning to flux the multi-view features. Additionally, the study in [21] investigated the key factors contributing to the success of the co-training method. The co-training learning model is not robust enough against noise pollution, which can lead to error exaggeration. Kumar et al. [17] proposed an MSC framework, in which the clustering hypotheses among views is co-regularized. Graph-based methods are another category MSC methods, which generally use the multiple graph fusion strategy to utilize the information among different views. Sa [22] developed a two-view clustering method, which utilizes different information between two views by constructing bipartite graphs. Moreover, the authors of [13] developed a multi-view spectral clustering algorithm with the help of low-rank and sparse decomposition (RMSC), achieving encouraging success in relation to several real datasets. In the work of Cao et al. [15], the diversity-induced MSC (DiMSC) was presented, leveraging the Hilbert–Schmidt Independence Criterion (HSIC), which plays a key role in utilizing complementary information among different views to enhance the clustering.

By assuming that the different views of an object come from a potential subspace, subspace learning MVC methods can be developed to capture the shared potential subspace. Blaschko and Lampert [23] introduced a novel spectral clustering technique that utilizes canonical correlation analysis (CCA) in its linear and kernel forms for dimensionality reduction. In [24], a low-rank common subspace (LRCS) MVC method is proposed, which can obtain compatible intrinsic information among views by using a common low-rank projection.

These MVC methods shows promising performance in clustering applications; however, they only use paired associations between different views, and may overlook the higher-order associations hidden in multi-view data [12,14,19,25,26,27]. Zhang et al. [12,19] developed a novel multi-view spectral clustering method named LTMSC, incorporating low-rank tensor constraints. In the method, the subspace representations are constructed into a single tensor. It is possible to explore higher-order relationships hidden in the multi-view data. Lu et al. [28] introduced an MSC method with hyper-Laplacian regularization and low-rank tensor constraints (HLR-MSCLRT), which can uncover the local information hidden in the data on the manifold. Nevertheless, the tensor norm employed in both LTMSC and HLR-MSCLRT lacks a clear physical interpretation.

Zhang et al. recently introduced the TNN [29] leveraging the tensor singular value decomposition (t-SVD). The TNN, defined as the summation of singular values, provides a rigorous measure of tensor data low-rankness. In [14], Xie et al. developed a t-SVD based MSC model, namely t-SVD-MSC, which preserves the low-rank property through TNN. With the use of TNN, t-SVD-MSC can more effectively explore the complementary information among all the views [30,31,32,33,34]. Furthermore, in [18], an essential tensor learning method for MSC using a TNN constraint, known as ETLMSC, is proposed. Pan et al. [35] proposed a non-negative non-convex low-rank tensor kernel function in an MSC model (NLRTGC) to reduce the bias from rank. To exploit high-dimensional hidden information, Pan et al. [36] proposed a low-rank fuzzy MSC learning algorithm with the TNN constraint (LRTGFL). Peng et al. [37] designed log-based non-convex functions to approximate tensor rank and tensor sparsity in the Finger-MVC model; these are more precise than the convex ones. Wang et al. [38] integrate noise elimination and subspace learning into a unified MSC framework, holding high-order associations of views constrained by the TNN. Du et al. [39] proposed a robust t-SVD-based multi-view clustering which simultaneously uses low rank and local smooth priors. Luo et al. [40] used an adaptively weighted tensor Schatten-p norm with an adjustable p-value to eliminate the biased estimate of rank.

The optimized BDR structure in affinity matrices inherently encodes cluster information, thereby substantially enhancing clustering efficacy. The low-rank tensor representations intrinsically capture latent high-order correlations across multi-view data through subspace embeddings, resulting in statistically significant clustering improvements. In this paper, inspired by the optimized BDR structure and low-rank tensor representations, we propose a novel MSC method called TMSC-TNNBDR, which integrates the advantages of TNN and BDR. The proposed model imposes BDR constraints on each subspace representation matrix, and all affinity matrices are combined into a tensor regularized by TNN. Finally, an efficient optimization algorithm based on ALM is developed.

The primary contributions of our work are as follows:

• The proposed TMSC-TNNBDR incorporates a BDR regularizer, which promotes a more pronounced block diagonal structure and improves clustering robustness.
• In the TMSC-TNNBDR model, the optimized architecture encodes a TNN constraint, under which TMSC-TNNBDR captures the global structure across all views, thereby effectively exploiting latent complementary information and high-order interactions among views.
• We proposed an ALM optimizer for TMSC-TNNBDR. This approach demonstrates superior clustering performance over comparative algorithms while maintaining competitive computational efficiency.

The remainder of this work is organized as follows. In Section 2, we summarize the notations used and some preliminary definitions. In Section 3, we briefly review two methods, namely the LRR [6] and the BDR [7]. Then, we propose the TMSC-TNNBDR and a solving procedure for TMSC-TNNBDR in Section 4. Subsequently we documented the experimental findings in Section 5. Ultimately, we conclude our work in Section 6.

2. Notations and Preliminaries

2.1. Notations

For a clear explanation of TMSC-TNNBDR, We summarize the notations in

Table 1. Notations summarized.

Notation	Description	Notation	Description
u	A scalar	U	A constant
u	A column vector	ui	The ith column of a matrix
U	A matrix	$\mathcal{U}$	A tensor
UT	The matrix transpose	$\mathcal{U}$T	The tensor transpose
U(v)	The vth frontal slice (matrix) of a tensor	$\mathcal{U}$(v)	The vth frontal slice (matrix) of $\mathcal{U}$ (tensor)
${‖U‖}_{\ast }$	The nuclear norm	${‖\mathcal{U}‖}_{TNN}$	The t-SVD-based TNN
${‖U‖}_{2,1}$	L2,1-norm	$\mathcal{I}$	The identity tensor
${‖U‖}_F$	The Frobenius norm	$\mathcal{U}\ast \mathcal{V}$	The t-product
${‖U‖}_{\boxed{k}}$	The k-block diagonal regularizer	$〈U, V〉$	The Frobenius inner product
$diag(U)$	Line up the diagonal elements of U as a column vector	$Diag(w)$	Expand w (column vector) into a diagonal matrix

. Bold calligraphy letters, e.g., $\mathcal{U}$, are deployed to denote tensors. Bold upper case letters, e.g., W, are deployed to denote matrices. Bold lower case letters, e.g., u, are deployed to denote vectors. Lower case letters, e.g., $u_{ij}$, are deployed to denote the entries. Among others, $1$ is assigned to denote the column vector of one. The diagonal elements of matrix W line up as a column vector, which is referred to as $diag(W)$. The column vector w is expanded into a diagonal matrix, which is labeled as $Diag(w)$.

Let ${‖W‖}_{\ast }$ denote the nuclear norm operator of matrix W, i.e., ${‖W‖}_{\ast }={\displaystyle {\sum }_i{\sigma }_i(W)}$, in which ${\sigma }_i(W)$ is the ith largest singular value of W. Assuming that the singular value decomposition of matrix W is expressed as $W=U\Sigma V^T$, then $D_{\tau }(W)$ denotes the singular-value thresholding operator applied to matrix W with boundary value $\tau$, i.e.,$D_{\tau }(W)=U{\Sigma }_{\tau }{V}^T$, where ${\Sigma }_{\tau }=diag\left\{\max ({\sigma }_i(W)-\tau ,0\right\}$.

Matrices extend naturally to tensors, which are multidimensional arrays. Mathematically, a matrix is a two-way tensor. Suppose $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$ is a three-way tensor, $\mathcal{Y}$ is likely to be regarded as the stack of matrices. Some block-based operators [31] for $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$ are defined as follows:

$$bcirc(\mathcal{Y})=\left[\begin{array}{cccc}{\mathcal{Y}}^{(1)}& {\mathcal{Y}}^{(n_3)}& \cdots & {\mathcal{Y}}^{(2)}\\ {\mathcal{Y}}^{(2)}& {\mathcal{Y}}^{(1)}& \cdots & {\mathcal{Y}}^{(n_3)}\\ ⋮& \ddots & \ddots & ⋮\\ {\mathcal{Y}}^{(n_3)}& {\mathcal{Y}}^{(n_3-1)}& \cdots & {\mathcal{Y}}^{(1)}\end{array}\right]$$

(1)

$$bvec(\mathcal{Y}):=\left(\begin{array}{c}{\mathcal{Y}}^{(1)}\\ {\mathcal{Y}}^{(2)}\\ ⋮\\ {\mathcal{Y}}^{(n_3)}\end{array}\right), \mathit{bvfold}(bvec(\mathcal{Y}))=\mathcal{Y}$$

(2)

$$\begin{array}{l}bdiag(\mathcal{Y}):=\left[\begin{array}{ccc}{\mathcal{Y}}^{(1)}& & \\ & \ddots & \\ & & {\mathcal{Y}}^{(n_3)}\end{array}\right],\\ \\ \mathit{bdfold}(bdiag(\mathcal{Y}))=\mathcal{Y}\end{array}$$

(3)

2.2. Preliminaries

We introduced some preliminary definitions [31] as follows:

Definition 1.

Suppose $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$ and $\mathcal{Z}\in {ℝ}^{n_2\times n_4\times n_3}$; the t-product $\mathcal{Y}\ast \mathcal{Z}$ is $\mathcal{M}\in {ℝ}^{n_1\times n_4\times n_3}$, i.e.,

$$\mathcal{M}=\mathcal{Y}\ast \mathcal{Z}=:\mathit{bvfold}\{bcirc(\mathcal{Y})bvec(\mathcal{Z})\}$$

(4)

Definition 2.

Suppose $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$; then, the tensor transpose of $\mathcal{Y}$ is ${\mathcal{Y}}^T\in {ℝ}^{n_2\times n_1\times n_3}$.

Definition 3.

Suppose $\mathcal{I}\in {ℝ}^{n_1\times n_1\times n_2}$; then, $\mathcal{I}$ is an identity tensor while its first frontal slice is a unit matrix ($I\in {ℝ}^{n_1\times n_1}$) and all others are zeros.

Definition 4.

If $\mathcal{P}\in {ℝ}^{n_1\times n_2\times n_3}$ satisfies

$${\mathcal{P}}^T\ast \mathcal{P}=\mathcal{P}\ast {\mathcal{P}}^T=\mathcal{I}$$

(5)

Then, $\mathcal{P}$ is an orthogonal tensor.

Definition 5.

Suppose $\mathcal{P}\in {ℝ}^{n_1\times n_2\times n_3}$; we define $\mathcal{P}$ as an f-diagonal tensor while all its frontal slices are diagonal.

Definition 6.

Suppose the tensor $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$; then, the t-SVD is defined as follows:

$$\mathcal{Y}=\mathcal{U}\ast \mathcal{L}\ast {\mathcal{V}}^T$$

(6)

where $\mathcal{L}, \mathcal{U}, \mathcal{V}\in {ℝ}^{n_1\times n_2\times n_3}$, $\mathcal{L}$ is f-diagonal, while $\mathcal{U}$ and $\mathcal{V}$ are both orthogonal.

Definition 7.

Suppose $\mathcal{Y}\in {ℝ}^{n_1\times n_2\times n_3}$; then, the TNN of $\mathcal{Y}$, i.e., ${‖\mathcal{Y}‖}_{TNN}$, is the summation of the singular values which are decomposed by t-SVD. It has been proven that t-SVD based TNN is the tightest convex relaxation to tensor tubal rank [29].

3. Related Work

Before presenting our method, this section establishes the theoretical foundation by first reviewing two classical methods: LRR [6] and BDR [7].

3.1. Low-Rank Representation (LRR)

Let us assume that $X=[x_1,x_2,\dots ,x_N]\in {ℝ}^{d\times N}$ is a set of N data points and the dimensionality of the data is d. LRR seeks to find a low-rank factorization of samples for clustering. The objective of LRR can be formulated as follows:

$$\begin{array}{l}\underset{Z,E}{\min }\lambda {‖E‖}_{2,1}+{‖Z‖}_{\ast }\\ s.t.X=XZ+E\end{array}$$

(7)

where $Z=[z_1,z_2,\dots ,z_N]\in {ℝ}^{N\times N}$ is the representation of dataset X, E refers to the approximation error, ${‖\cdot ‖}_{2,1}$ refers to the $L_{2,1}$-norm, and ${‖\cdot ‖}_{\ast }$ refers to the nuclear norm.

LRR executes spectral clustering via the affinity matrix W, where $W={\displaystyle \frac{\left|Z\right|+{\left|Z\right|}^T}{2}}$.

3.2. Block Diagonal Representation (BDR)

The authors of [7] provide the following block diagonal regularizer to chase the optimal representation.

Definition 8.

The k-block diagonal regularizer of the affinity matrix $W\in {ℝ}^{N\times N}$ can be formulated as follows:

$${‖W‖}_{\boxed{k}}={\displaystyle \sum _{i=N-k+1}^N{\delta }_i(L_W)}$$

(8)

where $L_W=Diag(W1)-W$, i.e., $L_W$ is the Laplacian matrix of W. ${\delta }_i(L_W)$ refers to the ith eigenvalue of $L_W$ and is in decreasing order. k is the number of subspaces. The loss function of BDR is defined as follows

$$\begin{array}{l}\underset{Z, W}{\min }{\displaystyle \frac{1}{2}}{‖X-XZ‖}_F^2+{\displaystyle \frac{\lambda }{2}}{‖Z-W‖}_F^2+\gamma {‖W‖}_{\boxed{k}}\\ s.t.diag(W)=0,W\ge 0,W=W^T\end{array}$$

(9)

The affinity matrix W is also defined as $W={\displaystyle \frac{\left|Z\right|+{\left|Z\right|}^T}{2}}$.

4. The Proposed TMSC-TNNBDR

In this section, we introduce the TMSC-TNNBDR framework that extends classical LRR and BDR approaches. Subsequently, we derive an ALM-based optimization scheme to solve the resulting non-convex problem.

4.1. Problem Formulation

Let $X^{(v)}=[x_1^{(v)},x_2^{(v)},\dots ,x_N^{(v)}]\in {ℝ}^{d\times N}$ and $H^{(v)}=[h_1^{(v)},h_2^{(v)},\dots ,h_N^{(v)}]\in {ℝ}^{N\times N}$ be, respectively, the feature matrix and subspace coefficient for the vth view. The loss function of TMSC-TNNBDR is demonstrated as follows:

$$\begin{array}{l}\underset{H^{(v)},B^{(v)}}{\min }{\displaystyle \sum _{v=1}^V\left(\frac{\lambda }{2}{‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2+\frac{\alpha }{2}{‖H^{(v)}-B^{(v)}‖}_F^2+\gamma {‖B^{(v)}‖}_{\boxed{k}}\right)}+{‖\mathcal{H}‖}_{TNN}\\ s.t.\mathcal{H}=\Psi (H^{(1)},H^{(2)},\dots ,H^{(V)}), diag(B^{(v)})=0,B^{(v)}\ge 0,B^{(v)}=B^{(v)T}\end{array}$$

(10)

where $\Psi (\cdot )$ represents an function that stacks all $H^{(v)}$(v = 1, 2, …, V) into a tensor in ${ℝ}^{N\times N\times V}$ then applying a rotation transformation to $\mathcal{H}\in {ℝ}^{N\times V\times N}$.

In Equation (10), ${‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2$ is the self-representation reconstruction error, ${‖B^{(v)}‖}_{\boxed{k}}$ denotes the BDR constraint to $B^{(v)}$, ${‖\mathcal{H}‖}_{TNN}$ denotes TNN low-rank constraint to $\mathcal{H}$, and ${‖H^{(v)}-B^{(v)}‖}_F^2$ can be seen as a Robust PCA term to remove the noise contained in the H^(v). Moreover, $\lambda$, $\alpha$, and $\gamma$ are tunable hyperparameters.

4.2. Optimization

The loss function of TMSC-TNNBDR, i.e., Equation (10), can be optimized through the ALM. The theorem relating to ${‖B^{(v)}‖}_{\boxed{k}}$ is described as follows:

Theorem 1 

([41]). Suppose $L\in {ℝ}^{n\times n}$, where L is semi-positive; then, the following holds:

$$\begin{array}{l}{\displaystyle \sum _{i=n-k+1}^n{\lambda }_i(L)=\underset{W}{\min }〈L,W〉}\\ s.t.0\preccurlyeq W\preccurlyeq I,tr(W)=k\end{array}$$

(11)

In accordance with Theorem 1, Equation (10) can be rewritten as Equation (12):

$$\begin{array}{l}\underset{H^{(v)},W^{(v)},B^{(v)}}{\min }{\displaystyle \sum _{v=1}^V\left(\frac{\lambda }{2}{‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2\right.}+{\displaystyle \frac{\alpha }{2}}{‖H^{(v)}-B^{(v)}‖}_F^2+\left.\gamma 〈Diag(B^{(v)}1)-B^{(v)},W^{(v)}〉\right)+{‖\mathcal{H}‖}_{TNN}\\ s.t.\mathcal{H}=\Psi (H^{(1)},H^{(2)},\dots ,H^{(V)}), diag(B^{(v)})=0,B^{(v)}\ge 0,B^{(v)}=B^{(v)T}, 0\preccurlyeq W^{(v)}\preccurlyeq I, tr(W^{(v)})=k\end{array}$$

(12)

To solve Equation (12), an auxiliary tensor variable $\mathcal{G}$ is introduced to replace $\mathcal{H}$. Then, the loss function of TMSC-TNNBDR is converted into the following:

$$\begin{array}{l}\underset{H^{(v)},W^{(v)},B^{(v)},\mathcal{G}}{\min }{\displaystyle \sum _{v=1}^V\left(\frac{\lambda }{2}{‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2\right.}+{\displaystyle \frac{\alpha }{2}}{‖H^{(v)}-B^{(v)}‖}_F^2+\left.\gamma 〈Diag(B^{(v)}1)-B^{(v)},W^{(v)}〉\right)+{‖\mathcal{G}‖}_{TNN}\\ s.t.\mathcal{G}=\mathcal{H}, \mathcal{H}=\Psi (H^{(1)},H^{(2)},\dots ,H^{(V)}), \mathcal{G}=\Psi (G^{(1)},G^{(2)},\dots ,G^{(V)}),\\ diag(B^{(v)})=0, B^{(v)}\ge 0, B^{(v)}=B^{(v)T}, 0\preccurlyeq W^{(v)}\preccurlyeq I, tr(W^{(v)})=k\end{array}$$

(13)

Equation (13) will be converted to the augmented Lagrangian formula, as follows:

$$\begin{array}{l}\mathcal{L}(H^{(1)},\dots ,H^{(V)};W^{(1)},\dots ,W^{(V)};B^{(1)},\dots ,B^{(V)};\mathcal{G})\\ ={\displaystyle \sum _{v=1}^V\left(\frac{\lambda }{2}{‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2+\frac{\alpha }{2}{‖H^{(v)}-B^{(v)}‖}_F^2\right.}+\left.\gamma 〈Diag(B^{(v)}1)-B^{(v)},W^{(v)}〉\right)\\ +{‖\mathcal{G}‖}_{TNN}+\left(\mathcal{P},\mathcal{H}-\mathcal{G}\right)+{\displaystyle \frac{\rho }{2}}{‖\mathcal{H}-\mathcal{G}‖}_F^2\\ s.t.diag(B^{(v)})=0, B^{(v)}\ge 0, B^{(v)}=B^{(v)T}, 0\preccurlyeq W^{(v)}\preccurlyeq I, tr(W^{(v)})=k\end{array}$$

(14)

where $\mathcal{P}$ denotes the Lagrange multiplier; $\rho$ is actually the penalty parameter.

We get the resolutions to $H^{(v)}$, $W^{(v)}$, $B^{(v)}$, and $\mathcal{G}$ by solving each variable alternately in Equation (14). The steps are described as follows:

$H^{(v)}$-subproblem: For computing $H^{(v)}$, we fix the other variables and tackle the following problem:

$$\begin{array}{l}{H}^{(v)\ast }=\arg \underset{H^{(v)}}{\min }{\displaystyle \frac{\lambda }{2}}{‖X^{(v)}-X^{(v)}{H}^{(v)}‖}_F^2+{\displaystyle \frac{\alpha }{2}}{‖H^{(v)}-B^{(v)}‖}_F^2+\left(P^{(v)},H^{(v)}-G^{(v)}\right)\\ +{\displaystyle \frac{\rho }{2}}{‖H^{(v)}-G^{(v)}‖}_F^2\end{array}$$

(15)

Differentiating by $H^{(v)}$, we can obtain the following:

$$H^{(v)\ast }={\left(\lambda X^{(v)T}{X}^{(v)}+(\alpha +\rho )I\right)}^{-1}\left(\lambda X^{(v)T}{X}^{(v)}+\alpha B^{(v)}+\rho G^{(v)}-P^{(v)}\right)$$

(16)

$W^{(v)}$-subproblem: $W^{(v)}$ will be computed as follows:

$$\begin{array}{l}{W}^{(v)\ast }=\underset{W^{(v)}}{\mathrm{argmin}}〈Diag(B^{(v)}1)-B^{(v)},W^{(v)}〉\\ s.t.0\preccurlyeq W^{(v)}\preccurlyeq I,tr(W^{(v)})=k\end{array}$$

(17)

For Equation (17), $W^{(v)\ast }=U{U}^T$, where $U\in {ℝ}^{N\times k}$ is a matrix concatenated from k eigenvectors that correspond to the k smallest eigenvalues of $Diag(B^{(v)}1)-B^{(v)}$ [7].

$B^{(v)}$-subproblem: $B^{(v)}$ can be computed as follows:

$$\begin{array}{l}{B}^{(v)\ast }=\arg \underset{B^{(v)}}{\min }{\displaystyle \frac{\alpha }{2}}{‖H^{(v)}-B^{(v)}‖}_F^2+\gamma 〈Diag(B^{(v)}1)-B^{(v)},W^{(v)}〉\\ s.t.diag(B^{(v)})=0,B^{(v)}\ge 0,B^{(v)}=B^{(v)T}\end{array}$$

(18)

Equation (18) can be converted into the following:

$$\begin{array}{l}{B}^{(v)\ast }=\arg \underset{B^{(v)}}{\min }{\displaystyle \frac{1}{2}}{‖B^{(v)}-H^{(v)}+{\displaystyle \frac{\alpha }{\gamma }}\left(diag(W^{(v)})1^T-W^{(v)}\right)‖}_F^2\\ s.t.diag(B^{(v)})=0,B^{(v)}\ge 0,B^{(v)}=B^{(v)T}\end{array}$$

(19)

The theorem in [7] enables the solution of Equation (19).

$\mathcal{G}$-subproblem: We fixed the other variables and update $\mathcal{G}$ as follows:

$${\mathcal{G}}^{\ast }=\arg \underset{\mathcal{G}}{\min }{‖\mathcal{G}‖}_{TNN}+{\displaystyle \frac{\rho }{2}}{‖\mathcal{G}-\left(\mathcal{H}+{\displaystyle \frac{\mathcal{P}}{\rho }}\right)‖}_F^2$$

(20)

The solution to Equation (20) can be obtained using the theorem in [14,29].

$\mathcal{P}$-subproblem: the Lagrange multiplier $\mathcal{P}$ can be updated as follows:

$${\mathcal{P}}^{\ast }=\mathcal{P}+\mu \left(\mathcal{H}-\mathcal{G}\right),$$

(21)

Finally, the TMSC-TNNBDR procedure is outlined in Algorithm 1.

Algorithm 1 TMSC-TNNBDR

Input: $X^{(1)},X^{(2)},\dots ,X^{(v)}$, $\lambda$, $\alpha$,$\gamma$ and cluster number k; Output: Clustering result Initialize:   $H^{(v)}=0$, $W^{(v)}=0$,$B^{(v)}=0$, $v=1,\dots ,V$, $\mathcal{G}=\mathcal{P}=0$, $\mu ={10}^{-5}$, $\rho ={10}^{-4}$, $\epsilon ={10}^{-7}$,   ${\mu }_{\max }={\rho }_{\max }={10}^{10}$ While not converged do   for $v=1,\dots ,V$ do      Update $H^{(v)}$ in accordance with Equation (16);      Update $W^{(v)}$ in accordance with Equation (17);      Update $B^{(v)}$ in accordance with Equation (19)   end   Update $\mathcal{G}$ in accordance with Equation (20);   Update $\mathcal{P}$ in accordance with Equation (21);   Update $\mu$ by $\mu =\min (\rho \mu ;{\mu }_{\max })$;   Check the convergence conditions: ${‖H^{(v)}-G^{(v)}‖}_{\infty }<\epsilon$. end Let $S={\displaystyle \frac{1}{V}}{\displaystyle {\sum }_{v=1}^Z\left|H^{(v)}\right|+\left|H^{{(v)}^T}\right|}$; Perform spectral clustering on S.

4.3. Computational Complexity and Convergence

To calculate $H^{(v)}$, it involves matrix multiplication and matrix inversion, whose complexities are $O(d{N}^2)$ and $O(N^3)$, respectively. For computing $W^{(v)}$, its complexity is $O(N^3+k{N}^2)$ because the main computational burdens are eigenvalue decomposition and matrix product. The computation of $B^{(v)}$ is $O(N^3)$ since it mainly depends on matrix multiplication. As for computing $\mathcal{G}$, the computational complexity is $O(N^{2}V\log (N))$. Thereafter, the total complexity of TMSC-TNNBDR is $O(V(d{N}^2+N^3)+N^{2}V\log (N))$.

The procedure of TMSC-TNNBDR is non-convex, which means it cannot achieve a global optimal solution. Nevertheless, TMSC-TNNBDR can converge to a local optimal point. In fact, each variable in Algorithm 1 has a closed-form solution. Following this, the value of the loss function decreases monotonically and remains bounded below. Clustering experiments are performed on some classic datasets, and the results showed that the TMSC-TNNBDR could converge stably.

5. Experimental Results

To evaluate the performance of the TMSC-TNNBDR model, we conduct experiments on abovementioned five image datasets. We compare TMSC-TNNBDR with some representative single-view-based methods, including SPC_best, LRR_best, BDR_best; multi-view-based methods, including Co-Reg SPC, RMSC, LTMSC, and DiMSC; and TNN-based method, i.e., t-SVD-MSC.

• SPC_best: SPC_best is a single-view clustering model using spectral clustering [42] to reach the best capability in all views.
• LRR_best: LRR_best is a single-view clustering model using LRR [6] to reach the best capability in all views.
• BDR_best: BDR_best is a single-view clustering model using BDR [7] to reach the best capability in all views.
• Co-Reg SPC [17]: A co-regularized MVSC method.
• RMSC [13]: RMSC is a multi-view spectral clustering algorithm with the help of low-rank and sparse de-composition.
• LTMSC [12]: LTMSC is a multi-view spectral clustering method incorporating low-rank tensor constraints.
• DiMSC [15]: DiMSC is a multi-view model utilizing HSIC to enhance diversity.
• t-SVD-MSC [14]: A MSC model using t-SVD.

5.1. Description of the Dataset

UCI-Digits: (https://archive.ics.uci.edu/dataset/80/optical+recognition+of+handwritten+digits (accessed on 17 August 2025)). There are a total of 2000 digit images that correspond to 10 classes in this dataset. Similarly to the work in [13], three feature types—morphological features, Fourier coefficients, and pixel averages—are extracted to assemble multi-view data. Ten UCI-Digits samples are shown in Figure 1.

ORL: (https://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html (accessed on 17 August 2025)). There are a total of 400 face images in ORL, which correspond to 40 people. Firstly, we zoom each image to a resolution of 64 × 64. Then, similarly to the work of [12,14], we retrieved LBP [43], intensity, and Gabor [44] features to formulate multi-view data. Some ORL samples are shown in Figure 2.

Yale: (https://gitcode.com/open-source-toolkit/885dd (accessed on 17 August 2025)). There are 165 face images in Yale, which correspond to 11 individuals. We zoom each image to a resolution of $64\times 64$. Similarly to the work of [12,14], we retrieved LBP [43], intensity, and Gabor [44] features in order to formulate multi-view data. Some Yale samples are shown in Figure 3.

Extended YaleB: (https://gitcode.com/open-source-toolkit/3d6b2 (accessed on 17 August 2025)). This dataset consists of pictures of 38 people; each individual has about 64 images under different illuminations. Similarly to the work in [12,14], we retrieved LBP [43], intensity, and Gabor [44] features to formulate multi-view data. We zoom each image to a resolution of 32 × 32. Ten examples of YaleB are shown in Figure 4.

COIL-20: (https://www.cs.columbia.edu/CAVE/software/softlib/coil-20.php (accessed on 17 August 2025)). There are a total of 1440 pictures in COIL-20, which are associated with 40 object classes. We zoom each picture to a resolution of 32 × 32. Similarly to the work in [12,14], we retrieved LBP [43], intensity, and Gabor [44] features to formulate multi-view data. Some COIL-20 samples are shown in Figure 5.

5.2. Evaluation Metrics

We employ ACC and NMI to compare the performance of various algorithms.

ACC, i.e., accuracy, is defined as follows:

$$\mathrm{ACC}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\delta (gn{d}_i,map(r_i))}}{n}}$$

(22)

where $gn{d}_i$ refers to the true category label of the i-th sample, $map(r_i)$ refers to clustering label, and $\delta (x,y)$ refers to Kronecker delta, which is defined as follows:

$$\delta (x,y)=\left\{\begin{array}{cc}1 ,\hfill & if x=y\\ 0 ,& otherwise\end{array}\right.$$

(23)

MI, i.e., mutual information, is defined as follows:

$$\mathrm{MI}(\mathit{D},\mathit{D}\prime )={\displaystyle \sum _{d_i\in D,d_j^{\prime }\in D\prime }p(d_i,d_j^{\prime })\log \frac{p(d_i{d}_j^{\prime })}{p(d_i)p(d_j^{\prime })}}$$

(24)

where D and $\mathit{D}\prime$ refer to two clusters. $p(d_i)$ and $p(d_j^{\prime })$ refer to the probabilities of belonging to D and $D\prime$. Correspondingly, $p(d_i,d_j^{\prime })$ refers to the joint probability.

NMI, i.e., normalize mutual information, is defined as follows:

$$\mathrm{NMI}(\mathit{D},\mathit{D}\prime )={\displaystyle \frac{MI(\mathit{D},\mathit{D}\prime )}{\max (H(\mathit{D}),H(\mathit{D}\prime ))}}$$

(25)

where $H(\cdot )$ is the entropy of the dataset.

5.3. Experiment Results

Notably, all evaluated clustering methods—specifically TMSC-TNNBDR, LRR_best, SPC_best, BDR_best, t-SVD-MSC, DiMSC, Co-Reg SPC, RMSC, and LTMSC—incorporate K-means procedures in the final step of spectral clustering. Randomness in K-means stems from both the random choice of initial cluster centers and the non-convex objective function. Twenty repetitions typically represent a sound compromise to mitigate the effects of randomness while maintaining computational feasibility. Therefore, we report median performance metrics after 20 experimental repetitions to ensure statistical robustness. Table 2, Table 3, Table 4, Table 5 and Table 6 report the clustering results from five public databases: UCI-digits, ORL, Yale, YaleB, COIL-20. Bold values in the tables indicate the best. The results clearly indicate that TMSC-TNNBDR exceeds the performance of other comparison algorithms. Our observation and analysis are delineated as follows:

• In comparative analyses of single-view clustering methodologies, LRR and BDR consistently outperform conventional spectral clustering (SPC). This performance advantage likely stems from their enhancement of the SPC framework through the incorporation of prior structural knowledge—specifically, low-rank constraints and block-diagonal regularization, respectively.
• Multi-view methods, including TMSC-TNNBDR, demonstrated a superior performance compared with single-view approaches. Our experimental results demonstrate that even selecting the best outcome from all individual single-view clustering procedures still underperforms multi-view clustering in the vast majority of cases. This confirms the superiority of multi-view clustering. It is generally believed that the success of existing multi-view clustering methods involve learning latent cross-view correlations, discovering underlying patterns, and integrating this summarized prior knowledge into clustering models. For instance, Co-Reg SPC embeds co-regularization of clustering consensus into spectral clustering. Similarly, RMSC incorporates both low-rank tensor and sparse constraints into the MSC framework. DiMSC leverages the Hilbert Schmidt Independence Criterion (HSIC) to extract complementary information across views, while LTMSC and t-SVD-MSC employ TNN constraint to explore such complementary information.
• In most cases, multi-view clustering based on tensor representation outperforms co-regularization-based Co-Reg SPC in clustering performance. This superiority is generally attributed to tensor representation’s ability to integrate multiple views into a unified structure, easily capturing complementary information and high-order interactions across views. Experimental results validate this explanation.
• TMSC-TNNBDR achieves superior performance compared to benchmark methods while maintaining favorable time efficiency. This advantage is likely attributed to the complementary interplay between the low-rank property and block-diagonal structure of the similarity matrix—where Tensor Nuclear Norm (TNN) and Block Diagonal Regularization (BDR), as two priori constraints, synergistically enhance the multi-view subspace clustering framework from distinct perspectives.

Table 2. Clustering results (mean ± standard deviation) on UCI-Digits.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.968 ± 0.001	0.740 ± 0.021	0.814 ± 0.006
NMI	0.769 ± 0.002	0.639 ± 0.013	0.765 ± 0.005
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.953 ± 0.001	0.719 ± 0.013	0.786 ± 0.007
NMI	0.929 ± 0.002	0.776 ± 0.008	0.801 ± 0.004
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.776 ± 0.008	0.912 ± 0.003	0.994 ± 0.001
NMI	0.791 ± 0.003	0.923 ± 0.002	0.984 ± 0.001

Table 2. Clustering results (mean ± standard deviation) on UCI-Digits.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.968 ± 0.001	0.740 ± 0.021	0.814 ± 0.006
NMI	0.769 ± 0.002	0.639 ± 0.013	0.765 ± 0.005
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.953 ± 0.001	0.719 ± 0.013	0.786 ± 0.007
NMI	0.929 ± 0.002	0.776 ± 0.008	0.801 ± 0.004
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.776 ± 0.008	0.912 ± 0.003	0.994 ± 0.001
NMI	0.791 ± 0.003	0.923 ± 0.002	0.984 ± 0.001

Table 3. Clustering results (mean ± standard deviation) on ORL.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.773 ± 0.003	0.726 ± 0.025	0.848 ± 0.003
NMI	0.895 ± 0.006	0.884 ± 0.002	0.938 ± 0.002
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.973 ± 0.003	0.837 ± 0.001	0.715 ± 0.000
NMI	0.992 ± 0.002	0.939 ± 0.003	0.853 ± 0.003
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.735 ± 0.006	0.793 ± 0.008	1.000 ± 0.000
NMI	0.873 ± 0.011	0.932 ± 0.993	1.000 ± 0.000

Table 3. Clustering results (mean ± standard deviation) on ORL.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.773 ± 0.003	0.726 ± 0.025	0.848 ± 0.003
NMI	0.895 ± 0.006	0.884 ± 0.002	0.938 ± 0.002
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.973 ± 0.003	0.837 ± 0.001	0.715 ± 0.000
NMI	0.992 ± 0.002	0.939 ± 0.003	0.853 ± 0.003
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.735 ± 0.006	0.793 ± 0.008	1.000 ± 0.000
NMI	0.873 ± 0.011	0.932 ± 0.993	1.000 ± 0.000

Table 4. Clustering results (mean ± standard deviation) on Yale.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.703 ± 0.002	0.634 ± 0.015	0.712 ± 0.004
NMI	0.706 ± 0.012	0.646 ± 0.009	0.716 ± 0.002
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.878 ± 0.013	0.703 ± 0.004	0.668 ± 0.002
NMI	0.913 ± 0.009	0.728 ± 0.009	0.715 ± 0.003
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.639 ± 0.038	0.743 ± 0.003	0.992 ± 0.002
NMI	0.685 ± 0.029	0.759 ± 0.09	0.994 ± 0.001

Table 4. Clustering results (mean ± standard deviation) on Yale.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.703 ± 0.002	0.634 ± 0.015	0.712 ± 0.004
NMI	0.706 ± 0.012	0.646 ± 0.009	0.716 ± 0.002
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.878 ± 0.013	0.703 ± 0.004	0.668 ± 0.002
NMI	0.913 ± 0.009	0.728 ± 0.009	0.715 ± 0.003
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.639 ± 0.038	0.743 ± 0.003	0.992 ± 0.002
NMI	0.685 ± 0.029	0.759 ± 0.09	0.994 ± 0.001

Table 5. Clustering results (mean ± standard deviation) on Extended YaleB.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.447 ± 0.023	0.283 ± 0.035	0.464 ± 0.012
NMI	0.408 ± 0.032	0.225 ± 0.043	0.432 ± 0.007
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.568 ± 0.003	0.470 ± 0.007	0.240 ± 0.001
NMI	0.605 ± 0.002	0.397 ± 0.006	0.148 ± 0.001
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.223 ± 0.011	0.626 ± 0.009	0.641 ± 0.002
NMI	0.161 ± 0.021	0.621 ± 0.005	0.631 ± 0.004

Table 5. Clustering results (mean ± standard deviation) on Extended YaleB.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.447 ± 0.023	0.283 ± 0.035	0.464 ± 0.012
NMI	0.408 ± 0.032	0.225 ± 0.043	0.432 ± 0.007
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.568 ± 0.003	0.470 ± 0.007	0.240 ± 0.001
NMI	0.605 ± 0.002	0.397 ± 0.006	0.148 ± 0.001
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.223 ± 0.011	0.626 ± 0.009	0.641 ± 0.002
NMI	0.161 ± 0.021	0.621 ± 0.005	0.631 ± 0.004

Table 6. Clustering results (mean ± standard deviation) on COIL-20.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.767 ± 0.002	0.682 ± 0.024	0.805 ± 0.002
NMI	0.870 ± 0.003	0.769 ± 0.011	0.872 ± 0.003
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.803 ± 0.004	0.774 ± 0.014	0.720 ± 0.007
NMI	0.865 ± 0.003	0.846 ± 0.002	0.809 ± 0.005
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.687 ± 0.043	0.802 ± 0.009	0.823 ± 0.004
NMI	0.802 ±0.016	0.853 ± 0.005	0.892 ± 0.003

Table 6. Clustering results (mean ± standard deviation) on COIL-20.

Algorithm	LRRbest	SPCbest	BDRbest
ACC	0.767 ± 0.002	0.682 ± 0.024	0.805 ± 0.002
NMI	0.870 ± 0.003	0.769 ± 0.011	0.872 ± 0.003
Algorithm	t-SVD-MSC	DiMSC	Co-Reg SPC
ACC	0.803 ± 0.004	0.774 ± 0.014	0.720 ± 0.007
NMI	0.865 ± 0.003	0.846 ± 0.002	0.809 ± 0.005
Algorithm	RMSC	LTMSC	TMSC-TNNBDR
ACC	0.687 ± 0.043	0.802 ± 0.009	0.823 ± 0.004
NMI	0.802 ±0.016	0.853 ± 0.005	0.892 ± 0.003

To evaluate the actual running time of our method, we conducted experiments on the aforementioned five datasets. For fairness in comparison, single-view clustering approaches (e.g., LRR_best, SPC_best, BDR_best) are intentionally excluded.

Table 7. Computational time (unit: seconds) of the comparative multi-view methods.

Datasets	Algorithms
	t-SVD-MSC	DiMSC	Co-Reg SPC	TMSC-TNNBDR
UCI-Digits	96.57	171.39	30.38	107.82
ORL	19.17	5.07	3.26	2.74
Yale	5.93	1.09	0.77	0.47
Extended YaleB	236.58	343.68	63.11	222.59
COIL-20	77.18	71.54	14.36	44.05

summarizes computational times of selected multi-view methods, with bold values indicating the best performance.

In the experiments, all compared multi-view clustering methods are based on subspace affinity matrix computations using complete graphs and spectral clustering, exhibiting cubic time complexity of N. Results across five datasets indicate that both Co-Reg SPC and our proposed TMSC-TNNBDR achieve optimal time efficiency. While significant performance gaps persist between algorithms, these gaps do not expand drastically with increasing dataset sizes. Considering the noticeable inferior clustering performance of Co-Reg SPC compared to tensor-based multi-view clustering methods, TMSC-TNNBDR delivers exceptionally outstanding results.

5.4. Convergence Analysis

Proving the global convergence of TMSC-TNNBDR is difficult. However, we did conduct some experiments to show the convergence properties. As per the convergence conditions in Algorithm 1, the match error is defined as follows:

$$ME={\displaystyle \frac{1}{V}}{\displaystyle \sum _{v=1}^V{‖H^{(v)}-G^{(v)}‖}_{\infty }<\epsilon }$$

(26)

The convergence curves of the TMSC-TNNBDR method on ORL and Yale are presented in Figure 6. As shown in Figure 6, the TMSC-TNNBDR method exhibits rapid convergence, achieving stable results within roughly 15 iterations.

5.5. Parametric Sensitivity

Three tunable hyperparameters characterize our proposed TMSC-TNNBDR: $\lambda$, $\alpha$ and $\gamma$, formally defined in Equation (10). During our optimization, candidate values of $\lambda$, $\alpha$ and $\gamma$ are selected from $\{0.002,0.006,0.02,0.06,0.2,0.6,2,6,12\}$. Where applicable, the search space is refined to narrowed ranges based on preliminary selections. The resulting configurations for tunable parameters are documented in

Table 8. Our configurations for tunable parameters.

Database	Parameters
	$\mathit{\lambda }$	$\mathit{\alpha }$	$\mathit{\gamma }$
UCI-Digits	0.2	0.03	0.003
ORL	0.2	0.06	0.006
Yale	0.2	0.02	0.002
YaleB	0.2	0.02	0.002
COIL-20	0.2	0.02	0.002

.

The clustering performance on the ORL and Yale databases is depicted in Figure 7. As shown in Figure 7, it is evident that the performance of the TMSC-TNNBDR is generally good and insensitive to varying values of $\alpha$ and $\gamma$, especially when $\alpha$ and $\gamma$ are relatively small.

6. Conclusions

In this paper, we propose a novel tensorized MSC with tensor nuclear norm and block diagonal representation (TMSC-TNNBDR). In this model, a BDR constraint is utilized to enforce the block diagonal structure of the learned similarity matrix. Additionally, low-rank tensor representations are applied to uncover the high-order correlations within multi-view data, ultimately advancing clustering objectives. Then, we developed an efficient augmented Lagrangian-based procedure for optimizing the TMSC-TNNBDR model. This method harnesses dual priors—block-diagonal structure and low-rank tensor representations—to steer subspace clustering, enabling more thorough exploitation of latent information in multi-view data while significantly alleviating noise interference.

Comparative evaluations demonstrate the superior clustering performance of TMSC-TNNBDR over baseline approaches. While the efficacy of TMSC-TNNBDR is empirically validated, significant challenges persist in multi-view subspace clustering. Our methodology leverages two distinct priors (block-diagonal structure and low-rank tensor constraints), raising new research questions: What additional priors could be incorporated? Is there demonstrable value in integrating more than two priors? Furthermore, algorithmic complexity remains a critical limitation; akin to other classical graph-based multi-view clustering techniques, TMSC-TNNBDR has a cubic time complexity of N. It is inevitably compromised by the curse of dimensionality under real-world data growth conditions. For larger-scale datasets, using anchor graphs instead of full-sample graphs may be an effective alternative. In such scenarios, leveraging prior knowledge to design regularizers remains highly valuable.