Adaptive graph regularized nonnegative tensor ring decomposition for data clustering

Xinyu Yao; Guimin Liu; Xinyu Yao; Guimin Liu

doi:10.3934/math.20261085

AIMS Mathematics

2026, Volume 11, Issue 8: 27085-27115. doi: 10.3934/math.20261085

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Research article

Adaptive graph regularized nonnegative tensor ring decomposition for data clustering

Xinyu Yao ¹,
Guimin Liu ^{2
,
,}

1.
School of Information Engineering and Artificial Intelligence, Lanzhou University of Finance and Economics, Lanzhou 730000, Gansu, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China

Received: 03 June 2026 Revised: 16 August 2026 Accepted: 19 August 2026 Published: 27 August 2026
MSC : 15A69, 65F15, 68W25

Nonnegative tensor ring (NTR) decomposition has been successfully applied to clustering tasks due to its ability to preserve the multilinear structure of tensor data while effectively extracting physically interpretable nonnegative features. Although existing graph-regularized NTR methods capture local manifold structures, their reliance on fixed predefined similarity graphs often leads to inaccurate manifold approximations, thereby degrading clustering performance. To address this limitation, this paper proposes an adaptive graph regularized NTR (AGRNTR) model, which integrates graph regularization and similarity matrix learning within the NTR framework. This design allows the similarity graph to be adaptively refined during tensor decomposition. By jointly learning the similarity matrix and the tensor ring factors, AGRNTR captures the intrinsic geometry of tensor data more accurately than methods using fixed predefined graphs, achieving superior clustering performance. To solve the AGRNTR model, an efficient block coordinate descent algorithm combined with multiplicative updates is developed, and its convergence is theoretically guaranteed. Numerical experiments conducted on benchmark datasets demonstrate the effectiveness and superiority of the proposed model.
- nonnegative tensor ring decomposition,
- adaptive graph regularization,
- block coordinate descent,
- clustering
Citation: Xinyu Yao, Guimin Liu. Adaptive graph regularized nonnegative tensor ring decomposition for data clustering[J]. AIMS Mathematics, 2026, 11(8): 27085-27115. doi: 10.3934/math.20261085

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Abstract

Nonnegative tensor ring (NTR) decomposition has been successfully applied to clustering tasks due to its ability to preserve the multilinear structure of tensor data while effectively extracting physically interpretable nonnegative features. Although existing graph-regularized NTR methods capture local manifold structures, their reliance on fixed predefined similarity graphs often leads to inaccurate manifold approximations, thereby degrading clustering performance. To address this limitation, this paper proposes an adaptive graph regularized NTR (AGRNTR) model, which integrates graph regularization and similarity matrix learning within the NTR framework. This design allows the similarity graph to be adaptively refined during tensor decomposition. By jointly learning the similarity matrix and the tensor ring factors, AGRNTR captures the intrinsic geometry of tensor data more accurately than methods using fixed predefined graphs, achieving superior clustering performance. To solve the AGRNTR model, an efficient block coordinate descent algorithm combined with multiplicative updates is developed, and its convergence is theoretically guaranteed. Numerical experiments conducted on benchmark datasets demonstrate the effectiveness and superiority of the proposed model.

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