Rank Constraint Research Articles

Multi-view subspace clustering with a consensus tensorized scaled simplex representation

Multi-view clustering has been at the forefront of research focused on identifying an ideal structure for the rational partitioning of sample points. Multi-view clustering faces two main challenges: identifying the optimal structure within each view and fusing these structures into a reliable consensus matrix. To overcome these obstacles, we introduce the Consensus Tensorized Scaled Simplex Representation (CTSSR). First, CTSSR uses the scaled simplex representation to reveal the structure within each view flexibly. Second, we apply a rank constraint strategy to ensure that the learned subspace structure embodies the optimal clustering structure. To fuse multiple views, we propose a normalized view weighting mechanism based on each view's reconstruction error, ensuring reasonable contributions to the consensus matrix. Finally, we employ tensor learning on weighted subspace views to explore high-order correlations and subtle differences between views, refining the exploration of inter-view relationships. CTSSR combines intra-view structure exploration with inter-view relationship analysis, resulting in an ideal consensus matrix. It shows excellent clustering performance on real-life datasets, significantly outperforming state-of-the-art methods in various experimental settings.

Machine Learning With Tree Tensor Networks, CP Rank Constraints, and Tensor Dropout.

Tensor networks developed in the context of condensed matter physics try to approximate order- N tensors with a reduced number of degrees of freedom that is only polynomial in N and arranged as a network of partially contracted smaller tensors. As we have recently demonstrated in the context of quantum many-body physics, computation costs can be further substantially reduced by imposing constraints on the canonical polyadic (CP) rank of the tensors in such networks. Here, we demonstrate how tree tensor networks (TTN) with CP rank constraints and tensor dropout can be used in machine learning. The approach is found to outperform other tensor-network-based methods in Fashion-MNIST image classification. A low-rank TTN classifier with branching ratio b=4 reaches a test set accuracy of 90.3% with low computation costs. Consisting of mostly linear elements, tensor network classifiers avoid the vanishing gradient problem of deep neural networks. The CP rank constraints have additional advantages: The number of parameters can be decreased and tuned more freely to control overfitting, improve generalization properties, and reduce computation costs. They allow us to employ trees with large branching ratios, substantially improving the representation power.

Hierarchical bipartite graph based multi-view subspace clustering

Multi-view subspace clustering has attracted much attention because of its effectiveness in unsupervised learning. The high time consumption and hyper-parameters are the main obstacles to its development. In this paper, we present a novel method to effectively solve these two defects. First, we employ the bisecting k-means method to generate anchors and construct the hierarchical bipartite graph, which greatly reduce the time consumption. Moreover, we adopt an auto-weighted allocation strategy to learn appropriate weight factors for each view, which can avoid the influence of hyper-parameters. Furthermore, by imposing low rank constraints on the fusion graph, our proposed method can directly obtained the cluster indicators without any post-processing operations. Finally, numerous experiments verify the superiority of proposed method.

A new graph-based clustering method with dual-feature regularization and Laplacian rank constraint

The performance of graph-based clustering is commonly limited by two-stage processing (Constructing and dividing similarity graph) and the quality of similar graphs. To this end, we propose a new graph-based clustering method with dual-feature regularization and Laplacian rank constraint. Specifically, our method reveals the clustering structure and unifies the two-stage process. It imposes a Laplacian rank constraint on the similarity graph to ensure that it has C connected components. In addition, a method based on dual-feature regularization is designed to capture local data feature information from both feature extraction and adaptive regression, and is applied to an accurate distance metric learning. A reweighting optimization is integrated to learn a high-quality robust similarity graph. Comprehensive experiments on Ecoli, Yale and Yeast datasets show that our method outperforms the existing graph-based clustering methods with an average improvement of about 4%, 5% and 7% on the evaluation metrics ACC, NMI and RI, respectively.

Measurement configuration for integrated electric–gas systems via observability analysis considering gas flow dynamics

AbstractState estimation plays an important role in the monitoring and control of integrated electric–gas systems (IEGSs), but it faces limitations due to insufficient measurement configurations and low data redundancy in these systems; additional measurement configurations are needed to increase the overall system observability. Owing to the lack of suitable observability analysis methods, optimal measurement configurations for IEGSs remain underexplored. This paper presents an IEGS observability analysis method that incorporates gas flow dynamics via the Lie derivative. This method incorporates the complex topological structure of the gas network and the dynamic process of gas flow into the IEGS observability analysis. Furthermore, the measurement configuration problem for IEGSs considering gas flow dynamics is formulated as a rank‐constrained optimization problem. To handle the rank constraints effectively, an iterative cutting method is developed with convergence guarantees. Finally, the efficacy and practicality of the proposed methods are validated through case studies of varying scales. The proposed optimal measurement configuration model reduces measurement configuration costs while maintaining system observability.

Examining negative transfer in Cantonese learners of English as a foreign language: an optimality theory approach to English syllable acquisition

Abstract The contrasting differences that exist in the English and Cantonese phonological systems result in divergent articulatory production between them. This article aims to investigate the negative transfer of Cantonese EFL learners in acquiring English consonant clusters and the constraint rankings of Cantonese and English syllable systems by comparing syllable structures and analyzing experimental statistics. The objective is to elucidate the reasons behind their negative transfer in English syllable acquisition. The research results demonstrated that Cantonese EFL learners found it easier to acquire consonant clusters in the onset position compared to the coda position. Additionally, an increase in consonants in the coda position posed greater difficulties for Cantonese EFL learners. The results also indicated that participants encountered the most problems with deletion, followed by substitution and epenthesis. The OT analysis reveals that the different rankings of faithfulness and markedness constraints in Cantonese and English led to transfer errors.

Economic study of Binadhan-22 production in some selected areas of Bangladesh

This study was conducted in four districts of Bangladesh, namely Mymensingh, Bogura, Naogaon and Rangpur to estimate the profitability and factors affecting gross return of the variety. This study was based on primary data which were collected from 200 Binadhan-22 growing farmers. In the sampled areas data were collected through pre-designed interview schedule from January-February, 2023. Tabular, descriptive statistics and Cobb-Douglas production function model were used to fulfill the objectives. It was found that Binadhan-22 production is profitable. The average net return per hectare was Tk. 28876. The net return was highest in Rangpur (Tk. 32480/ha) followed by Naogaon (Tk. 28554/ha), Mymensingh (Tk. 27331/ha) and Bogura (Tk. 27139/ha), respectively. Benefit cost ratio was estimated at 1.39 and 2.04 on full cost and variable cost basis. Cobb-Douglas production function was chosen to determine the factor affecting gross return of Binadhan-22 production. All of the factors namely, human labour cost, power tiller cost, seed cost, fertilizer cost, irrigation cost, insecticides cost were statistically significant and positive. The regression coefficients for farming experience and agricultural training under all areas had negative but significant relationship at 5% and 10% level, respectively. The regression coefficient of age was positive and significant at 10% level. Under all areas, the regression coefficients of education and farm size were positive but not significant. Farmers faced some constraints in cultivating of the variety. The first ranked constraint was lack of availability of seeds (93%). Other constraints were lack of training (61%), lack of capital (38%) and lack of technical know-how (16%). In addition to helping farmers use a better combination of inputs that will create higher productivity and return, effective extension services may also assist farmers in cultivating rice in a way that will contribute to food security and self-sufficiency. Bangladesh J. Nuclear Agric, 38(1): 81-91, 2024

Addressing Constraint Coupling and Autonomous Decision-Making Challenges: An Analysis of Large-Scale UAV Trajectory-Planning Techniques

With the increase in UAV scale and mission diversity, trajectory planning systems faces more and more complex constraints, which are often conflicting and strongly coupled, placing higher demands on the real-time and response capabilities of the system. At the same time, conflicts and strong coupling pose challenges the autonomous decision-making capability of the system, affecting the accuracy and efficiency of the planning system in complex environments. However, recent research advances addressing these issues have not been fully summarized. An in-depth exploration of constraint handling techniques and autonomous decision-making issues will be of great significance to the development of large-scale UAV systems. Therefore, this paper aims to provide a comprehensive overview of this topic. Firstly, the functions and application scenarios of large-scale UAV trajectory planning are introduced and classified in detail according to the planning method, realization function and the presence or absence of constraints. Then, the constraint handling techniques are described in detail, focusing on the priority ranking of constraints and the principles of their fusion and transformation methods. Then, the importance of autonomous decision-making in large-scale UAV trajectory planning is described in depth, and related dynamic adjustment algorithms are introduced. Finally, the future research directions and challenges of large-scale UAV trajectory planning are outlooked, providing directions and references for future research in the fields of UAV clustering and UAV cooperative flight.

Recover then aggregate: unified cross-modal deep clustering with global structural information for single-cell data.

Single-cell cross-modal joint clustering has been extensively utilized to investigate the tumor microenvironment. Although numerous approaches have been suggested, accurate clustering remains the main challenge. First, the gene expression matrix frequently contains numerous missing values due to measurement limitations. The majority of existing clustering methods treat it as a typical multi-modal dataset without further processing. Few methods conduct recovery before clustering and do not sufficiently engage with the underlying research, leading to suboptimal outcomes. Additionally, the existing cross-modal information fusion strategy does not ensure consistency of representations across different modes, potentially leading to the integration of conflicting information, which could degrade performance. To address these challenges, we propose the 'Recover then Aggregate' strategy and introduce the Unified Cross-Modal Deep Clustering model. Specifically, we have developed a data augmentation technique based on neighborhood similarity, iteratively imposing rank constraints on the Laplacian matrix, thus updating the similarity matrix and recovering dropout events. Concurrently, we integrate cross-modal features and employ contrastive learning to align modality-specific representations with consistent ones, enhancing the effective integration of diverse modal information. Comprehensive experiments on five real-world multi-modal datasets have demonstrated this method's superior effectiveness in single-cell clustering tasks.

Nonconvex Factorization and Manifold Formulations Are Almost Equivalent in Low-Rank Matrix Optimization

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish a sandwich relation on the spectrum of Riemannian and Euclidean Hessians at first-order stationary points (FOSPs). As a result of that, we obtain an equivalence on the set of FOSPs, second-order stationary points, and strict saddles between the manifold and factorization formulations. In addition, we show that the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another. Similarities and differences in the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and factorization formulations for handling rank constraints, and it provides a geometric explanation for the similar empirical performance of factorization and manifold approaches in low-rank matrix optimization observed in the literature. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections (in particular, the sandwich relation), we are able to solve unanswered questions in the literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing. Funding: This work was supported by the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107 and 62141407], and the Shanghai Science and Technology Program [Grant 21JC1400600]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0030 .

An Intelligent Proofreading for Remote Skiing Actions Based on Variable Shape Basis

AbstractThe current proofreading algorithms for action regulation mainly recover the 3D structure and action information of non-rigid objects from image sequences by factorization. Most of algorithms assume that the camera model is an affine model. This assumption only holds if the size and depth of the object change very little relative to the distance from the object to the camera, which is in the case of fixed-shape basis. When the object is very close to the camera, this assumption causes a large reconstruction error. This paper solves this problem by the intelligent proofreading algorithms for remote skiing teaching actions based on variable shape basis. Firstly, the improved Retinex algorithm is used to enhance the multi-frame video images of skiing actions to make the action details more prominent. Then, measurement matrix is calculated after eliminating the translation vector by coordinate transformation. Under the condition of rank constraint, the measurement matrix is decomposed by singular value decomposition algorithm, and the correct shape basis structure of 3D action features can be obtained by using the variable shape basis. Finally, by randomly initializing a parameter, the optimized parameter and the least square algorithm are used to optimize the randomly initialized parameter further. The iteration until the convergence of the objective function can be used to calculate the deformation degree of the actions. The test results show that this algorithm improves the proofreading accuracy of action regulation in skiing teaching, and the proofreading results of various uploaded sliding actions are correct, which can be applied to remote skiing teaching and community learning.

Multi-view clustering via high-order bipartite graph fusion

Multi-view clustering is widely applied in engineering and scientific research. It helps reveal the underlying structures and correlations behind complex multi-view data. Graph-based multi-view clustering stands as a prominent research frontier within the multi-view clustering field, yet faces persistent challenges. Firstly, typically constructed initial input graphs for each view yields sparse clustering structure, hindering clustering performance. Secondly, as data sources proliferate, algorithms encounter escalating time complexities, notably in methods relying on n×n fully connected graphs. Thirdly, prevailing graph fusion strategies struggle to mitigate the impact of low-quality graphs, impeding overall efficacy. In this paper, we present a novel Multi-View Clustering method based on High-Order Bipartite Graph fusion (MCHBG). For the first two challenges, the introduced high-order bipartite graphs in MCHBG reveal richer clustering structures, effectively alleviating sparse clustering structure of the input graph, while keeping the overall algorithm’s computational complexity controlled within O(n). For the third challenge, our graph fusion mechanism selectively integrates high-order bipartite graphs, and implicitly weights the selected bipartite graphs to mitigate the impact of low-quality bipartite graphs. MCHBG learns a structured fusion bipartite graph under the Laplacian rank constraint, which directly indicates the clusters of data. Extensive experimental results demonstrate the effectiveness and superiority of MCHBG. Code available: https://anonymous.4open.science/r/MCHBG.

Efficient Multi-View Clustering via Unified and Discrete Bipartite Graph Learning.

Although previous graph-based multi-view clustering (MVC) algorithms have gained significant progress, most of them are still faced with three limitations. First, they often suffer from high computational complexity, which restricts their applications in large-scale scenarios. Second, they usually perform graph learning either at the single-view level or at the view-consensus level, but often neglect the possibility of the joint learning of single-view and consensus graphs. Third, many of them rely on the k -means for discretization of the spectral embeddings, which lack the ability to directly learn the graph with discrete cluster structure. In light of this, this article presents an efficient MVC approach via u nified and d iscrete b ipartite g raph l earning (UDBGL). Specifically, the anchor-based subspace learning is incorporated to learn the view-specific bipartite graphs from multiple views, upon which the bipartite graph fusion is leveraged to learn a view-consensus bipartite graph with adaptive weight learning. Furthermore, the Laplacian rank constraint is imposed to ensure that the fused bipartite graph has discrete cluster structures (with a specific number of connected components). By simultaneously formulating the view-specific bipartite graph learning, the view-consensus bipartite graph learning, and the discrete cluster structure learning into a unified objective function, an efficient minimization algorithm is then designed to tackle this optimization problem and directly achieve a discrete clustering solution without requiring additional partitioning, which notably has linear time complexity in data size. Experiments on a variety of multi-view datasets demonstrate the robustness and efficiency of our UDBGL approach. The code is available at https://github.com/huangdonghere/UDBGL.

Achieving food security: Household perception and adoption of home gardening techniques in Ghana

As global food crises and high food prices continue to impact communities, there is a growing focus on strengthening local food systems and improving livelihoods. Home gardening techniques have gained renewed attention as a means of food production and enhancing household food security. However, there is a lack of empirical studies on the perception and adoption of home gardening techniques. This study therefore fills this gap by examining the perception and adoption of home gardening techniques among households in Ghana. A multistage sampling technique was used to select 348 households and data was collected through a structured questionnaire. Using a 5-point Likert scale, we analyzed the households' perception of home gardening techniques, and utilized a multivariate probit regression model to examine the factors influencing multiple adoption of home gardening techniques. The Kendall’s coefficient of concordance was used to test the level of agreement among ranked constraints in adopting home gardening techniques. The generalized Poisson regression was used to examine the number of adoption of home gardening techniques. The results revealed that most households perceived home gardening techniques as a way to improve household food security. Households were found to adopt multiple home gardening techniques, such as use of pots, sacks, and tires. Empirically, factors such as sex, education, occupation, income, technology cost, and social network had a significant influence on multiple adoption of home gardening techniques. Also, age, education, household size, income, technology cost, social network, inadequate space and insect and disease damages significantly influenced the number of adoptions of home gardening techniques. Furthermore, lack of time and inadequate space were the main constraints limiting the adoption of these techniques. We suggest government interventions to spur the adoption of home gardening techniques to improve household food nutrition.

Diverse representation-guided graph learning for multi-view metric clustering

Multi-view graph clustering has garnered tremendous interest for its capability to effectively segregate data by harnessing information from multiple graphs representing distinct views. Despite the advances, conventional methods commonly construct similarity graphs straightway from raw features, leading to suboptimal outcomes due to noise or outliers. To address this, latent representation-based graph clustering has emerged. However, it often hypothesizes that multiple views share a fixed-dimensional coefficient matrix, potentially resulting in useful information loss and limited representation capabilities. Additionally, many methods exploit Euclidean distance as a similarity metric, which may inaccurately measure linear relationships between samples. To tackle these challenges, we develop a novel diverse representation-guided graph learning for multi-view metric clustering (DRGMMC). Concretely, raw sample matrix from each view is first projected into diverse latent spaces to capture comprehensive knowledge. Subsequently, a popular metric is leveraged to adaptively learn similarity graphs with linearity-aware based on attained coefficient matrices. Furthermore, a self-weighted fusion strategy and Laplacian rank constraint are introduced to output clustering results directly. Consequently, our model merges diverse representation learning, metric learning, consensus graph learning, and data clustering into a joint model, reinforcing each other for holistic optimization. Substantial experimental findings substantiate that DRGMMC outperforms most advanced graph clustering techniques.

Lattice-Based Polynomial Commitments: Towards Asymptotic and Concrete Efficiency

Polynomial commitments schemes are a powerful tool that enables one party to commit to a polynomial p of degree d, and prove that the committed function evaluates to a certain value z at a specified point u, i.e. p(u)=z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(u) = z$$\\end{document}, without revealing any additional information about the polynomial. Recently, polynomial commitments have been extensively used as a cryptographic building block to transform polynomial interactive oracle proofs (PIOPs) into efficient succinct arguments. In this paper, we propose a lattice-based polynomial commitment that achieves succinct proof size and verification time in the degree d of the polynomial. Extractability of our scheme holds in the random oracle model under a natural ring version of the BASIS assumption introduced by Wee and Wu (EUROCRYPT 2023). Unlike recent constructions of polynomial commitments by Albrecht et al. (CRYPTO 2022), and by Wee and Wu, we do not require any expensive preprocessing steps, which makes our scheme particularly attractive as an ingredient of a PIOP compiler for succinct arguments. We further instantiate our polynomial commitment, together with the Marlin PIOP (EUROCRYPT 2020), to obtain a publicly-verifiable trusted-setup succinct argument for Rank-1 Constraint System (R1CS). Performance-wise, we achieve 17\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$17$$\\end{document}MB proof size for 220\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{20}$$\\end{document} constraints, which is 15\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$15$$\\end{document}X smaller than currently the only publicly-verifiable lattice-based SNARK proposed by Albrecht et al.

Strong consistency of an estimator by the truncated singular value decomposition for an errors-in-variables regression model with collinearity

In this paper, we prove strong consistency of an estimator by the truncated singular value decomposition for a multivariate errors-in-variables linear regression model with collinearity. This result is an extension of Gleser's proof of the strong consistency of total least squares solutions to the case with modern rank constraints. While the usual discussion of consistency in the absence of solution uniqueness deals with the minimal norm solution, the contribution of this study is to develop a theory that shows the strong consistency of a set of solutions. The proof is based on properties of orthogonal projections, specifically properties of the Rayleigh-Ritz procedure for computing eigenvalues. This makes it suitable for targeting problems where some row vectors of the matrices do not contain noise. Therefore, this paper gives a proof for the regression model with the above condition on the row vectors, resulting in a natural generalization of the strong consistency for the standard TLS estimator.

Explicit unsupervised feature selection based on structured graph and locally linear embedding

Numerous redundant and irrelevant features are usually contained in high-dimensional data whose presence, if ignored, can bring detrimental effects on the performance of data processing tasks. Unsupervised feature selection (UFS) as a trending topic in a great deal of domains has been greatly concerned, which can effectively filter out redundant and irrelevant features in the unlabeled data. The majority of existing UFS methods mainly focus on building the parameterized model in the original feature space or some certain low-dimensional feature subspace, which cannot fully capture the information of the target space. In this work, we propose a novel UFS method that unifies the explicit feature selective matrix and structured graph into a learning framework. An explicit selection matrix, whose scale, bound of the elements and structure are considered, is tailored for UFS. Furthermore, we take into account the local consistency so that the low dimensional manifold structures can be preserved to a better degree. Instead of using the pre-defined similarity matrix to construct graph, we make use of the way of learning to obtain the adaptive graph. Besides, for the purpose of exploiting the cluster structure of data, we impose a rank constraint on the graph. Notably, both the explicit selection matrix and intrinsic structure are learned in the target feature subspace, and thus the proposed methodology is more straightforward and effective. We present an effective and efficient algorithm for the resulting problem based on alternative optimization strategy, and establish its convergence and computational complexity analysis. The experimental results show that our proposed approach has improved by over 5.28% and 5.79% on average in terms of clustering accuracy (ACC) and Normalized Mutual Information (NMI) compared with comparison methods, which demonstrates its superiority.

Adoption of post-harvest practices and their influencing factors: A study of ginger growers in Syangja, Nepal

Proper ginger post-harvest practices enhance quality, extend shelf life, and ameliorate market access. Prioritizing post-harvest practices in ginger can tremendously increase the market value of ginger in Nepal. Nevertheless, this cannot be done without thorough understanding about the current adoption of post-harvest practices among farmers. So, this study was carried out to know the status of adoption of post-harvest practices in ginger, examine factors associated with adoption, identify and rank major constraints of adoption. The collection of primary data involved 80 ginger growers chosen randomly from altogether 6 wards in Galyang, Waling, and Chapakot municipalities. Data entry and analysis were performed using two software: MS Excel 2021 and SPSS Version 27. The adoption status of ginger post-harvest practices was assessed and quantified using frequency distribution. The chi-square test revealed a significant association between adoption and membership (5%), extension (1%) and training (1%). However, the association between demographic factors like age, gender, education level, experiences in ginger cultivation, and the adoption of post-harvest practices in ginger was statistically non-significant. Moreover, an independent sample t-test and indexing technique were employed. Average adoption index was computed based on which high and low adopters were categorized. Notable advantages emerged for high adopters as they cultivated ginger in larger areas (p < 0.01) and gained higher economic returns (p < 0.05). Five constraints regarding the adoption were ranked through indexing, where a poor marketing system had the highest index value of 0.83, followed by the unavailability of improved processing technologies. In contrast, labor crisis was the least ranked constraint. This study concluded that farmers still follow the traditional methods of ginger post-harvest practices, denoting significant potential for improvement. Thus, offering impactful trainings and support to farmers in line with their recommendations can be the pivotal step towards enhancing the post-harvest practices in ginger sector in Syangja district.

New approach for learning structured graph with Laplacian rank constraint

Many existing graph-clustering methods get results in a two-stage manner, including constructing a graph from data and partitioning it. It always leads to sub-optimal performances. To address the drawbacks, imposing rank constraint on the graph’s Laplacian matrix is an effective way. It can learn a structured graph that connected components are exactly equal to the cluster numbers. According to separate connected components, it directly obtains clustering assignments. However, previous works heuristically solve it during the optimization without any theory guarantee. To handle this issue, we propose a novel solver to optimize this problem with theoretical advantages, enabling the structured bipartite graphs to directly indicate clustering results. Initially, we discover that pairwise and bipartite graphs exhibit the same connected components after undergoing a specific operation. Then, we derive an efficient algorithm to optimize two different clustering models with rank constraints based on this property. Moreover, our proposed method exhibits linear computational complexity to the data scale. Extensive empirical results on both synthetic and real benchmark data demonstrate that the proposed method outperforms existing methods.