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Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors

Suyuan LiuSiwei WangPei ZhangKai XuXinwang LiuChangwang ZhangFeng Gao

article2022AAAI223 citations

Proposes a scalable multi-view subspace clustering method that jointly learns consensus anchors and a fused graph with exact connected components, achieving linear time complexity and directly generating cluster labels without heuristic anchor sampling or post-processing steps.

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Modern data applications frequently gather information across diverse formats and sources, such as text, images, and video representing the same underlying event. Multi-view subspace clustering groups these complex, high-dimensional datasets into meaningful categories by uncovering shared low-dimensional structures. However, conventional multi-view clustering methods suffer from cubic computational complexity relative to sample size, making them impractical for large-scale operations. Recent attempts to accelerate processing using representative "anchor points" rely on fixed, heuristic sampling methods and require multi-stage post-processing steps. These limitations degrade clustering quality, add hyper-parameter tuning overhead, and hinder real-time scalability.

The article introduces and evaluates a scalable, parameter-free algorithm named Efficient One-pass Multi-view Subspace Clustering with Consensus Anchors (EOMSC-CA). The primary objective is to demonstrate a unified framework that learns shared anchor representations and graph structures simultaneously, directly outputting discrete cluster labels without secondary processing steps.

To evaluate this approach, the authors integrated anchor learning, graph construction, and adaptive view weighting into a single mathematical optimization problem. A strict graph connectivity constraint ensures the resulting anchor graph contains exactly the required number of connected clusters. The researchers benchmarked the algorithm against eight state-of-the-art multi-view clustering methods across nine widely recognized image and object datasets, ranging from 400 to over 101,000 samples, measuring performance via accuracy, normalized mutual information, and F-score.

The evaluation yielded several key findings. First, the proposed method reduces computational complexity to scale linearly with the number of data points, allowing it to process the 101,499-sample YouTubeFace dataset where traditional methods fail due to out-of-memory errors. Second, the algorithm achieved top-tier clustering quality, attaining the highest accuracy on datasets such as Caltech101-7 (83.51%) and YouTubeFace (26.50%), and remaining competitive across all others. Third, it eliminates manual hyper-parameter tuning by learning view weights adaptively, whereas competing methods require tuning up to four parameters. Finally, sensitivity analyses confirmed that clustering performance remains highly stable across different anchor quantities.

These findings indicate substantial practical benefits for enterprise data processing. Organizations can lower computational infrastructure costs and shorten execution timelines when categorizing massive, multi-modal data streams. By eliminating post-processing discretization and manual parameter calibration, the framework reduces operational risk and deployment complexity while maintaining high clustering accuracy across diverse domains.

Organizations handling large-scale multi-view clustering tasks should consider adopting this unified consensus-anchor approach. Technical teams can leverage the authors' publicly accessible implementation to run pilot benchmarks against existing clustering pipelines, especially for workloads exceeding tens of thousands of records.

Confidence in these findings is supported by rigorous mathematical proofs of linear complexity and consistent experimental performance across nine standardized datasets. A slight limitation is that the user must still specify the target cluster count and search for the baseline anchor matrix dimensions, though performance shows minimal sensitivity to these selections.

China Computer FederationNational University of Defense TechnologyPeking Universityclustering algorithmsmulti-viewsviewpointmulti-view subspace clusteringconsensus anchorsspectral clustering
Tracesource/EOMSC-CALiu et al (2022).pdf
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Abstract

Multi-view subspace clustering (MVSC) optimally integrates multiple graph structure information to improve clustering performance. Recently, many anchor-based variants are proposed to reduce the computational complexity of MVSC. Though achieving considerable acceleration, we observe that most of them adopt fixed anchor points separating from the sub-sequential anchor graph construction, which may adversely affect the clustering performance. In addition, post-processing is required to generate discrete clustering labels with additional time consumption. To address these issues, we propose a scalable and parameter-free MVSC method to directly output the clustering labels with optimal anchor graph, termed as Efficient One-pass Multi-view Subspace Clustering with Consensus Anchors (EOMSC-CA). Specially, we combine anchor learning and graph construction into a uniform framework to boost clustering performance. Meanwhile, by imposing a graph connectivity constraint, our algorithm directly outputs the clustering labels without any post-processing procedures as previous methods do. Our proposed EOMSC-CA is proven to be linear complexity respecting to the data size. The superiority of our EOMSC-CA over the effectiveness and efficiency is demonstrated by extensive experiments. Our code is publicly available at https://github.com/Tracesource/EOMSC-CA.

Topics

clustering algorithmsmulti-viewsviewpoint

Keywords

multi-view subspace clusteringconsensus anchorsspectral clusteringoptimal anchor graphanchor point selectionsubspace representationgraph learningspectral embedding

Table of Contents

  • Introduction
  • Background
  • Subspace Clustering
  • Multi-view Subspace Clustering
  • Anchor-based Multi-view Clustering
  • The Proposed Methodology
  • Motivation
  • Formulation of Problem
  • Optimization
  • Complexity Analysis
  • Experiments
  • Benchmark Datasets
  • Experimental Setup
  • Experimental Results
  • Running Time Comparison
  • Sensitivity Analysis
  • Conclusion
  • Acknowledgements
  • References

Knowls

  1. Knowl 1 — Joint consensus-anchor and graph-learning formulation

    model/method

    EOMSC-CA jointly learns view-specific projections, a shared consensus-anchor matrix, view weights, and a nonnegative anchor graph rather than selecting anchors before graph construction. For vv views, let Xp∈Rdp×nX_p\in\mathbb{R}^{d_p\times n} be the data matrix of view pp, Wp∈Rdp×lW_p\in\mathbb{R}^{d_p\times l} its projection, A∈Rl×mA\in\mathbb{R}^{l\times m} the shared anchor matrix with mm anchors, Z∈Rm×nZ\in\mathbb{R}^{m\times n} the fused anchor graph, and βp\beta_p the nonnegative weight of view pp. The optimization problem is

    min⁡{Wp},A,Z,β∑p=1vβp2∥Xp−WpAZ∥F2s.t.βp≥0,∑p=1vβp=1,Wp⊤Wp=Il,A⊤A=Im,Z≥0,Z⊤1m=1n,rank⁡(L~)=n+m−k.\begin{aligned} \min_{\{W_p\},A,Z,\beta}\quad &\sum_{p=1}^{v}\beta_p^2\left\|X_p-W_pAZ\right\|_F^2\\ \text{s.t.}\quad &\beta_p\ge 0,\quad \sum_{p=1}^{v}\beta_p=1,\\ &W_p^{\top}W_p=I_l,\quad A^{\top}A=I_m,\\ &Z\ge 0,\quad Z^{\top}\mathbf{1}_m=\mathbf{1}_n,\\ &\operatorname{rank}(\widetilde L)=n+m-k. \end{aligned}

    Here nn is the number of data points, dpd_p is the dimension of view pp, ll is the consensus-anchor dimension, kk is the desired number of clusters, and 1r\mathbf{1}_r is an rr-dimensional all-ones vector. The reconstruction WpAZW_pAZ forces every view to use the same latent anchors and the same fused data-to-anchor graph, while βp2\beta_p^2 adaptively controls each view's contribution.

  2. Knowl 2 — Connectivity constraint guarantees the cluster structure

    theoretical result

    EOMSC-CA converts the rectangular anchor graph Z∈Rm×nZ\in\mathbb{R}^{m\times n} into an undirected bipartite graph

    S=[0n×nZ⊤Z0m×m]∈R(n+m)×(n+m).S=\begin{bmatrix}0_{n\times n}&Z^{\top}\\Z&0_{m\times m}\end{bmatrix}\in\mathbb{R}^{(n+m)\times(n+m)}.

    Let DD be the diagonal degree matrix of SS, with Dii=∑jSijD_{ii}=\sum_jS_{ij}, and let the normalized Laplacian be L~=I−D−1/2SD−1/2\widetilde L=I-D^{-1/2}SD^{-1/2}. The number of connected components of SS equals the multiplicity of the zero eigenvalue of L~\widetilde L. Consequently, imposing rank⁡(L~)=n+m−k\operatorname{rank}(\widetilde L)=n+m-k gives L~\widetilde L exactly kk zero eigenvalues and therefore exactly kk connected components. Because SS is formed from the data-to-anchor edges in ZZ, the induced anchor graph has the same cluster connectivity structure.

  3. Knowl 3 — Connectivity-aware optimization of the fused anchor graph

    algorithm

    With WpW_p, AA, and β\beta fixed, EOMSC-CA updates the anchor graph ZZ using an adaptive normalized-Laplacian penalty. Define

    C=∑p=1vβp2Xp⊤WpA∈Rn×m,s=∑p=1vβp2.C=\sum_{p=1}^{v}\beta_p^2X_p^{\top}W_pA\in\mathbb{R}^{n\times m},\qquad s=\sum_{p=1}^{v}\beta_p^2.

    For each data point ii, let z:,i∈Rmz_{:,i}\in\mathbb{R}^{m} be column ii of ZZ, and let ci,:⊤∈Rmc_{i,:}^{\top}\in\mathbb{R}^{m} be row ii of CC transposed. During the inner iterations, compute the data degrees dn(i)=∑j=1mzjid_n(i)=\sum_{j=1}^{m}z_{ji} and anchor degrees dm(j)=∑i=1nzjid_m(j)=\sum_{i=1}^{n}z_{ji}. Let Fn∈Rn×kF_n\in\mathbb{R}^{n\times k} and Fm∈Rm×kF_m\in\mathbb{R}^{m\times k} be the data and anchor parts of the normalized-Laplacian indicator matrix, obtained from the top kk singular vectors of Dn−1/2Z⊤Dm−1/2D_n^{-1/2}Z^{\top}D_m^{-1/2}, with the normalization Fn⊤Fn+Fm⊤Fm=IkF_n^{\top}F_n+F_m^{\top}F_m=I_k. Define

    tji=∥Fn(i,:)dn(i)−Fm(j,:)dm(j)∥22.t_{ji}=\left\|\frac{F_n(i,:)}{\sqrt{d_n(i)}}-\frac{F_m(j,:)}{\sqrt{d_m(j)}}\right\|_2^2.

    For each column ii, update ZZ by projecting onto the probability simplex Δm={z∈Rm:z≥0,1m⊤z=1}\Delta_m=\{z\in\mathbb{R}^{m}:z\ge0,\mathbf{1}_m^{\top}z=1\}:

    z:,i=ΠΔm(ci,:⊤−(λ/2)t:,is),z_{:,i}=\Pi_{\Delta_m}\left(\frac{c_{i,:}^{\top}-(\lambda/2)t_{:,i}}{s}\right),

    where λ\lambda is the adaptive connectivity-penalty coefficient and t:,it_{:,i} contains the values tjit_{ji}. The procedure initializes the indicator factors, alternates graph-column updates, degree updates, and singular-vector updates, adapts λ\lambda according to the normalized-Laplacian constraint, and stops when ZZ has exactly kk connected components.

  4. Knowl 4 — Alternating optimization pipeline

    algorithm

    EOMSC-CA alternates the four blocks of the joint objective until convergence. The inputs are the multi-view data {Xp}p=1v\{X_p\}_{p=1}^{v} and the desired cluster number kk; the output is an anchor graph Z∈Rm×nZ\in\mathbb{R}^{m\times n} with exactly kk connected components.

    Input: Multi-view data {X_p}_{p=1}^v and cluster number k
    Output: Anchor graph Z with exactly k connected components
    Initialize W_p, A, and Z; set beta_p = 1/v for every view p
    repeat
        Compute C = sum over p of beta_p^2 X_p^T W_p A
        Update Z using the connectivity-aware anchor-graph procedure
        For each view p, update W_p from its orthogonal Procrustes problem
        Update A from the shared orthogonal Procrustes problem
        Update beta from the closed-form view-weight formula
    until the objective converges
    Return Z

    The learned graph itself is the final clustering representation; no separately sampled anchors, graph-fusion stage, spectral embedding stage, or kk-means post-processing stage is required.

  5. Knowl 5 — Closed-form updates for projections, anchors, and view weights

    equation

    The non-graph blocks of EOMSC-CA have closed-form singular-value-decomposition updates. With ZZ, AA, and β\beta fixed, define Gp=XpZ⊤A⊤∈Rdp×lG_p=X_pZ^{\top}A^{\top}\in\mathbb{R}^{d_p\times l}. If Gp=UpΣpVp⊤G_p=U_p\Sigma_pV_p^{\top} is its thin SVD, the optimal view projection is

    Wp=UpVp⊤,p=1,…,v.W_p=U_pV_p^{\top},\qquad p=1,\ldots,v.

    With ZZ, all WpW_p, and β\beta fixed, define B=∑p=1vβp2Wp⊤XpZ⊤∈Rl×mB=\sum_{p=1}^{v}\beta_p^2W_p^{\top}X_pZ^{\top}\in\mathbb{R}^{l\times m}. If B=UBΣBVB⊤B=U_B\Sigma_BV_B^{\top}, the optimal consensus-anchor matrix is

    A=UBVB⊤.A=U_BV_B^{\top}.

    With ZZ, AA, and all WpW_p fixed, let Mp=∥Xp−WpAZ∥F2M_p=\|X_p-W_pAZ\|_F^2. Subject to nonnegative weights summing to one, the optimal view coefficient is

    βp=1/Mp∑q=1v1/Mq,p=1,…,v.\beta_p=\frac{1/M_p}{\sum_{q=1}^{v}1/M_q},\qquad p=1,\ldots,v.

    Thus, views with smaller reconstruction error receive larger weights, without introducing a manually tuned view-balancing parameter.

  6. Knowl 6 — Direct clustering from connected components

    model/method

    After optimization, EOMSC-CA assigns each data point the label of the connected component containing its corresponding data vertex in the augmented bipartite graph SS. The rank constraint guarantees exactly kk such components, so the graph directly encodes the kk clusters. This replaces the usual eigen-decomposition of an n×nn\times n similarity matrix, spectral embedding, and subsequent discretization or kk-means step. The paper reports complexity O(nml)O(nml) for this final graph-labeling step, which is linear in the number of data points when the anchor number mm and latent dimension ll are small relative to nn.

  7. Knowl 7 — Linear complexity from anchor compression

    theoretical result

    Let tt be the number of inner iterations used to optimize ZZ, let d=∑p=1vdpd=\sum_{p=1}^{v}d_p be the sum of view dimensions, and let m,l≪nm,l\ll n. The connectivity-aware ZZ optimization costs

    O(nm2t+m3t+nmlt+nmdt).O\left(nm^2t+m^3t+nmlt+nmdt\right).

    Including the projection, consensus-anchor, and view-weight updates, the main optimization cost is reported as

    O(n(m2t+mlt+mdt+dl)+m3t+dl2+mdl+ml2).O\left(n\left(m^2t+mlt+mdt+dl\right)+m^3t+dl^2+mdl+ml^2\right).

    Since mm, dd, ll, and tt are treated as much smaller than the sample count nn, the optimization is linear in nn. The subsequent connected-component labeling costs O(nml)O(nml), so the complete method avoids the O(n3)O(n^3) graph construction and spectral-clustering costs associated with dense multi-view subspace-clustering methods.

  8. Knowl 8 — Benchmark protocol and datasets

    experimental setup

    EOMSC-CA was evaluated on nine multi-view benchmarks using accuracy (ACC), normalized mutual information (NMI), and Fscore. The datasets span 400 to 101,499 samples and 2 to 6 views:

    Could not parse LaTeX table

    The comparisons used eight methods: MLRSSC, AMGL, SFMC, RMKM, BMVC, LMVSC, MSGL, and FPMVS. EOMSC-CA has no tuned regularization hyperparameters, but its anchor count mm and anchor dimension ll were each searched over {k,2k,…,7k}\{k,2k,\ldots,7k\}, where kk is the dataset's class count. Competing methods were given their best searched parameters, and their clustering results used the best of 50 kk-means runs. Experiments were run in MATLAB 2019b on an Intel Core i9-10900X machine with 64 GB RAM.

  9. Knowl 9 — Clustering performance across nine datasets

    data/table

    The reported comparison evaluates EOMSC-CA against the eight baselines using ACC, NMI, and Fscore. The table below gives EOMSC-CA's exact score and the strongest non-EOMSC-CA score for each dataset and metric; the baseline name is included to identify the relevant comparison.

    Could not parse LaTeX table

    EOMSC-CA is the best ACC method on Caltech101-7, SUNRGBD, NUSWIDEOBJ, and YoutubeFace, the best Fscore method on ORL_mtv, Caltech101-7, NUSWIDEOBJ, and YoutubeFace, and the best NMI method on ORL_mtv. It remains competitive on the other datasets, while several baselines become unavailable on the largest datasets because of memory limits.

  10. Knowl 10 — Runtime, convergence, and anchor-number behavior

    empirical result

    The runtime comparison shows EOMSC-CA is generally faster than most of the evaluated multi-view clustering methods across the nine datasets, including the largest datasets. BMVC and MSGL are faster in some cases, but they use four and two hyperparameters respectively and obtain weaker clustering results overall. On ORL_mtv and NUSWIDEOBJ, ACC, NMI, and Fscore increase monotonically over the alternating iterations and stabilize during the final iterations, indicating empirical convergence of the optimization. Varying the anchor count from 2k2k through 7k7k while fixing the anchor dimension produces only small changes in the three clustering metrics on both datasets, suggesting that the method is not highly sensitive to the number of anchors. The practical qualification is that the anchor count and anchor dimension still must be selected for an experiment, even though the method does not tune additional objective-balancing hyperparameters.

Coverage note — No substantial contributed material was omitted; proof-only details such as the singular-vector auxiliary theorem and algebraic intermediate transformations were excluded.

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Citation

MLA
Liu, S., et al. “Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors”. Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 7, 2022, pp. 7576–84, https://doi.org/10.1609/AAAI.V36I7.20723.
APA
Liu, S., Wang, S., Zhang, P., Xu, K., Liu, X., Zhang, C., & Gao, F. (2022). Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors. Proceedings of the AAAI Conference on Artificial Intelligence, 36(7), 7576–7584. https://doi.org/10.1609/AAAI.V36I7.20723
Chicago
Liu, S., S. Wang, P. Zhang, et al. 2022. “Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors”. Proceedings of the AAAI Conference on Artificial Intelligence 36 (7): 7576–84. https://doi.org/10.1609/AAAI.V36I7.20723.
Harvard
Liu, S. et al. (2022) “Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors”, Proceedings of the AAAI Conference on Artificial Intelligence, 36(7), pp. 7576–7584. Available at: https://doi.org/10.1609/AAAI.V36I7.20723.
Vancouver
1. Liu S, Wang S, Zhang P, Xu K, Liu X, Zhang C, Gao F (2022) Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors. Proceedings of the AAAI Conference on Artificial Intelligence 36:7576–7584

BibTeX

@article{Liu_2022, title={Efficient One-Pass Multi-View Subspace Clustering with Consensus Anchors}, volume={36}, ISSN={2159-5399}, url={http://dx.doi.org/10.1609/AAAI.V36I7.20723}, DOI={10.1609/aaai.v36i7.20723}, number={7}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, publisher={Association for the Advancement of Artificial Intelligence (AAAI)}, author={Liu, Suyuan and Wang, Siwei and Zhang, Pei and Xu, Kai and Liu, Xinwang and Zhang, Changwang and Gao, Feng}, year={2022}, month=June, pages={7576–7584} }
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