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Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering

Pei ZhangSiwei WangLiang LiChangwang ZhangXinwang LiuEn ZhuZhe LiuLu ZhouLei Luo

article2023AAAI79 citations

Proposes a scalable multi-view clustering framework that automatically weights varied anchor graph sizes across different views to avoid costly hyperparameter tuning while achieving linear computational complexity.

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Modern data applications frequently gather information from multiple distinct perspectives or feature sets, such as combining text, image, and facial data. Grouping this multi-view data without pre-existing labels is a vital task for modern analytics. Traditional graph-based clustering techniques deliver strong accuracy by mapping relationships between all data points, but their computational demands grow steeply with data volume, making them impractical for large-scale enterprise data. Existing scalable alternatives approximate relationships using a small subset of representative points, termed anchors. However, these methods force every data view to use an identical number of anchors and require extensive, manual hyperparameter searches to find that number, creating substantial computational bottlenecks and ignoring the unique diversity of each data source.

The article develops and evaluates a scalable multi-view clustering framework called Flexible and Diverse Anchor Graph Fusion. The main objective is to eliminate manual anchor tuning and accommodate varied data structures by automatically weighting and fusing anchor representations of different sizes across multiple views.

To demonstrate this method, the authors established a mathematically proven fusion approach that scales linearly with the number of data samples, avoiding costly full-graph constructions. They evaluated the framework across ten public benchmark datasets ranging from small collections of 165 instances to large-scale sets containing up to 280,000 samples, comparing its performance and speed against ten existing baseline methods under standardized testing conditions.

The analysis produced several key findings. First, the proposed method consistently achieved superior clustering accuracy, outperforming the best existing alternatives by margins between 2.35% and 13.03% on small- to medium-sized datasets. Second, compared to a standard linear-time baseline, the framework improved accuracy by up to 44.24% on medium datasets and up to 11.65% on large datasets. Third, the model scaled effectively to massive datasets without encountering the out-of-memory errors that caused several conventional algorithms to fail. Finally, the iterative optimization converged rapidly, typically within 20 iterations, while maintaining stable clustering quality across varied parameter ranges.

These findings indicate that organizations can achieve state-of-the-art data clustering performance at a fraction of the computational and labor costs associated with traditional manual model tuning. By automatically determining the importance of varied anchor sizes per view, the method reduces total processing overhead, mitigates memory risks on big data, and shortens deployment timelines.

Organizations seeking to cluster large-scale, complex multi-view datasets should consider adopting flexible anchor fusion strategies to replace costly full-graph methods and rigid single-anchor pipelines. The authors have open-sourced the implementation code, enabling technical teams to run pilot evaluations on proprietary data. Potential limitations to keep in mind include the need to define a reasonable range of candidate anchor counts and the algorithm's convergence to local rather than guaranteed global mathematical optima. Nevertheless, the broad experimental validation across varied datasets supports a high degree of confidence in the model's scalability and clustering quality.

HuaweiNanjing University of Aeronautics and AstronauticsNational University of Defense Technologyclustering methodsmulti-viewsviewpointscalable multi-view clusteringanchor graph fusionmulti-view anchor graph
Jeaninezpp/FDAGFZhang et al (2023).pdf
  • Paper: Co-regularized Multi-view Spectral Clustering, Abhishek Kumar et al. (2011). This paper establishes foundational principles for multi-view spectral clustering and consensus graph formulation that motivate scalable anchor graph approaches.
  • Paper: A tutorial on spectral clustering, Ulrike von Luxburg (2007). This tutorial covers essential spectral graph theory and graph Laplacian properties required to understand graph-based clustering and anchor graph approximations.
  • Paper: On Spectral Clustering: Analysis and an algorithm, Andrew Y. Ng et al. (2001). This foundational paper analyzes normalized spectral clustering algorithms, providing the baseline spectral framework that anchor-based methods aim to scale.
  • Paper: Self-Tuning Spectral Clustering, Lihi Zelnik-Manor et al. (2004). This paper introduces self-tuning mechanisms in spectral clustering to handle multi-scale data, directly addressing hyperparameter sensitivity similar to anchor selection challenges.
  • Paper: Kernel k-means: spectral clustering and normalized cuts, Inderjit S. Dhillon et al. (2004). This work establishes the mathematical equivalence between spectral graph partitioning and kernel clustering, underpinning the linear optimization formulations used in anchor graph fusion.
  • Paper: Graph Regularized Nonnegative Matrix Factorization for Data Representation, Deng Cai et al. (2011). This paper provides core techniques for graph regularization and manifold preservation that are central to multi-view affinity matrix construction.
  • Paper: Robust Subspace Segmentation by Low-Rank Representation, Guangcan Liu et al. (2010). This paper details low-rank representation and spectral grouping methods that inform low-rank anchor graph representations.

No sufficiently relevant recommendations were found.

Cover for Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering

Abstract

In the past few years, numerous multi-view graph clustering algorithms have been proposed to enhance the clustering performance by exploring information from multiple views. Despite the superior performance, the high time and space expenditures limit their scalability. Accordingly, anchor graph learning has been introduced to alleviate the computational complexity. However, existing approaches can be further improved by the following considerations: (i) Existing anchor-based methods share the same number of anchors across views. This strategy violates the diversity and flexibility of multi-view data distribution. (ii) Searching for the optimal anchor number within hyper-parameters takes much extra tuning time, which makes existing methods impractical. (iii) How to flexibly fuse multi-view anchor graphs of diverse sizes has not been well explored in existing literature. To address the above issues, we propose a novel anchor-based method termed Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering (FDAGF) in this paper. Instead of manually tuning optimal anchor with massive hyper-parameters, we propose to optimize the contribution weights of a group of pre-defined anchor numbers to avoid extra time expenditure among views. Most importantly, we propose a novel hybrid fusion strategy for multi-size anchor graphs with theoretical proof, which allows flexible and diverse anchor graph fusion. Then, an efficient linear optimization algorithm is proposed to solve the resultant problem. Comprehensive experimental results demonstrate the effectiveness and efficiency of our proposed framework. The source code is available at https://github.com/Jeaninezpp/FDAGF.

Topics

clustering methodsmulti-viewsviewpoint

Keywords

scalable multi-view clusteringanchor graph fusionmulti-view anchor graphanchor graph learninglarge-scale multi-view subspace clusteringspectral clustering

Table of Contents

  • Introduction
  • Related Work
  • Anchor-based MVGC
  • Large-scale Multi-view Subspace Clustering in Linear Time (LMVSC)
  • Methodology
  • Multi-view Diversity
  • Generate Multi-size Multi-view Anchor Graph
  • Flexible Multi-view Anchor Graph Fusion
  • Optimization and Analysis
  • Optimization
  • Analysis and Extensions
  • Experiment
  • Datasets and Baselines
  • Experiment Setup
  • Clustering Performance
  • Running Time
  • Convergence and Sensitivity
  • Conclusion
  • Acknowledgments
  • References

Knowls

  1. Knowl 1 — Flexible multi-choice anchor graph learning objective

    model/method

    For a multi-view dataset, view vv is represented by Xv∈Rdv×nX_v\in\mathbb{R}^{d_v\times n}, where dvd_v is its feature dimension and nn is the number of samples. FDAGF predefines RR anchor-count choices. Choice rr uses mrm_r anchors and learns an anchor matrix Avr∈Rdv×mrA_v^r\in\mathbb{R}^{d_v\times m_r} together with an anchor graph Zvr∈Rmr×nZ_v^r\in\mathbb{R}^{m_r\times n}. The column jj of ZvrZ_v^r gives a nonnegative assignment of sample jj to the mrm_r anchors.

    The method jointly learns all anchor matrices, anchor graphs, and per-view/per-choice importance weights β∈RV×R\beta\in\mathbb{R}^{V\times R} by solving

    min⁡{Avr,Zvr}, β∑v=1V∑r=1Rβvr(∥Xv−AvrZvr∥F2+α∥Zvr∥F2)+λ∥β∥F2s.t.(Avr)⊤Avr=Imr,Zvr≥0,(Zvr)⊤1mr=1n,β≥0,β1R=1V.\begin{aligned} \min_{\{A_v^r,Z_v^r\},\,\beta}\quad &\sum_{v=1}^{V}\sum_{r=1}^{R}\beta_v^r\left(\left\|X_v-A_v^rZ_v^r\right\|_F^2+\alpha\left\|Z_v^r\right\|_F^2\right)+\lambda\left\|\beta\right\|_F^2\\ \text{s.t.}\quad &(A_v^r)^\top A_v^r=I_{m_r},\quad Z_v^r\geq 0,\quad (Z_v^r)^\top\mathbf{1}_{m_r}=\mathbf{1}_n,\\ &\beta\geq 0,\quad \beta\mathbf{1}_R=\mathbf{1}_V . \end{aligned}

    Here, α>0\alpha>0 controls anchor-graph regularization, λ>0\lambda>0 regularizes the importance weights, ImrI_{m_r} is the mr×mrm_r\times m_r identity matrix, and 1q\mathbf{1}_q is an all-ones vector of length qq. Thus, different views can favor different anchor counts, while the weights softly combine all choices rather than selecting a single manually tuned anchor number. The paper notes that setting λ=0\lambda=0 degenerates toward a solution in which only one anchor choice receives weight.

  2. Knowl 2 — Hybrid fusion of unequal-size anchor graphs

    model/method

    FDAGF fuses anchor graphs with different numbers of rows without padding them or constructing their full n×nn\times n affinity matrices. For view vv and choice rr, define

    Σvr=diag⁡(Zvr1n),Z^vr=(Σvr)−1/2Zvr,\Sigma_v^r=\operatorname{diag}(Z_v^r\mathbf{1}_n),\qquad \widehat Z_v^r=(\Sigma_v^r)^{-1/2}Z_v^r,

    where Σvr∈Rmr×mr\Sigma_v^r\in\mathbb{R}^{m_r\times m_r} is diagonal and Z^vr∈Rmr×n\widehat Z_v^r\in\mathbb{R}^{m_r\times n} is the normalized anchor graph. The corresponding full graph would be

    Svr=(Zvr)⊤(Σvr)−1Zvr=(Z^vr)⊤Z^vr∈Rn×n.S_v^r=(Z_v^r)^\top(\Sigma_v^r)^{-1}Z_v^r=(\widehat Z_v^r)^\top\widehat Z_v^r\in\mathbb{R}^{n\times n}.

    Instead of explicitly forming these full graphs, FDAGF vertically concatenates the normalized graphs after weighting them:

    Z‾=[β11Z^11;…;β1RZ^1R;…;βVRZ^VR].\overline Z=\left[\sqrt{\beta_1^1}\widehat Z_1^1;\ldots;\sqrt{\beta_1^R}\widehat Z_1^R;\ldots;\sqrt{\beta_V^R}\widehat Z_V^R\right].

    All blocks have the same nn columns, so this construction is valid even when their row counts mrm_r differ. Its implicit fused graph is

    Z‾⊤Z‾=∑v=1V∑r=1RβvrSvr.\overline Z^\top\overline Z=\sum_{v=1}^{V}\sum_{r=1}^{R}\beta_v^r S_v^r.

    This is the paper’s hybrid fusion strategy: it preserves view- and choice-specific weights while representing the consensus graph through a tall matrix rather than an n×nn\times n matrix.

  3. Knowl 3 — Spectral equivalence of the concatenated anchor representation

    theoretical result

    Let Svr=(Z^vr)⊤Z^vrS_v^r=(\widehat Z_v^r)^\top\widehat Z_v^r be the normalized full graph for view vv and anchor choice rr, let βvr≥0\beta_v^r\geq 0 be its fusion weight, and let Z‾\overline Z be the vertically concatenated weighted matrix defined by stacking βvrZ^vr\sqrt{\beta_v^r}\widehat Z_v^r. The adaptive fused graph is

    S=∑v=1V∑r=1RβvrSvr=Z‾⊤Z‾.S=\sum_{v=1}^{V}\sum_{r=1}^{R}\beta_v^rS_v^r=\overline Z^\top\overline Z.

    If the singular value decomposition of Z‾\overline Z is Z‾=UΛQ⊤\overline Z=U\Lambda Q^\top, with orthonormal UU and QQ, then

    S=QΛ2Q⊤.S=Q\Lambda^2Q^\top.

    Consequently, the eigenvectors of the fused n×nn\times n graph SS are exactly the right singular vectors of Z‾\overline Z. FDAGF therefore obtains the common spectral embedding from the top kk right singular vectors of Z‾\overline Z, where kk is the desired number of clusters, without eigendecomposing the full fused graph.

  4. Knowl 4 — Alternating optimization procedure for FDAGF

    algorithm

    FDAGF takes the multi-view data {Xv}v=1V\{X_v\}_{v=1}^{V}, the number of clusters kk, the number of anchor choices RR, and positive parameters α\alpha and λ\lambda as input. It initializes feasible AvrA_v^r, ZvrZ_v^r, and β\beta, then alternates exact block updates until convergence.

    For each view vv and choice rr, the anchor-matrix update forms Bvr=Xv(Zvr)⊤B_v^r=X_v(Z_v^r)^\top. If Bvr=UbΣbVb⊤B_v^r=U_b\Sigma_bV_b^\top is its singular value decomposition, the optimal orthonormal anchor matrix is Avr=UbVb⊤A_v^r=U_bV_b^\top.

    With AvrA_v^r fixed, define Gvr=(1+α)−1(Avr)⊤XvG_v^r=(1+\alpha)^{-1}(A_v^r)^\top X_v. Each sample column is updated independently by projecting Gvr[:,j]G_v^r[:,j] onto the probability simplex:

    Z_v^r[:,j]=\max\left(G_v^r[:,j}+\eta_j\mathbf{1}_{m_r},0\right),

    where ηj\eta_j is chosen so that the resulting column is nonnegative and sums to one.

    With all anchor matrices and graphs fixed, define the reconstruction cost

    ξvr=∥Xv−AvrZvr∥F2+α∥Zvr∥F2.\xi_v^r=\left\|X_v-A_v^rZ_v^r\right\|_F^2+\alpha\left\|Z_v^r\right\|_F^2.

    For each view, the weight row βv[:]\beta_v[:] is obtained by projecting the vector [−ξv1/(2λ),…,−ξvR/(2λ)][-\xi_v^1/(2\lambda),\ldots,-\xi_v^R/(2\lambda)] onto the RR-dimensional probability simplex. After convergence, FDAGF constructs Z‾\overline Z, computes its top kk right singular vectors, and applies kk-means to those vectors to obtain cluster labels.

    Because every block update minimizes the objective with the other blocks fixed, the objective decreases monotonically during the alternating iterations; the paper states that the procedure converges to a local minimum.

  5. Knowl 5 — Linear dependence on the number of samples

    theoretical result

    Assume that the feature dimensions dvd_v, anchor counts mrm_r, number of views VV, and number of anchor choices RR are fixed independently of the sample count nn. For one FDAGF optimization iteration, the stated computational cost over all views and choices is

    O(∑v=1V∑r=1R(dvmr2+nmr2+dvmrn)).O\left(\sum_{v=1}^{V}\sum_{r=1}^{R}\left(d_vm_r^2+nm_r^2+d_vm_rn\right)\right).

    The three terms correspond respectively to updating the orthonormal anchor matrix, updating the anchor graph, and evaluating the costs needed for the importance-weight update. The post-processing singular-value decomposition of the concatenated anchor matrix and the final kk-means step are also linear in nn under the same fixed-anchor-size assumption. FDAGF consequently has linear complexity with respect to the number of samples and avoids the quadratic storage and cubic spectral computation associated with explicitly constructing an n×nn\times n fused graph.

  6. Knowl 6 — Experimental protocol and benchmark coverage

    experimental setup

    FDAGF was evaluated on ten public multi-view datasets. Their sample count, number of views, and number of clusters were: yaleA (165,3,15)(165,3,15); MSRCV1 (210,6,7)(210,6,7); Flower17 (1360,7,17)(1360,7,17); UCI-Digit (2000,3,10)(2000,3,10); Caltech101 (9144,5,102)(9144,5,102); Reuters (18758,5,5)(18758,5,5); VGGFace (36287,4,100)(36287,4,100); CIFAR100 (60000,4,99)(60000,4,99); YTF20 (63896,4,20)(63896,4,20); and EMNIST (280000,4,9)(280000,4,9).

    The comparison included RMKM, AMGL, FMR, PMSC, BMVC, LMVSC, SMVSC, FMCNOF, FPMVS, and SFMC. Clustering quality was measured by Accuracy (ACC), Normalized Mutual Information (NMI), Purity, and Fscore, with larger values better. K-means initialization and final post-processing were each repeated 50 times, and the reported result was the highest ACC over the tested parameters together with its corresponding NMI, Purity, and running time.

    For FDAGF, α\alpha was searched over {10−5,10−1,101,103}\{10^{-5},10^{-1},10^{1},10^{3}\}, λ\lambda over {101,103,105}\{10^{1},10^{3},10^{5}\}, and R=4R=4. The four anchor choices used anchor counts ranging from kk to 4k4k. Experiments used MATLAB R2020b on an Intel Core i7-7820X CPU with 64 GB RAM.

  7. Knowl 7 — Performance on small and medium datasets

    empirical result

    On the four small or medium datasets, FDAGF achieved the highest value among all compared methods for every reported metric. The exact FDAGF results were:

    • yaleA: ACC 0.89090.8909, NMI 0.91610.9161, Purity 0.92730.9273, Fscore 0.83990.8399; the best non-FDAGF ACC was 0.76060.7606 from PMSC.
    • MSRCV1: ACC 0.90000.9000, NMI 0.81420.8142, Purity 0.90000.9000, Fscore 0.81280.8128; the best non-FDAGF ACC was 0.77480.7748 from FMR.
    • Flower17: ACC 0.44120.4412, NMI 0.43190.4319, Purity 0.49630.4963, Fscore 0.29280.2928; the best non-FDAGF ACC was 0.33750.3375 from LMVSC.
    • UCI-Digit: ACC 0.93500.9350, NMI 0.86700.8670, Purity 0.93500.9350, Fscore 0.87430.8743; the best non-FDAGF ACC was 0.91150.9115 from RMKM.

    The benchmark matrix reported on page 6 therefore supports the paper’s claim that flexible fusion of multiple anchor counts improves clustering quality over both full-graph and conventional fixed-anchor baselines on these datasets.

  8. Knowl 8 — Performance on large-scale datasets

    empirical result

    On the six large-scale datasets, FDAGF obtained the highest ACC among the seven large-scale-oriented methods in every case. The exact FDAGF metric values and the strongest competing ACC were:

    • Caltech101: FDAGF ACC 0.27180.2718, NMI 0.44790.4479, Purity 0.34630.3463, Fscore 0.24420.2442; strongest competing ACC 0.25910.2591 from FPMVS.
    • Reuters: FDAGF ACC 0.61190.6119, NMI 0.42920.4292, Purity 0.68980.6898, Fscore 0.47540.4754; strongest competing ACC 0.57780.5778 from SMVSC.
    • VGGFace: FDAGF ACC 0.08880.0888, NMI 0.16270.1627, Purity 0.12980.1298, Fscore 0.03850.0385; strongest competing ACC 0.07900.0790 from SMVSC.
    • CIFAR100: FDAGF ACC 0.11060.1106, NMI 0.18360.1836, Purity 0.18640.1864, Fscore 0.04480.0448; strongest competing ACC 0.09530.0953 from LMVSC.
    • YouTubeFace20: FDAGF ACC 0.69780.6978, NMI 0.78940.7894, Purity 0.78120.7812, Fscore 0.60090.6009; strongest competing ACC 0.67260.6726 from LMVSC.
    • EMNIST: FDAGF ACC 0.73400.7340, NMI 0.65430.6543, Purity 0.74780.7478, Fscore 0.58880.5888; strongest competing ACC 0.68990.6899 from BMVC.

    The large-scale results reported on page 7 show that FDAGF consistently leads in ACC, although it is not best on every secondary metric for every dataset. SFMC produced an out-of-memory error on EMNIST under the experimental hardware.

  9. Knowl 9 — Running-time advantage without anchor-count traversal

    empirical result

    The running-time comparison on page 6 measured the main algorithm under each method’s optimal parameters rather than including the full hyper-parameter-tuning cost. FDAGF required substantially less time than most compared methods across the evaluated datasets and retained the linear-in-nn behavior predicted by its complexity analysis. On the largest datasets, BMVC, LMVSC, SMVSC, and FMCNOF were sometimes faster than FDAGF, but FDAGF did not need a separate traversal to discover an optimal anchor count: it optimized the contributions of four preselected anchor counts jointly. The paper therefore reports a favorable performance–efficiency tradeoff, especially relative to methods whose anchor-number tuning adds extra time.

  10. Knowl 10 — Fast convergence and parameter stability

    empirical result

    The convergence plots on page 7 show that the FDAGF objective decreases sharply at first and then stabilizes within approximately 20 alternating-optimization iterations on both MSRCV1 and EMNIST. The parameter-sensitivity experiments on the same datasets varied α∈{10−5,10−1,101,103}\alpha\in\{10^{-5},10^{-1},10^{1},10^{3}\} and λ∈{101,103,105}\lambda\in\{10^{1},10^{3},10^{5}\}. Purity remained relatively stable across the tested combinations, indicating that the observed clustering performance was not confined to a very narrow choice of the two trade-off parameters.

Coverage note — The brief single-view extension, the qualitative UCI-Digit graph visualization, and the detailed baseline formulations were omitted because they are secondary to the proposed multi-view model, fusion theorem, optimization, scalability analysis, and benchmark results.

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Citation

MLA
Zhang, P., et al. “Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering”. Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 9, 2023, pp. 11262–69, https://doi.org/10.1609/AAAI.V37I9.26333.
APA
Zhang, P., Wang, S., Li, L., Zhang, C., Liu, X., Zhu, E., Liu, Z., Zhou, L., & Luo, L. (2023). Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering. Proceedings of the AAAI Conference on Artificial Intelligence, 37(9), 11262–11269. https://doi.org/10.1609/AAAI.V37I9.26333
Chicago
Zhang, P., S. Wang, L. Li, et al. 2023. “Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering”. Proceedings of the AAAI Conference on Artificial Intelligence 37 (9): 11262–69. https://doi.org/10.1609/AAAI.V37I9.26333.
Harvard
Zhang, P. et al. (2023) “Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering”, Proceedings of the AAAI Conference on Artificial Intelligence, 37(9), pp. 11262–11269. Available at: https://doi.org/10.1609/AAAI.V37I9.26333.
Vancouver
1. Zhang P, Wang S, Li L, Zhang C, Liu X, Zhu E, Liu Z, Zhou L, Luo L (2023) Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering. Proceedings of the AAAI Conference on Artificial Intelligence 37:11262–11269

BibTeX

@article{Zhang_2023, title={Let the Data Choose: Flexible and Diverse Anchor Graph Fusion for Scalable Multi-View Clustering}, volume={37}, ISSN={2159-5399}, url={http://dx.doi.org/10.1609/AAAI.V37I9.26333}, DOI={10.1609/aaai.v37i9.26333}, number={9}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, publisher={Association for the Advancement of Artificial Intelligence (AAAI)}, author={Zhang, Pei and Wang, Siwei and Li, Liang and Zhang, Changwang and Liu, Xinwang and Zhu, En and Liu, Zhe and Zhou, Lu and Luo, Lei}, year={2023}, month=June, pages={11262–11269} }
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