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Multi-View Spectral Clustering via Structured Low-Rank Matrix Factorization
Yang Wang, Lin Wu
TL;DR
Multi-view spectral clustering must reconcile complementary views without losing their distinct local manifold structures, but common low-rank representations and primal similarity matrices are limited for this purpose. The paper factorizes low-rank similarities into latent data-cluster representations, applies Laplacian regularization, and iteratively minimizes between-view divergences. Extensive real-world experiments demonstrate advantages over state-of-the-art methods, while the approach still relies on tuned parameters and leaves extensions such as out-of-sample clustering for future work.
Problem
Multi-view clustering requires consensus across complementary views, but common low-rank representations can overlook distinct manifolds and primal similarity matrices may poorly reveal clustering structure.
Method
The method learns clustered low-rank representations through structured matrix factorization, Laplacian regularization, and iterative between-view agreement over latent data-cluster representations.
Results
Extensive experiments on real-world multi-view datasets demonstrate advantages over state-of-the-art methods, including LRRGL and common-subspace models.
Takeaways & Limitations
Latent data-cluster representations provide the paper’s basis for coordinating views while preserving view-specific nonlinear manifold structure.
Takeaways & Limitations
The method still requires tuned parameters, and out-of-sample multi-view clustering remains future work.
Abstract
from arXiv · showhide
Multi-view data clustering attracts more attention than their single view counterparts due to the fact that leveraging multiple independent and complementary information from multi-view feature spaces outperforms the single one. Multi-view Spectral Clustering aims at yielding the data partition agreement over their local manifold structures by seeking eigenvalue-eigenvector decompositions. However, as we observed, such classical paradigm still suffers from (1) overlooking the flexible local manifold structure, caused by (2) enforcing the low-rank data correlation agreement among all views; worse still, (3) LRR is not intuitively flexible to capture the latent data clustering structures. In this paper, we present the structured LRR by factorizing into the latent low-dimensional data-cluster representations, which characterize the data clustering structure for each view. Upon such representation, (b) the laplacian regularizer is imposed to be capable of preserving the flexible local manifold structure for each view. (c) We present an iterative multi-view agreement strategy by minimizing the divergence objective among all factorized latent data-cluster representations during each iteration of optimization process, where such latent representation from each view serves to regulate those from other views, such intuitive process iteratively coordinates all views to be agreeable. (d) We remark that such data-cluster representation can flexibly encode the data clustering structure from any view with adaptive input cluster number. To this end, (e) a novel non-convex objective function is proposed via the efficient alternating minimization strategy. The complexity analysis are also presented. The extensive experiments conducted against the real-world multi-view datasets demonstrate the superiority over state-of-the-arts.
1 Introduction
Multi-view spectral clustering seeks agreement across complementary views while preserving view-specific manifold structure. The paper addresses limitations of common low-rank and fusion strategies by factorizing representations into latent data-cluster matrices and coordinating them across views.
- Motivation: Multi-view clustering seeks consensus by leveraging complementary information from multiple feature spaces.Early fusion concatenates views, while other methods optimize multi-view objectives or combine view-specific clusterings.
- Limitations of Existing Methods: Common-subspace methods cannot flexibly characterize heterogeneous local manifold structures across views.CCA-based approaches project views into one lower-dimensional space, while LRR enforces a common low-rank representation.
- Limitations of Existing Methods: LRRGL models view-dependent nonlinear manifolds, but its primal similarity matrix is less intuitive and less effective for revealing ideal clustering structures.Its consensus loss is imposed directly on the primal matrices Zi across views.
- Our Contributions: The proposed method factorizes each low-rank similarity matrix into a symmetric latent data-cluster indicator representation.This factorization is intended to preserve cluster structure while retaining flexible manifold information for each view.
- Our Contributions: Laplacian regularization preserves nonlinear local manifold structure, while consensus loss minimizes divergence among latent data-cluster matrices across views.The method uses a shared clustering number across views and coordinates representations iteratively during optimization.
2 Structured Low-Rank Matrix Factorization to Spectral Clustering
The paper factorizes each view’s low-rank representation into a data-cluster matrix, then combines manifold regularization and consensus learning to preserve view-specific structure while coordinating views.
- Data-cluster factorization: Structured low-rank factorization represents each view with latent data-cluster factors intended to encode its underlying clustering structure.The factors are constrained to be low-dimensional and can be set with an adaptive cluster number.
- Factor constraints and initialization: The factors are initialized by K-means, normalized to have orthonormal columns, and constrained to share the same dimension across views.The shared dimension d is interpreted as the clustering number, while setting U_i = V_i avoids additional parameters.
- Manifold preservation: The factorized representation preserves each view’s nonlinear local manifold structure through a graph Laplacian regularizer.The graph is built from view-specific similarities, with L_i = H_i − W_i.
- Multi-view agreement: Rather than enforcing one identical representation, the model learns distinct view-specific factors and minimizes their divergence to achieve multi-view agreement.The consensus term coordinates the latent representations while retaining flexible structure in each view.
- Objective formulation: The resulting objective jointly models reconstruction, sparse corruption, graph structure, nonnegativity, and consensus among factorized representations.Auxiliary variables D_i and G_i support the constrained formulation and alternating optimization.
3 Optimization Strategy
The non-convex objective is optimized by alternating updates of view-specific variables and auxiliary variables until convergence, with row-wise updates making the factor optimization efficient when the cluster dimension is small.
- Alternating optimization: The non-convex objective is solved by alternately updating U_i and E_i across views, with auxiliary variables D_i and G_i introduced for optimization.Each variable is updated while the others are fixed, and the process repeats until convergence.
- U_i update: Direct inversion for the U_i update costs O(n^3), which is computationally prohibitive when the sample number n is large.The bottleneck arises from inversion of an n × n matrix.
- U_i update: Updating each row of U_i reduces the inversion cost to O(d^3), where d is the shared cluster number and is typically small.The paper identifies this row-wise procedure as much more efficient than the closed-form update.
- Iterative view agreement: During each iteration, each view’s latent factor influences the updates of the other views, progressively decreasing their divergence toward agreement.After convergence, K-means assigns data objects to clusters using the updated U_i.
- Initialization: The algorithm initializes graph quantities and variables, including sparse noise with 20% corrupted entries drawn uniformly from [-5,5].The initialization also sets μ[0] = 10^-3, ϵ1 = 10^-3, and ϵ2 = 10^-1.
4 Solve Ui:
The method solves the non-convex objective through alternating updates of view-specific variables, beginning from initialized latent data-cluster representations and producing final similarities for clustering.
- Alternating optimization: Each Ui row is sequentially updated, after which its columns are orthonormalized.
- Alternating optimization: The algorithm updates Lagrange multipliers and adapts µ during optimization to promote faster convergence.
- Alternating optimization: The optimization alternately updates each view's variables while fixing the others until convergence, using auxiliary variables for the ith-view subproblem.The strategy is summarized as an alternating optimization procedure and applies the same optimization pattern across views.
- Initialization: Ui is initialized so each entry measures similarity between a data object and one of d anchors representing cluster centers.The initialization uses spectral-clustering outcomes with d clusters, and the graph Laplacians are computed offline from each view's original features.
- Initialization: All views use the same initial cluster number d, reflecting the assumed shared underlying clustering structure and reducing initial divergence among views.
- Convergence: The alternating procedure converges to a stationary point, but global optimality is not guaranteed because the objective is not jointly convex.
- Similarity construction: The converged Ui matrices are thresholded, converted into view-specific similarities, averaged into a final matrix W, and clustered into d groups.
4 Experiments
Experiments evaluate the method on multiple multi-view datasets using ACC and NMI, parameter studies, baseline comparisons, noise corruption, and affinity-matrix visualizations. The method outperforms LRRGL and common-subspace approaches, shows robustness to noise, and achieves the best reported PASCAL VOC 2012 performance under selected views.
- Experimental settings: Experiments use UCI handwritten Digits, AwA, and additional benchmark datasets, comparing the method with established multi-view spectral clustering baselines.The baselines include feature concatenation, similarity aggregation, CCA, co-regularization, co-training, and robust or low-rank representation methods.
- Experimental settings: ACC and NMI are evaluated under repeated runs, with the clustering number set to the ground-truth class count and 20% feature corruption used for robustness testing.Results are averaged over 10 runs; corrupted feature elements follow a uniform distribution over [5,- 5].
- Parameter study: ACC increases as both λ2 and β increase, leading to the selected parameter pair λ2 = 0.7 and β = 0.2.The study varies each parameter over [0.001,10] while fixing the other parameter.
- Results: The method outperforms LRRGL, indicating that factorized latent data-cluster representations better encode clustering structure across views than LRRGL’s primal affinity matrix.The reported comparison attributes the advantage to improved encoding of per-view and multi-view data-cluster representations.
- Results: Both the method and LRRGL outperform common-low-dimensional-subspace models, while the two low-rank methods are more effective under noise corruptions than other methods.The paper relates the common-subspace limitation to its inability to encode local graph structures within a single subspace.
- Results: The method achieves the best performance on PASCAL VOC 2012 with the selected two views, and its affinity matrices show clearer diagonal and non-diagonal block structure than LRRGL.The block-structure observations are reported for UCI, AwA, and NUS datasets.
5 Conclusion
The paper learns clustered low-rank representations through structured matrix factorization for multi-view spectral clustering. It coordinates views by minimizing divergences between latent representations, preserves view-specific manifolds with graph Laplacian regularization, and identifies extensions for broader applicability.
- Structured matrix factorization learns clustered low-rank representations for multi-view spectral clustering.The representation is designed to capture data-cluster structure across views.
- Iterative minimization of between-view divergences coordinates factorized latent data-cluster representations across views.The method seeks multi-view agreement directly in the factorized representations.
- Graph Laplacian regularization is imposed on low-dimensional data-cluster representations to adapt to multi-view spectral clustering.
- Future work targets multi-view out-of-sample embedding, jointly learned consensus graphs, non-parametric graph construction, fewer tuned parameters, and view weighting.These directions are presented as ways to extend or improve the method’s practice.