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Localized Sparse Incomplete Multi-view Clustering
Chengliang Liu, Zhihao Wu, Jie Wen, Chao Huang, Yong Xu
TL;DR
Incomplete multi-view clustering must handle arbitrary missing views, information imbalance, and insufficient use of local structure. LSIMVC addresses these issues with sparse matrix factorization, graph-embedded consensus learning, and adaptive view weighting, and it outperforms state-of-the-art methods across six databases.
Problem
Incomplete multi-view clustering lacks methods that flexibly handle arbitrary missing views while addressing information imbalance and local structure across incomplete views.
Method
LSIMVC learns sparse individual and consensus representations using graph-embedded multi-view matrix factorization with adaptive view weighting.
Results
LSIMVC outperforms state-of-the-art incomplete multi-view clustering methods on six databases.
Takeaways & Limitations
Integrating localized structure, sparse representations, and adaptive weighting yields a flexible consensus representation for incomplete multi-view clustering.
Abstract
from arXiv · showhide
Incomplete multi-view clustering, which aims to solve the clustering problem on the incomplete multi-view data with partial view missing, has received more and more attention in recent years. Although numerous methods have been developed, most of the methods either cannot flexibly handle the incomplete multi-view data with arbitrary missing views or do not consider the negative factor of information imbalance among views. Moreover, some methods do not fully explore the local structure of all incomplete views. To tackle these problems, this paper proposes a simple but effective method, named localized sparse incomplete multi-view clustering (LSIMVC). Different from the existing methods, LSIMVC intends to learn a sparse and structured consensus latent representation from the incomplete multi-view data by optimizing a sparse regularized and novel graph embedded multi-view matrix factorization model. Specifically, in such a novel model based on the matrix factorization, a l1 norm based sparse constraint is introduced to obtain the sparse low-dimensional individual representations and the sparse consensus representation. Moreover, a novel local graph embedding term is introduced to learn the structured consensus representation. Different from the existing works, our local graph embedding term aggregates the graph embedding task and consensus representation learning task into a concise term. Furthermore, to reduce the imbalance factor of incomplete multi-view learning, an adaptive weighted learning scheme is introduced to LSIMVC. Finally, an efficient optimization strategy is given to solve the optimization problem of our proposed model. Comprehensive experimental results performed on six incomplete multi-view databases verify that the performance of our LSIMVC is superior to the state-of-the-art IMC approaches. The code is available in https://github.com/justsmart/LSIMVC.
I. INTRODUCTION
Incomplete multi-view clustering addresses data with missing views and uneven information, while prior methods impose restrictive requirements or underuse local structure. LSIMVC learns a flexible sparse, structured consensus representation by combining graph-regularized consensus learning, adaptive view weighting, and sparse representations.
- Motivation: Incomplete multi-view data creates information imbalance across samples and views and weakens complementary information when views are unpaired or massively missing.The paper motivates incomplete multi-view clustering with missing text, images, video, audio, and medical examinations.
- Limitations of prior work: Existing approaches include kernel recovery, multiple-kernel learning, and matrix factorization, but impose requirements such as an intact view or dependence on kernel quality.KCCA-IV requires at least one complete view, while kernel-learning methods depend strongly on kernel quality.
- Proposed method: LSIMVC handles arbitrary view-missing cases, including incomplete views and samples, without the data-distribution requirements of methods such as PMVC.The model targets all kinds of partial multiview data and learns a sparse, structured consistent latent representation for clustering.
- Contributions: LSIMVC is reported to outperform state-of-the-art methods on six databases.The paper describes this as sufficient experimental evidence supporting the proposed flexible incomplete multi-view clustering model.
- Proposed method: Its graph-regularized consensus term jointly performs consensus representation learning and local-structure exploration, applying the graph constraint to discrepancies between unified and individual representations.This design is intended to obtain a more discriminative unified representation.
B. Kernel CCA with incomplete views (KCCA-IV)
KCCA-IV applies kernel canonical correlation analysis to incomplete views by reconstructing the incomplete view’s kernel matrix through Laplacian regularization. Its central limitation is the requirement for at least one complete view, making it unusable when every view is incomplete.
- Method: KCCA-IV assumes one complete view and one incomplete view, using the complete view as basic information for incomplete-kernel recovery.The method constructs the incomplete view’s kernel matrix from available and missing instances before applying kernel CCA.
- Method: The incomplete kernel matrix is reconstructed by Laplacian regularization before kernel CCA extracts consistent low-dimensional features for clustering.The recovered kernel is aligned with the complete-view kernel for subsequent correlation analysis.
- Limitation: KCCA-IV is powerless when all views are incomplete.Its complete-view requirement defines the method’s practical scope boundary.
C. Partial multi-view clustering
LSIMVC addresses partial multi-view clustering by learning sparse, structured consensus representations while accommodating arbitrary missing views and view information imbalance.
- PMVC relies mainly on paired samples and becomes ineffective when incomplete views have few or no shared available samples.
- LSIMVC targets arbitrary view-missing cases by learning a sparse and structured consistent latent representation from partial multi-view data.
- The consensus representation Q aggregates individual view representations through view-presence indicator matrices G^(v).
- The model uses adaptive view weights and sparse representations to account for information imbalance across views.
- A fusion graph embedding constraint preserves local structure from all views within consensus representation learning.
- The graph regularizes distances between individual and consensus representations, encouraging corresponding representations of neighboring instances to remain close.
B. Solution to LSIMVC
LSIMVC solves its constrained optimization problem by alternating updates over basis matrices, individual representations, consensus representation, and adaptive view weights.
- The optimization problem is decomposed into four subproblems for U^(v), P^(v), Q, and α.
- The U^(v) update uses the orthogonal constraint and an SVD of X^(v)P^(v)ᵀ to obtain the optimal basis matrix.
- The P^(v) update is solved as sparse regularized column-wise optimization using a shrinkage operator.
- The Q update is obtained by differentiating its subproblem and setting the partial derivative to zero.
- Alternating updates of U^(v), P^(v), Q, and α produce local optimal solutions for the objective.
C. Computational complexity analysis
The algorithm initializes view-specific factors and graphs, then iteratively updates its variables; its main computational costs arise from SVD and consensus updates.
- The P1 update requires SVD of X^(v)P^(v)ᵀ, while P2 and P4 have no major computational operations.
- Algorithm 1 takes incomplete multi-view data, view-presence information, and parameters λ, β, and r as input.
- Initialization constructs view-specific similarity graphs, random orthogonal U^(v), P^(v)=U^(v)ᵀX^(v), and unit view weights.
- The Q update involves a matrix inverse, but the relevant matrix is diagonal and its inverse is obtained by reciprocating diagonal elements.
IV. EXPERIMENTS AND ANALYSIS
The experiments compare LSIMVC with multiple top-performing incomplete multi-view clustering methods on diverse datasets to assess its effectiveness.
- Experiments compare multiple top-performing IMC methods on diverse datasets to verify LSIMVC's effectiveness.
A. Experiment settings
The experiments evaluate LSIMVC against eleven IMC methods on six multiview databases under constructed missing-view settings, using ACC, NMI, and purity.
- Eleven IMC methods, spanning baseline, kernel-learning, spectral-clustering, matrix-decomposition, and deep approaches, are selected for comparison.
- Six datasets—Handwritten, BBCSport, 3 Sources, Caltech7, NH face, and Animal—provide the experimental evaluation.
- Table I reports detailed statistics for the six multiview databases used in the experiments.
- Incomplete data are constructed by randomly removing view instances while ensuring every sample retains at least one available view.
- Performance is evaluated with clustering accuracy, normalized mutual information, and purity, where higher values indicate better clustering.
B. Experimental results and analysis
LSIMVC achieves nearly the best performance across the incomplete-data benchmarks, with substantial gains on Handwritten and Animal, while local-structure methods are also competitive.
- LSIMVC almost achieves the best performance across the state-of-the-art methods in the reported incomplete-dataset tables.The tables report mean and standard deviation for ACC, NMI, and purity.
- Approximately 10%, 3%, and 8% gains over PIC are reported for ACC, NMI, and purity on Handwritten with a 30% missing rate.
- PIC and UEAF also perform competitively, supporting the positive role of exploiting internal data structure in consensus representation learning.
- All methods improve as Animal’s paired-sample rate increases, whereas randomly removed-view datasets generally worsen as the missing rate increases.
- On Handwritten with 30% removed instances, LSIMVC’s t-SNE representation shows better intra-class compactness and inter-class separation.
C. Parameter sensitivity analysis
LSIMVC’s clustering accuracy is evaluated across penalty and smoothing parameter ranges on incomplete BBCSport and Caltech7 data. The results indicate broad parameter regions with satisfactory or relatively optimal performance, alongside dataset-dependent sensitivity.
- The study searches λ, β, and r on BBCSport with a 30% missing rate and Caltech7 with a 50% missing rate.
- LSIMVC achieves relatively optimal ACC on BBCSport for λ ∈ [10^-1, 10^5] and β ∈ [10^-8, 10^-2], while Caltech7 favors λ ∈ [10^-7, 10^2] and β ∈ [1, 10^5].
- ACC is largely insensitive to λ and β over the tested candidate sets, although the favorable ranges differ between BBCSport and Caltech7.
- For smoothing parameter r, BBCSport maintains satisfactory clustering for r ∈ [2, 7], whereas Caltech7 ACC changes little across the tested values.
D. Ablation experiments
Ablation experiments remove LSIMVC’s adaptive weighting, sparsity, and graph embedding components on BBCSport and 3Sources. Each degraded variant performs worse than the complete model, with graph removal producing the largest reported decline.
- The ablations are conducted on BBCSport and 3Sources under different incomplete ratios.
- All three degraded variants perform worse than complete LSIMVC, confirming contributions from adaptive weighting, sparsity, and graph embedding.
- Removing graph embedding greatly lowers performance, indicating that modeling within-view local structure substantially improves clustering.
E. Convergence analysis
LSIMVC’s block-coordinate optimization has a non-increasing objective with lower bound zero, while experiments show rapid objective decrease and ACC stabilization. Early ACC fluctuations can occur because intermediate variables change substantially during optimization.
- The objective value is monotonically non-increasing during iterative optimization and is bounded below by zero.
- As iterations increase, the objective decreases rapidly and ACC rises toward a stable level on BBCSport and Caltech7.
- BBCSport ACC is not strictly monotonic early in optimization because substantial variable changes make intermediate k-means clustering results fluctuate.
V. CONCLUSION
LSIMVC combines sparse individual and consensus representations, localized structure preservation, and adaptive view weighting for incomplete multi-view clustering. Experiments on six incomplete multi-view databases outperform compared state-of-the-art IMC methods, while high missing rates and large datasets remain important future challenges.
- LSIMVC integrates sparse representations, localized graph structure, and adaptive view weighting into one incomplete multi-view clustering framework.The method jointly models sparse individual views and consensus representations, embeds localized structure, and weights views according to discriminability.
- Experiments on six incomplete multi-view databases show LSIMVC exceeds the compared state-of-the-art incomplete multi-view clustering methods.
- Improving performance and stability under high missing rates and on large datasets remains a crucial problem.