Explainable K-means Neural Networks for Multi-view Clustering

Conference: ICLR 2026
OpenReview: https://openreview.net/forum?id=ljM1HTSH9c
Code: To be confirmed
Area: Explainability / Multi-view Clustering / Representation Learning
Keywords: Multi-view clustering, K-means, Kernel K-means, Explainable neural networks, Subspace learning, Three-layer optimization

TL;DR

Decomposes multi-view clustering into three layer optimization sub-problems: "linear clustering → non-linear clustering → multi-view fusion." Each layer is implemented by K-means / Kernel K-means objectives, assembled into an EKNN network where the function of each layer is explainable, thereby achieving a balance across effectiveness, efficiency, completeness, and consistency.

Background & Motivation

Background: Multi-view clustering aims to cluster data points from different views (different features/modalities). Classic K-means has low computational cost but can only handle linearly separable clusters; non-linear methods such as Kernel K-means and spectral clustering identify arbitrary-shaped clusters by calculating similarities between all point pairs. While effective, they incur high space/time overhead and are impractical for large-scale data.

Limitations of Prior Work: ① Most methods focus solely on effectiveness (clustering quality), with no significant improvement in efficiency; ② Non-linear methods represent a cluster using all data points, leading to high overhead, while "multi-center" approximations are sensitive to center selection; ③ In multi-view scenarios, consistency and completeness are recognized as two major principles, yet existing work cannot simultaneously balance effectiveness, efficiency, consistency, and completeness.

Key Challenge: To achieve good results for non-linearly separable clusters, one must calculate all-pair similarities (expensive); to achieve efficiency, one must use linear approximations (inaccurate). In a multi-view setting, these two must be balanced alongside cross-view consistency and completeness, resulting in four mutually constraining objectives.

Goal: To balance effectiveness, efficiency, completeness, and consistency within an explainable framework for non-linear multi-view clustering on large-scale data.

Key Insight: [Key Observation] Different clusters in complex datasets are non-linearly separable in the global geometric space but linearly separable in the local geometric space—that is, "several small linearly separable clusters compose one non-linearly separable cluster." [Three-layer Decomposition] Based on this hypothesis, multi-view clustering is explicitly decomposed into three sub-problems: 1) Linear clustering of raw data (to reduce scale and maintain efficiency); 2) Non-linear clustering on the set of linear clusters (to maintain effectiveness); 3) Integrating partition matrices across views for multi-view clustering (to maintain completeness and consistency). [Mechanism] Each sub-problem is defined by K-means / Kernel K-means objectives, where K-means plays the role of convolution in a CNN. This constructs the EKNN, where each layer has a known physical meaning and the whole system is explainable.

Method

Overall Architecture

EKNN models multi-view clustering as a three-layer optimization problem, chaining three K-means-style layers for each view \(v\): first, self-representation produces a subspace representation \(\Theta^v\), followed sequentially by a linear clustering layer, a non-linear clustering layer, and a multi-view clustering layer. Finally, a shared partition matrix \(H\) binds the results of all views. The joint objective of the three layers is solved alternatingly using a K-means-style iterative algorithm. Since the inputs, variables, and losses of each layer have clear physical meanings, the network is explainable.

flowchart LR
    X["Multi-view Data X^v"] --> SR["Self-representation Subspace<br/>Θ^v (X^v=X^vΘ^v)"]
    SR --> L["Linear Clustering Layer (K-means)<br/>f_L = ‖Θ^v − W^v V^v‖²"]
    L --> N["Non-linear Clustering Layer (Kernel K-means)<br/>f_N = Tr(K^v) − Tr(Û^vᵀ K^v Û^v)"]
    N --> M["Multi-view Clustering Layer (K-means)<br/>f_M = ‖W^v Û^v − H G^v‖²"]
    M --> H["Shared Partition Matrix H<br/>(Completeness + Consistency)"]
    H --> R["Final Clustering Result"]

Key Designs

1. Linear Clustering Layer: Compressing data with local linear clusters to embed "efficiency" into the framework. For each view, self-representation subspace learning is first performed, constrained by \(X^v = X^v\Theta^v\) to obtain a subspace representation \(\Theta^v\) that reveals the underlying cluster structure. Then, K-means is applied: \(f_{Lv} = \lVert \Theta^v - W^v V^v \rVert_F^2\), where \(W^v \in \mathbb{R}^{n\times p}\) is the partition matrix from data points to \(p\) linear clusters, and \(V^v\) is the cluster center matrix. A key constraint is that the number of linear clusters \(p\) is much larger than the true number of clusters \(k\). This step compresses \(n\) points into \(p\) local linear clusters ("several linear clusters form one non-linear cluster"). Subsequent non-linear clustering only operates on \(p\) clusters instead of \(n\) points, yielding efficiency. The authors note this layer actually comprises two sub-layers: one for \(X^v = X^v\Theta^v\) and one for the reconstruction loss.

2. Non-linear Clustering Layer: Performing Kernel K-means on the set of linear clusters to embed "effectiveness" into the framework. The second layer performs Kernel K-means on the linear cluster centers \(V^v\): \(f_{Nv} = \lVert \Phi(V^v) - U^v Z^v \rVert_F^2 = \mathrm{Tr}(K^v) - \mathrm{Tr}\big((\hat U^v)^\top K^v \hat U^v\big)\), where \(U^v \in \mathbb{R}^{p\times k}\) partitions \(p\) linear clusters into \(k\) non-linear clusters, \(K^v\) is the Gaussian kernel matrix of \(V^v\), and \(\hat U^v = U^v (D^v)^{-1/2}\) is the normalized partition. Because the kernel operation acts only on \(p\) linear clusters rather than all \(n\) points, it preserves the non-linear ability to identify arbitrary-shaped clusters while reducing the heavy \(n \times n\) cost of Kernel K-means to \(p \times p\). This is the pivotal compromise for "both effectiveness and efficiency."

3. Multi-view Fusion Layer + Shared \(H\): Using reconstruction to embed "completeness" and "consistency" into the framework. The third layer aligns the linear + non-linear partition \(W^v\hat U^v\) of each view via a linear mapping \(G^v\) to a partition matrix \(H \in \mathbb{R}^{n\times k}\) shared by all views: \(f_{Mv} = \lVert W^v\hat U^v - HG^v \rVert_F^2\). When \(W^v\hat U^v\) is fixed, this term degrades to classic K-means. "Forcing each view to reconstruct the same \(H\)" indirectly achieves consistency, while \(H\) aggregating information from all views brings completeness. The three layers are combined into a weighted total objective \(\min_H F = \sum_v \alpha f_{Lv} + \beta f_{Nv} + \gamma f_{Mv}\), plus a self-representation reconstruction term \(\eta\lVert X^v - X^v\Theta^v\rVert_F^2\) and a nuclear norm regularization \(\mu\lVert\Theta^v\rVert_*\) (for noise suppression and intra-class homogeneity preservation), optimized alternatingly via K-means iteration.

4. Explainability and Subspace Extension: Grounding "explainable" in model design rather than post-hoc explanation. Unlike most XAI using post-hoc methods (visualization/text/feature correlation) to explain black boxes, EKNN's explainability stems from the model design itself, expressed in two levels of transparency: model decomposability (inputs are raw data, learned variables have clear physical meanings, losses are K-means/Kernel K-means) and algorithmic transparency (dynamic behavior and error surfaces are mathematically derivable). K-means and Kernel K-means are special cases of EKNN. The authors further extend EKNN to multi-view subspace learning: introducing a shared subspace representation \(\Theta^C\) based on \(H\) (constrained by \(H = H\Theta^C\)), solving it using ADMM with auxiliary variables \(R'\) and multipliers \(Y'\), and finally running spectral clustering on the learned \(\Theta^C\)—more complex mappings yield better results.

Key Experimental Results

Main Results Table (ACC% ± STD%)

Across 7 datasets (COIL20/ORL/BBCSport/Reuters/Football/ANIMAL/NoisyMNIST), compared with COMIC, DMF, MVEC, BMVC, DiMSC, RMSL, and SSSL-M.

Dataset	RMSL	SSSL-M	Ours (EKNN)	Ours (Extended)
COIL20	82.19	84.36	80.17	85.22
ORL	88.10	91.56	86.28	91.73
BBCSport	97.61	93.25	92.00	94.21
Reuters	54.04	59.46	42.17	60.55
Football	92.29	89.78	80.15	90.25
ANIMAL	73.19	77.49	70.76	79.00
NoisyMNIST	84.28	85.91	72.30	86.20

The extended version (EKNN for multi-view subspace learning) achieved the highest ACC on 5 out of 7 datasets; F-score and RI trends remained consistent (extended version mostly optimal).

Ablation Study

• Number of linear clusters \(p\): Scanned in \(\{20, \dots, 200\}\); ACC/NMI increased with \(p\) before plateauing once a certain value was reached, though computational cost increased simultaneously; \(p=160\) was chosen.
• Convergence: On COIL20/ORL/BBCSport, the objective function decreased monotonically and converged within ~50 iterations.
• Running Time: Benefiting from "linear clustering for data compression," EKNN and its extended version take significantly less time than RMSL and SSSL-M.

Key Findings

• Extended > Basic: The introduction of latent representation and shared subspace representation yields consistently better results, suggesting that more complex mappings lead to superior outcomes.
• Three-layer Decomposition is Effective: Using linear clustering as a layer can reduce overfitting of the final clustering to some extent, indirectly boosting performance.
• Efficiency Stems from Structure: Compressing \(n\) points into \(p\) clusters via the linear layer first, allowing expensive kernel operations to act only on \(p \times p\) instead of \(n \times n\), is the fundamental reason for significantly lower time costs compared to RMSL/SSSL-M.
• Robustness: Performance is stable across various data types and metrics (ACC/F-score/RI/NMI), validating the universality of the three-layer decomposition.

Highlights & Insights

• The "locally linear, globally non-linear" hypothesis is elegant: It directly converts the geometric intuition that "non-linear clusters = an assembly of several linear clusters" into a computable three-layer structure, decoupling efficiency (linear layer compression) and effectiveness (kernel layer shape recognition) at the structural level.
• Explainability comes from design rather than post-hoc: EKNN does not provide explanations for a black box; instead, every layer's loss and variable have physical meanings. K-means and Kernel K-means are its special cases. This "white-box" explainability is more trustworthy in practical deployment.
• K-means ≈ Convolution Analogy: Analogizing K-means components to CNN convolutions and component connections to network connections provides a unified narrative for "building neural networks with classic clustering."

Limitations & Future Work

• Competitiveness of Basic EKNN is Average: On several datasets, the basic ACC is significantly lower than RMSL/SSSL-M (e.g., only 42.17 vs. 59.46 on Reuters). The version that truly beats the SOTA is the Extended version with subspace learning and spectral clustering, indicating that the gain of the three-layer framework partially relies on additional components.
• High Number of Hyperparameters: \(\alpha, \beta, \gamma, \eta, \mu, \eta', \mu'\) and the number of linear clusters \(p\) all require tuning. While the framework is explainable, the tuning burden is notable.
• Solver and Complexity Details in Appendix: Iterative algorithms and complexity analysis are not expanded in the main text; optimization quality under discrete constraints (\(w, u, h \in \{0, 1\}\)) deserves further scrutiny.
• Scalability: Although efficiency is emphasized, the experimental datasets are limited in scale. Performance in ultra-large or online multi-view scenarios remains to be verified.

Related Work & Insights

• K-means / Kernel K-means / Spectral Clustering: This work reuses these classic objectives as "network layers," continuing the subspace multi-view clustering lineage of LMSC, DiMSC, RMSL, and SSSL-M, but reorganized with three-layer decomposition. Kernel K-means provides the non-linear capacity to recognize arbitrary shapes, while spectral clustering performs the final partition on the shared subspace representation \(\Theta^C\) in the extended version.
• Self-representation Subspace Clustering (Elhamifar & Vidal): Revealed cluster structure through a self-representation coefficient matrix \(\Theta^v\) with the constraint \(X^v = X^v\Theta^v\). This is the basis for allowing the linear layer to cluster "refined data points" rather than raw noisy features.
• Multi-view Consistency / Completeness Principles: Current work often achieves consistency by "maximizing correlation between views." This paper instead achieves it indirectly by "forcing each view to reconstruct the same shared \(H\)," and ensures completeness by letting \(H\) aggregate information from all views—a new implementation of these two classic principles.
• Explainable AI (XAI): The authors clearly distinguish between post-hoc explanations (visualization/text/feature correlation) and design-time transparency, arguing that the latter (model decomposability + algorithmic transparency) is more practical—providing a clear methodological reference for "explainable clustering/explainable representation learning."

One-sentence Evaluation

A multi-view clustering framework that engineers the "local linear assembly into non-linear" geometric intuition into a three-layer K-means network: the logic is elegant, explainability is solid, and true competitiveness comes from the extended version with subspace learning and spectral clustering.

Rating

• Novelty: ⭐⭐⭐⭐ The three-layer optimization decomposition of "local linear to non-linear" + the networked narrative of "each layer is a type of K-means, explainable overall" is a fresh perspective.
• Experimental Thoroughness: ⭐⭐⭐ Covers 7 datasets × multiple metrics + relatively complete \(p\)/convergence/time analysis, but the basic version is weaker than comparison methods in several places, with highlights mainly supported by the extended version.
• Writing Quality: ⭐⭐⭐ The framework and motivation are clearly explained, but the formulas are dense, solution details are moved to the appendix, and some expressions and formatting are slightly unrefined.
• Value: ⭐⭐⭐⭐ Provides a structural framework for "explainable + efficiency/effectiveness decoupling" in multi-view clustering, offering reference value for applications requiring white-box clustering.

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