LAG-Net: A Deep Unfolding Network for Multi-View Clustering via Learnable Anchor Graph

Abstract

Multi-view clustering aims to discover consistent cluster structures from heterogeneous features without supervision. Anchor-based methods improve scalability by representing samples through a compact set of anchors, but fixed anchors may be misaligned with the evolving cluster geometry. This mismatch is the main problem addressed here: the shared sample–anchor graph and the global anchor geometry need to be refined together across heterogeneous views, rather than in two disconnected stages. This paper proposes Learnable Anchor Graph Network (LAG-Net), a deep unfolding framework that jointly learns a shared anchor graph, view-specific anchor indicators, and global anchor alignment within a unified model. The global anchor alignment provides geometric guidance for shared anchor graph learning and promotes anchor consistency across heterogeneous views. By unfolding the derived optimization procedure into a trainable network, LAG-Net enables the anchor representations and sample–anchor relationships to be progressively refined. Experiments on six benchmark datasets demonstrate the effectiveness and scalability of the proposed method compared with representative multi-view clustering approaches.

1. Introduction

Multi-view data describe the same objects from different feature spaces and provide complementary information that cannot be fully captured by a single view [1,2,3]. For example, an image can be characterized by color, texture, shape, and deep semantic features, while a multimedia object may simultaneously contain visual, textual, and acoustic information [4,5]. Multi-view clustering aims to integrate these descriptions and discover consistent cluster structures without label supervision [6,7,8]. By exploiting both cross-view consistency and view-specific complementarity, multi-view clustering has been widely applied to multimedia analysis, pattern recognition, and large-scale data mining [9,10,11,12].

A central problem in multi-view clustering is how to efficiently model structural relationships among samples. Conventional graph-based methods construct pairwise similarity graphs for individual views and then fuse them into a consensus graph [13,14,15,16]. Although these methods can preserve local neighborhood structures, constructing and storing an $n\times n$ graph generally requires quadratic complexity with respect to the number of samples. The subsequent spectral decomposition further limits their applicability to large-scale datasets [17]. Anchor graph learning alleviates this problem by introducing a compact set of representative anchors and representing each sample through its relationships with these anchors [18,19,20]. When the number of anchors is much smaller than the number of samples, the resulting sample–anchor graph provides an efficient approximation to the full sample graph. This strategy has been successfully applied to large-scale clustering, bipartite graph learning, and consensus graph construction.

Despite their scalability, existing anchor-based methods still face a key limitation [21,22,23]: Many methods obtain anchors through an independent preprocessing procedure, such as k-means, and keep them fixed during subsequent graph learning [24,25,26,27]. These predefined anchors are determined by the original feature distribution and cannot adapt to the progressively evolving cluster geometry, so the learned sample–anchor relationships may remain constrained by semantically ambiguous anchor positions. In multi-view clustering, heterogeneous feature spaces make the problem more severe. When anchor geometry and graph structure are optimized separately, it is difficult to coordinate shared sample–anchor assignments with view-specific anchor realizations. Consequently, existing anchor-based models rarely enable the shared anchor graph and global anchor structure to be collaboratively refined within a unified framework.

To address these limitations, Learnable Anchor Graph Network (LAG-Net) is proposed as a deep unfolding framework for scalable multi-view clustering. LAG-Net formulates a unified optimization model that jointly learns a shared sample–anchor graph $\mathbf{H}$, view-specific anchor indicators ${\mathbf{P}}_v$, row-sparse feature noise ${\mathbf{E}}_v$, adaptive view weights $w_v$, and globally aligned anchors $\mathbf{A}$. The shared anchor graph captures the consensus assignment structure across views, while ${\mathbf{P}}_v$ realizes this graph in heterogeneous feature spaces; ${\mathbf{E}}_v$ and $w_v$ further suppress corrupted samples and emphasize more reliable views during reconstruction. A key component is the global anchor alignment mechanism: sample–anchor distances defined by $\mathbf{A}$ provide geometric guidance for learning $\mathbf{H}$, and the learned assignments, in turn, refine the global anchor locations, so anchor geometry and cluster structure can evolve collaboratively rather than in two disconnected stages. An alternating minimization procedure is derived to cyclically update all coupled variables with closed-form or proximal solutions. The iterative procedure is then unfolded into a feed-forward network with five interpretable modules, namely, AnchorGraphModule, FeatureNoiseModule, AnchorIndicatorModule, AnchorAlignmentModule, and ViewWeightModule. Selected thresholds and linear transformations are replaced by learnable parameters, allowing LAG-Net to refine the anchor graph, view-specific representations, and global anchor geometry layer by layer through end-to-end training. Figure 1 illustrates the overall pipeline, and the main contributions of this work are summarized as follows:

Figure 1. Overall architecture of LAG-Net. Each unfolding layer takes the multi-view features $\left\{{\mathbf{X}}_v\right\}$ and the current states $(\mathbf{H},{\mathbf{E}}_v,{\mathbf{P}}_v,\mathbf{A},w_v)$. AnchorGraphModule updates the shared graph $\mathbf{H}$ from view-wise residuals and the sample–anchor distances induced by $\mathbf{A}$. FeatureNoiseModule then estimates ${\mathbf{E}}_v$; AnchorIndicatorModule realizes $\mathbf{H}$ in each view through SVD Procrustes to obtain ${\mathbf{P}}_v$; AnchorAlignmentModule refreshes $\mathbf{A}$ from $\mathbf{H}$ and the concatenated features $\mathbf{X}$; ViewWeightModule returns $w_v$ from the reconstruction residuals. The five updated variables are passed to the next layer, and the final $\mathbf{H}$ is used for clustering. The inputs are the unlabeled multi-view features $\left\{{\mathbf{X}}_v\right\}$. Ground-truth class labels are not fed into the network and are used only after inference to compute ACC, NMI, and ARI.

• Formulate a unified multi-view anchor learning model that jointly captures shared sample–anchor relationships, view-specific anchor representations, feature noise, adaptive view contributions, and global anchor geometry.
• Demonstrate that global anchor alignment provides a geometric reference across heterogeneous views, and that the shared graph and global anchors can be refined together.
• Validate the unfolded network on six benchmarks: LAG-Net attains the best ACC on five of the six datasets, remains competitive on Flickr, and scales linearly in the sample size up to YTF-50 ($n>{10}^5$).

The remainder of this paper is organized as follows: Section 2 reviews related studies on multi-view clustering and anchor graph learning. Section 3 presents the proposed optimization model, solution procedure, and unfolding architecture. Section 4 reports the experimental settings and results. Finally, Section 5 concludes this paper.

2. Related Work

This section reviews the studies most closely related to the proposed method from two perspectives: the first concerns multi-view clustering methods for integrating heterogeneous information, while the second focuses on anchor graph learning for scalable structural representation.

2.1. Multi-View Clustering

Multi-view clustering aims to exploit the consistent and complementary information contained in heterogeneous views to discover intrinsic cluster structures without label supervision. Rather than merely distinguishing between shallow and deep architectures, recent multi-view clustering studies can be organized according to their primary research objectives, including structural modeling and efficient optimization, semantic coordination and robust representation, and adaptation to generalized clustering scenarios.

Structure Modeling and Efficient Optimization: This line of research explicitly models sample relationships through factorization, graphs, tensors, kernels, or anchors, with particular emphasis on structural preservation and computational scalability. Liu et al. [28] proposed a one-pass framework that unifies multi-view matrix factorization and partition learning, producing clustering assignments with linear complexity. It does not learn a bipartite sample–anchor graph whose geometry is refined together with view-specific indicators. Wang et al. [29] investigated the anchor-unaligned problem and established cross-view anchor correspondences before graph fusion, thereby improving the reliability of large-scale multi-view clustering. The correspondence is obtained as a preprocessing stage, so the anchors and the shared graph are not updated in one unfolded loop. Xia et al. [30] developed a tensorized bipartite graph learning method to capture intra-view neighborhood structures and inter-view high-order correlations without constructing full sample-level graphs. The solver remains a hand-designed alternating procedure without learnable unfolding parameters. More recently, Wang et al. [31] introduced a deep multi-view anchor clustering model in which anchors are jointly optimized with sample–anchor relations rather than being manually selected and fixed. It does not couple a shared graph $\mathbf{H}$, view-specific ${\mathbf{P}}_v$, row-sparse ${\mathbf{E}}_v$, adaptive $w_v$, and a global alignment variable $\mathbf{A}$ inside one optimization-inspired network. Xu et al. [32] further combined classical optimization with lightweight neural representation learning, providing an efficient and interpretable alternative to conventional parameter-intensive deep clustering networks. The network is not obtained by unfolding a five-variable anchor objective with global geometric guidance.

Semantic Coordination and Robust Representation: Another major direction focuses on coordinating shared and complementary semantics across views while suppressing irrelevant, conflicting, or noisy information. Yan et al. [33] proposed a differentiable information bottleneck framework that analytically measures mutual information and jointly explores feature-level, cluster-level, and joint consistency. Mutual-information coordination does not produce an interpretable sample–anchor graph. Wang et al. [34] formulated multi-view clustering as an unsupervised cooperative game and employed Shapley values to quantify individual view contributions and enhance cross-view cooperation. View importance is estimated after the fact, rather than as a closed-form weight inside an unfolded reconstruction step. Guo et al. [35] addressed dual noisy correspondence in contrastive multi-view clustering by refining false positive and false negative pairs through contextual similarity and spectral denoising. The robustness target is pairwise correspondence, not row-sparse feature noise on an anchor graph. Dong et al. [36] designed a feature-channel attention encoder and a view-graph-based progressive fusion mechanism to enhance discriminative view-specific information while reducing conflicts among views. Attention fusion does not expose a module-wise correspondence to alternating optimization updates. Considering imperfect cross-view correspondence, Yang et al. [37] modeled partially aligned multi-view data from a causal intervention perspective and learned invariant representations for generalized clustering. The setting is partial alignment, which is outside the complete-view protocol of LAG-Net, but it motivates treating view inconsistency as an explicit variable rather than ignoring it. Chen et al. [38] introduced a disentangle-then-fuse framework that explicitly separates shared and private latent factors and adaptively integrates them through gated attention. Shared/private latents are not realized as a shared anchor graph plus view-specific indicators.

Adaptive and Generalized Multi-View Clustering: Recent studies have also extended multi-view clustering beyond the conventional assumption that all views are complete, static, aligned, and equally reliable. Wang et al. [39] studied continual multi-view clustering with incrementally arriving views and developed a self-training and structure-alignment framework to absorb new-view knowledge while alleviating catastrophic forgetting. Continual arrival of views is a different protocol from the transductive complete-view setting considered here. Xu et al. [40] investigated fairness in multi-view clustering and combined attention-enhanced Kolmogorov–Arnold networks with distribution alignment to reduce the dominance of sensitive attributes while retaining discriminative representations. Fairness with respect to sensitive attributes is not the objective of LAG-Net, which instead reweights views by reconstruction quality.

Taken together, these works remain relevant: factorization and bipartite graphs motivate linear structure learning, learnable anchors motivate moving beyond k-means initialization, view-weighting and noise modeling motivate $w_v$ and ${\mathbf{E}}_v$, and deep fusion motivates end-to-end training. The specific gap left for LAG-Net is to put these pieces into one coupled objective and unfold it into five interpretable modules with global anchor alignment.

2.2. Anchor Graph Learning

Anchor graph learning represents a large number of samples using a compact set of representative anchors and models their relationships through a bipartite sample–anchor graph. Compared with conventional sample-level graphs, anchor graphs substantially reduce the storage and computational costs associated with graph construction and spectral decomposition while retaining the essential neighborhood and cluster structures. Owing to these advantages, anchor graph learning has been applied to a wide range of representation and clustering tasks.

For multi-view graph fusion, Li et al. [41] proposed a scalable and parameter-free bipartite graph fusion method that jointly learns view weights, a consensus bipartite graph, and cluster assignments, avoiding the construction of full sample-level similarity matrices. The fusion remains a shallow alternating solver, and the anchors are not refined by a learnable unfolding of a global alignment variable. For semi-supervised learning, Zhang et al. [42] incorporated feature and label information into a unified sample–anchor graph and jointly predicted the labels of samples and anchors, enabling efficient learning on large-scale multi-view data. The method uses labels, whereas LAG-Net is unsupervised. For dimensionality reduction, Zhu et al. [43] developed an unsupervised optimized bipartite graph embedding framework that simultaneously learns representative anchors, sample–anchor relationships, and low-dimensional projections while preserving local and global geometric structures. The goal is embedding rather than unfolding a clustering objective into network modules.

Anchor graphs have also been introduced into graph neural representation learning for non-graph data. Zhang et al. [44] dynamically constructed bipartite graphs between samples and anchors and performed graph convolution with linear complexity, allowing graph neural networks to cluster large-scale non-graph datasets. The GNN layers are not derived from the proximal and Procrustes updates of a five-variable model. From an auto-encoding perspective, Li et al. [45] proposed BGAE, which integrates multi-view representation encoding, bipartite graph construction, and cluster-oriented decoding into a unified self-supervised framework. The encoder–decoder is trained as a black-box network rather than as unfolded optimization steps. For fast multi-view clustering, Qin et al. [46] designed a dual-consensus anchor learning model that maintains consistency between the learned anchor graph and the resulting cluster partition through structural and orthogonality constraints. Dual consensus is still obtained by alternating optimization without learnable layer-wise maps.

Beyond clustering itself, anchor graph learning has been employed to support feature selection and structural representation. Liu et al. [47] incorporated latent semantic learning and multi-layer anchor graph construction into multi-view unsupervised feature selection, enabling feature importance to be progressively refined while suppressing redundant and noisy information. The output is a feature subset, not a cluster assignment from an unfolded shared graph. For large-scale structured clustering, Ma et al. [48] developed an anchor-based constrained Laplacian rank model that learns compact sample–anchor affinities and explicitly imposes cluster-connected structures without operating on a full similarity graph. The Laplacian-rank constraint is not combined with view-specific indicators, noise, and global alignment in a trainable unfolding. To capture more complex neighborhood relationships, Zhao et al. [49] extended conventional first-order sample–anchor associations to high-order bipartite proximity learning and adaptively integrated structural information from different orders. High-order proximity still uses a fixed-order construction rather than layer-wise learned maps. More recently, Li et al. [50] formulated multi-view bipartite graph clustering from a generalized probabilistic graphical perspective, explicitly characterizing cross-view commonality, view-specific variations, and noise within a unified probabilistic framework. The probabilistic factorization is not unfolded into the five modules used by LAG-Net.

Each of these anchor graph studies is kept because it is close to one ingredient of LAG-Net: bipartite fusion, view weights, noise, structural consistency, or scalable GNN/auto-encoding. The remaining gap is that they do not jointly learn $\mathbf{H}$, ${\mathbf{P}}_v$, ${\mathbf{E}}_v$, $w_v$, and $\mathbf{A}$ by unfolding a single coupled objective.

3. The Proposed Framework: LAG-Net

This section presents the proposed Learnable Anchor Graph Network (LAG-Net) for scalable multi-view clustering. LAG-Net models sample–anchor relations by a shared anchor graph $\mathbf{H}$, realizes this graph in each view through view-specific anchor indicators ${\mathbf{P}}_v$, and introduces a global anchor alignment variable $\mathbf{A}$ to align anchors across views and provide geometric guidance for learning $\mathbf{H}$. The optimization objective that jointly learns the shared anchor graph, view-specific anchor indicators, global anchor alignment, and view weights is first formulated; the alternating solution for each variable block is then derived; and the iterative procedure is finally unfolded into a deep network architecture with interpretable modules. The main notation is summarized in Table 1.

Table 1. Summary of key notations used throughout this paper.

Notation	Description
V, n, m	Number of views, samples, and anchors
$d_v$	Feature dimension of the v-th view
d	Total multi-view feature dimension $d={\sum }_v{d}_v$
${\mathbf{X}}_v\in {\mathbb{R}}^{n\times d_v}$	Input data matrix for the v-th view
$\mathbf{H}\in {\mathbb{R}}^{n\times m}$	Shared anchor graph matrix
${\mathbf{P}}_v\in {\mathbb{R}}^{m\times d_v}$	View-specific anchor indicator matrix for the v-th view
${\mathbf{E}}_v\in {\mathbb{R}}^{n\times d_v}$	Row-sparse noise matrix for the v-th view
$\mathbf{A}\in {\mathbb{R}}^{m\times d}$	Global anchor alignment matrix
$w_v$	Adaptive view weight with $w_v\ge 0$ and ${\sum }_v{w}_v=1$, hence $w_v\in [0,1]$

3.1. Multi-View Anchor Learning Optimization Problem

Given multi-view data ${\{{\mathbf{X}}_v\in {\mathbb{R}}^{n\times d_v}\}}_{v=1}^V$, anchor-based clustering represents all samples through a compact set of m anchors, where $m\ll n$ ensures linear complexity. The relations between samples and anchors are encoded by a shared anchor graph $\mathbf{H}\in {\mathbb{R}}^{n\times m}$, whose entry $h_{ij}$ measures the affinity between sample i and anchor j. Each view is decomposed as ${\mathbf{X}}_v\approx \mathbf{H}{\mathbf{P}}_v+{\mathbf{E}}_v$, where ${\mathbf{P}}_v$ characterizes the anchors in the v-th view and determines how the shared anchor graph $\mathbf{H}$ is realized in that view, while ${\mathbf{E}}_v$ absorbs sample-level corruptions. However, ${\left\{{\mathbf{P}}_v\right\}}_{v=1}^V$ only describe view-specific anchor indicators; they do not explicitly align anchors across views or provide a global geometric reference for learning $\mathbf{H}$. As a result, when anchor semantics are inconsistent across views, the shared anchor graph may be learned from view-wise reconstruction alone and may fail to fully capture cross-view consistency.

To address this limitation, a global anchor alignment matrix $\mathbf{A}\in {\mathbb{R}}^{m\times d}$ is introduced in the multi-view global feature space. The matrix $\mathbf{A}$ is used to align the anchors indicated by different views and provides an explicit geometric reference for learning the shared anchor graph $\mathbf{H}$. Specifically, $\mathbf{A}$ defines sample–anchor distances $\parallel {\mathbf{x}}_i-{\mathbf{a}}_j{\parallel }^2$, which are coupled with the affinities $h_{ij}$ in $\mathbf{H}$ through the anchor regularizer below. Additionally, since views may differ in quality or discriminative power, adaptive weights ${\left\{w_v\right\}}_{v=1}^V$ are incorporated to automatically emphasize high-quality views. The resulting five-variable optimization problem is

$$\begin{array}{cc}{\displaystyle \underset{\mathbf{H},{\mathbf{P}}_v,{\mathbf{E}}_v,\mathbf{A},w_v}{\min }}\hfill & {\displaystyle {\displaystyle \sum _{v=1}^V}{w}_v^r\left(\frac{1}{2}\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v{\parallel }_F^2+{\alpha \parallel \mathbf{H}\parallel }_1+\beta {\parallel {\mathbf{E}}_v\parallel }_{2,1}\right)}\hfill \\ \hfill & + \mu ·\mathcal{R}(\mathbf{A},\mathbf{H},\mathbf{X}),\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & {\displaystyle \mathbf{H}\ge 0, {\displaystyle \sum _{v=1}^V}{w}_v=1, w_v\ge 0,}\hfill \end{array}$$

(1)

where $r>1$ controls the smoothness of the weight distribution. Both ${\alpha \parallel \mathbf{H}\parallel }_1$ and $\beta \parallel {\mathbf{E}}_v{\parallel }_{2,1}$ lie inside the view-weighted sum. They are therefore multiplied by $w_v^r$, whereas the anchor regularizer $\mu ·\mathcal{R}$ is written outside this sum. The view-specific indicators $\left\{{\mathbf{P}}_v\right\}$ are updated by orthogonal Procrustes via economy SVD. When $m\le d_v$, this yields ${\mathbf{P}}_v{\mathbf{P}}_v^{\top }={\mathbf{I}}_m$; when $m>d_v$, it yields ${\mathbf{P}}_v^{\top }{\mathbf{P}}_v={\mathbf{I}}_{d_v}$, which is the tightest orthogonality feasible in ${\mathbb{R}}^{m\times d_v}$. The objective in Equation (1) consists of two complementary parts: The first term is a view-weighted multi-view reconstruction objective, for each view v, the Frobenius loss $\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v{\parallel }_F^2$ requires the shared anchor graph $\mathbf{H}$ and the view-specific anchor indicator ${\mathbf{P}}_v$ to explain the observed features, while ${\alpha \parallel \mathbf{H}\parallel }_1$ promotes sparse anchor usage and $\beta \parallel {\mathbf{E}}_v{\parallel }_{2,1}$ encourages row-sparse noise. The coefficient $w_v^r$ reweights views so that cleaner views contribute more to learning the common representation. The second term, $\mu ·\mathcal{R}(\mathbf{A},\mathbf{H},\mathbf{X})$, is an anchor regularizer that couples the global anchor alignment variable $\mathbf{A}$ with the shared anchor graph $\mathbf{H}$; it does not replace the reconstruction term but instead supplies global geometric guidance for sample–anchor affinity learning. The anchor learning regularizer is defined as

$$\mathcal{R}(\mathbf{A},\mathbf{H},\mathbf{X})={\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{i=1}^n}{h}_{ij}{\parallel {\mathbf{x}}_i-{\mathbf{a}}_j\parallel }^2=\mathrm{Tr}\left({\mathbf{H}}^{\top }\mathbf{D}\right),$$

(2)

with ${\mathbf{x}}_i\in {\mathbb{R}}^d$ denoting the multi-view global feature of sample i and $\mathbf{D}\in {\mathbb{R}}^{n\times m}$ denoting the sample–anchor squared distance matrix, whose $(i,j)$-th entry is ${\left[\mathbf{D}\right]}_{ij}={\parallel {\mathbf{x}}_i-{\mathbf{a}}_j\parallel }^2$. Each view is independently standardized by a per-feature z-score before this concatenation, so that coordinates within a view have zero mean and unit variance. In AnchorGraphModule, $\mathbf{D}$ is further divided by its mean, which removes the global scale of the concatenated Euclidean distances. The reconstruction terms remain view-wise through ${\mathbf{P}}_v$, ${\mathbf{E}}_v$, and $w_v$, so a high-dimensional view cannot dominate the Frobenius losses. The regularizer $\mathcal{R}$ computes a weighted sum of squared distances from every sample to every anchor, where the weight $h_{ij}$ is the assignment strength. Minimizing $\mathcal{R}$ moves each aligned global anchor ${\mathbf{a}}_j$ toward a soft centroid of the samples assigned to it (Equation (14)), while simultaneously encouraging samples with large $h_{ij}$ to stay close to their anchors in feature space. In matrix form, $\mathrm{Tr}\left({\mathbf{H}}^{\top }\mathbf{D}\right)$ feeds the distance signal back into the update of the shared anchor graph $\mathbf{H}$ (Equation (5)), so graph learning and global anchor geometry are optimized in a coupled manner.

3.2. Alternating Optimization

Problem (1) involves five coupled variable blocks with non-convex bilinear terms and manifold constraints. An alternating minimization strategy is adopted to cyclically update $\mathbf{H}$, ${\mathbf{E}}_v$, ${\mathbf{P}}_v$, $\mathbf{A}$, and $w_v$ while fixing the remaining variables. Each subproblem admits either a closed-form solution or a simple proximal update, as detailed below.

Update of $\mathbf{H}$ (Shared Anchor Graph). Fixing the remaining variables, the $\mathbf{H}$-dependent terms in Equation (1) are ${\sum }_v{w}_v^r·{\displaystyle \frac{1}{2}}\parallel {\tilde{\mathbf{X}}}_v-\mathbf{H}{\mathbf{P}}_v{\parallel }_F^2+\alpha {\parallel \mathbf{H}\parallel }_1{\sum }_v{w}_v^r+\mu \mathrm{Tr}\left({\mathbf{H}}^{\top }\mathbf{D}\right)$. The shared sparsity coefficient therefore collapses to $\overline{\alpha }=\alpha {\sum }_v{w}_v^r$, and the subproblem is

$$\underset{\mathbf{H}\ge 0}{\min } {\displaystyle \sum _{v=1}^V}{w}_v^r·{\displaystyle \frac{1}{2}}\parallel {\tilde{\mathbf{X}}}_v-\mathbf{H}{\mathbf{P}}_v{\parallel }_F^2+\overline{\alpha }{\parallel \mathbf{H}\parallel }_1+\mu \mathrm{Tr}\left({\mathbf{H}}^{\top }\mathbf{D}\right),$$

(3)

where ${\tilde{\mathbf{X}}}_v={\mathbf{X}}_v-{\mathbf{E}}_v$ is the noise-corrected residual. The gradient of the smooth part with respect to $\mathbf{H}$ is

$${\nabla }_{\mathbf{H}}f={\displaystyle \sum _{v=1}^V}{w}_v^r(\mathbf{H}{\mathbf{P}}_v-{\tilde{\mathbf{X}}}_v){\mathbf{P}}_v^{\top }+\mu \mathbf{D}.$$

(4)

When $m\le d_v$, exploiting the semi-orthogonality ${\mathbf{P}}_v{\mathbf{P}}_v^{\top }={\mathbf{I}}_m$, the Lipschitz constant is $L_H={\sum }_v{w}_v^r$. The proximal gradient step with soft-thresholding ${\mathcal{S}}_{\theta }(·)$ yields

$${\mathbf{H}}^{(l+1)}\leftarrow {\mathcal{S}}_{\theta }\left({\displaystyle \sum _{v=1}^V}{w}_v^r\left[{\mathbf{H}}^{\left(l\right)}(\mathbf{I}-L_H^{-1}{\mathbf{P}}_v{\mathbf{P}}_v^{\top })+L_H^{-1}{\tilde{\mathbf{X}}}_v{\mathbf{P}}_v^{\top }\right]-\frac{\mu }{L_H}{\mathbf{D}}^{\left(l\right)}\right),$$

(5)

where $\theta =\overline{\alpha }/L_H$ and the nonnegativity constraint is enforced by clipping.

Update of ${\mathbf{E}}_v$ (Feature Noise Separation). For each view v, the ${\mathbf{E}}_v$-dependent terms in Equation (1) are $w_v^r\left({\displaystyle \frac{1}{2}}\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v{\parallel }_F^2+\beta {\parallel {\mathbf{E}}_v\parallel }_{2,1}\right)$. Since $w_v^r>0$, dividing by $w_v^r$ does not change the minimizer, and the noise subproblem reduces to

$$\underset{{\mathbf{E}}_v}{\min } {\displaystyle \frac{1}{2}}\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v{\parallel }_F^2+\beta {\parallel {\mathbf{E}}_v\parallel }_{2,1}.$$

(6)

This is the proximal operator of the ${\mathsf{\ell }}_{2,1}$-norm, solved by the row-wise shrinkage operator ${\mathcal{D}}_{\rho }(·)$:

$${\mathbf{E}}_v^{(l+1)}\leftarrow {\mathcal{D}}_{{\rho }_v}\left({\mathbf{X}}_v-{\mathbf{H}}^{(l+1)}{\mathbf{P}}_v^{\left(l\right)}\right),$$

(7)

where ${\rho }_v=\beta$ and the operator acts on each row ${\mathbf{e}}^{\left(i\right)}$ as

$${\mathcal{D}}_{\rho }\left({\mathbf{e}}^{\left(i\right)}\right)=\max \left(1-\rho /\parallel {\mathbf{e}}^{\left(i\right)}{\parallel }_2,0\right)·{\mathbf{e}}^{\left(i\right)}.$$

(8)

Update of ${\mathbf{P}}_v$ (View-Specific Anchor Indicator). For views with $m\le d_v$, the subproblem for the view-specific anchor indicator ${\mathbf{P}}_v$ under ${\mathbf{P}}_v{\mathbf{P}}_v^{\top }={\mathbf{I}}_m$ is

$$\underset{{\mathbf{P}}_v{\mathbf{P}}_v^{\top }=\mathbf{I}}{\min } {\displaystyle \frac{1}{2}}{\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v\parallel }_F^2.$$

(9)

Expanding the Frobenius norm and dropping ${\mathbf{P}}_v$-independent terms, this reduces to the orthogonal Procrustes problem:

$$\underset{{\mathbf{P}}_v{\mathbf{P}}_v^{\top }=\mathbf{I}}{\max } \mathrm{Tr}\left({\mathbf{P}}_v{({\mathbf{X}}_v-{\mathbf{E}}_v)}^{\top }\mathbf{H}\right).$$

(10)

Let ${\mathbf{H}}^{\top }({\mathbf{X}}_v-{\mathbf{E}}_v)={\mathbf{B}}_v{\Sigma }_v{\mathbf{C}}_v^{\top }$ be the economy SVD; the closed-form solution is

$${\mathbf{P}}_v^{(l+1)}={\mathbf{B}}_v{\mathbf{C}}_v^{\top },$$

(11)

which coincides with the economy SVD Procrustes step implemented in AnchorIndicatorModule; when $m>d_v$, the same update attains ${\mathbf{P}}_v^{\top }{\mathbf{P}}_v={\mathbf{I}}_{d_v}$ instead.

Update of $\mathbf{A}$ (Global Anchor Alignment). The global anchor alignment subproblem isolates the regularizer:

$$\underset{\mathbf{A}}{\min } \mu {\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{i=1}^n}{h}_{ij}{\parallel {\mathbf{x}}_i-{\mathbf{a}}_j\parallel }^2.$$

(12)

Setting the gradient with respect to ${\mathbf{a}}_j$ to zero yields

$${\displaystyle \frac{\partial \mathcal{R}}{\partial {\mathbf{a}}_j}}=-2\mu {\displaystyle \sum _{i=1}^n}{h}_{ij}({\mathbf{x}}_i-{\mathbf{a}}_j)=0,$$

(13)

which admits the closed-form solution

$${\mathbf{a}}_j^{(l+1)}={\displaystyle \frac{{\sum }_{i=1}^n{h}_{ij}^{(l+1)}{\mathbf{x}}_i}{{\sum }_{i=1}^n{h}_{ij}^{(l+1)}}}.$$

(14)

In matrix form, ${\mathbf{A}}^{(l+1)}=\mathrm{diag}{\left({\mathbf{H}}^{\top }{\mathbf{1}}_n\right)}^{-1}{\mathbf{H}}^{\top }\mathbf{X}$, where $\mathbf{X}\in {\mathbb{R}}^{n\times d}$ is the multi-view global feature matrix and each row of $\mathbf{A}$ stores one aligned global anchor in ${\mathbb{R}}^d$. This update is analogous to the centroid step in k-means: each aligned global anchor is pulled toward a weighted average of the samples assigned to it. Equivalently, Equation (14) uses the column-normalized weights ${\tilde{h}}_{ij}=h_{ij}/{\sum }_{i^{\prime }}{h}_{i^{\prime }j}$. In the unfolded network (Section 3.3), this step is approximated by replacing ${\tilde{h}}_{ij}$ with $\mathrm{softmax}\left(h_{ij}\right)$ along the sample dimension. The two weightings coincide only in special cases and are generally different when $\mathbf{H}\ge \mathbf{0}$. If column j of $\mathbf{H}$ is empty, i.e., ${\sum }_i{h}_{ij}=0$, the denominator in Equation (14) is undefined. The unfolded AnchorAlignmentModule therefore uses softmax along the sample dimension, whose denominator ${\sum }_i\exp \left(h_{ij}\right)$ is always positive. An all-zero column then receives uniform weights $1/n$, so the corresponding centroid is the mean of $\mathbf{X}$. A nearly empty column remains well defined under softmax and is not amplified by a vanishing linear mass. When the exact centroid rule is used at initialization and in the ablation of Section 4.3, the column sum is replaced by $\max ({\sum }_i{h}_{ij},\epsilon )$ with $\epsilon ={10}^{-8}$.

Update of $w_v$ (Adaptive View Weights). The weight subproblem is

$$\underset{w_v\ge 0,{\sum }_v{w}_v=1}{\min } {\displaystyle \sum _{v=1}^V}{w}_v^r{\mathcal{Q}}_v,$$

(15)

where ${\mathcal{Q}}_v={\displaystyle \frac{1}{2}}\parallel {\mathbf{X}}_v-\mathbf{H}{\mathbf{P}}_v-{\mathbf{E}}_v{\parallel }_F^2+{\alpha \parallel \mathbf{H}\parallel }_1+\beta {\parallel {\mathbf{E}}_v\parallel }_{2,1}$ is exactly the bracketed view cost in Equation (1). Using Lagrange multipliers, the KKT conditions give

$$w_v^{(l+1)}={\displaystyle \frac{{\left({\mathcal{Q}}_v\right)}^{1/(1-r)}}{{\sum }_{v^{\prime }=1}^V{\left({\mathcal{Q}}_{v^{\prime }}\right)}^{1/(1-r)}}}.$$

(16)

When $r=2$, this simplifies to $w_v\propto 1/{\mathcal{Q}}_v$, assigning higher weights to views with smaller errors. Thus, $w_v$ lies on the probability simplex rather than the unbounded orthant ${\mathbb{R}}_{+}$: $w_v\in [0,1]$ and ${\sum }_v{w}_v=1$. In the implementation, the closed-form weights are clipped at $0.05$ and then renormalized, so that no view is assigned an exact zero weight. The smoothness $r>1$ is selected per dataset, as listed in Section 4.1: $r=3$ on Animals, $r=5$ on Caltech102, $r=2$ on Flickr, $r=5$ on MITIndoor, $r=2$ on Scene15, and $r=5$ on YTF-50.

3.3. Deep Unfolding Network Architecture

The alternating procedure described above treats all algorithmic parameters (thresholds, step sizes) as fixed constants. To enable data-driven adaptation, the L iterations are unfolded into a feed-forward network where each layer corresponds to one round of updates and the fixed parameters are replaced by learnable counterparts optimized via backpropagation. The resulting architecture, LAG-Net, comprises five modules executed sequentially in each unfolding layer. Throughout this subsection, Algorithm 1, and Equations (17)–(25), l indexes unfolding layers and t indexes training epochs. Initialization is written with superscript $\left(0\right)$. Layer $l\in \{0,\dots ,L-1\}$ maps states with superscript $\left(l\right)$ to $(l+1)$, using the layer parameters ${\theta }^{\left(l\right)}$, ${\rho }_v^{\left(l\right)}$, ${\mathbf{R}}^{\left(l\right)}$, ${\mathbf{U}}^{\left(l\right)}$, and ${\mathbf{W}}^{\left(l\right)}$. Each epoch $t=1,\dots ,T$ restarts from the same initialization and updates Θ after one L-layer forward pass. The training loss is evaluated at the last layer, i.e., on ${\mathbf{H}}^{\left(L\right)}$, ${\mathbf{P}}_v^{\left(L\right)}$, ${\mathbf{A}}^{\left(L\right)}$, and $w_v^{\left(L\right)}$. The epoch index t is not attached to the layer states, because $\mathbf{H}$ is not carried from one epoch to the next.

Algorithm 1 LAG-Net: Training and Inference
Require: Multi-view data ${\left\{{\mathbf{X}}_v\right\}}_{v=1}^V$; epochs T; layers L; anchors m; clusters c; smoothness $r>1$; learning rate $\eta$; alignment weight ${\lambda }_{\mathrm{align}}$.
Ensure: Cluster assignments $\mathbf{y}\in {\{1,\dots ,c\}}^n$.
1:	Stage 1: Initialization
2:	Standardize each view independently by a per-feature z-score; concatenate $\mathbf{X}=[{\mathbf{X}}_1,\dots ,{\mathbf{X}}_V]\in {\mathbb{R}}^{n\times d}$;
3:	Compute the top-m left singular vectors of $\mathbf{X}$ and run k-means (10 restarts) to obtain a binary one-hot ${\mathbf{H}}^{\left(0\right)}$;
4:	Initialize ${\mathbf{P}}_v^{\left(0\right)}$ by economy SVD Procrustes of ${\left({\mathbf{H}}^{\left(0\right)}\right)}^{\top }{\mathbf{X}}_v$, $\forall v$;
5:	Set ${\mathbf{E}}_v^{\left(0\right)}=\mathbf{0}$, $\forall v$, and $w_v^{\left(0\right)}=1/V$;
6:	Set ${\mathbf{A}}^{\left(0\right)}=\mathrm{diag}{\left({\left({\mathbf{H}}^{\left(0\right)}\right)}^{\top }{\mathbf{1}}_n\right)}^{-1}{\left({\mathbf{H}}^{\left(0\right)}\right)}^{\top }\mathbf{X}$, with column sums replaced by $\max ({\sum }_i{h}_{ij},\epsilon )$;
7:	Initialize Θ: ${\theta }^{\left(l\right)}\leftarrow 0.01$, ${\rho }_v^{\left(l\right)}\leftarrow 0.01$, ${\mathbf{R}}^{\left(l\right)},{\mathbf{U}}^{\left(l\right)},{\mathbf{W}}^{\left(l\right)}$ from the proximal maps, and $\Delta \mathbf{A}\leftarrow \mathbf{0}$;
8:	Stage 2: End-to-End Training
9:	for  $t=1\to T$  do
10:	Reload $({\mathbf{H}}^{\left(0\right)},{\mathbf{P}}_v^{\left(0\right)},{\mathbf{E}}_v^{\left(0\right)},{\mathbf{A}}^{\left(0\right)},w_v^{\left(0\right)})$ from Stage 1;
11:	for $l=0\to L-1$ do
12:	AnchorGraphModule: ${\mathbf{H}}^{(l+1)}\leftarrow {\mathcal{S}}_{{\theta }^{\left(l\right)}}(·)$ as in Equation (17), then clip ${\mathbf{H}}^{(l+1)}\ge 0$;
13:	FeatureNoiseModule: ${\mathbf{E}}_v^{(l+1)}\leftarrow {\mathcal{D}}_{{\rho }_v^{\left(l\right)}}({\mathbf{X}}_v-{\mathbf{H}}^{(l+1)}{\mathbf{P}}_v^{\left(l\right)})$, $\forall v$;
14:	AnchorIndicatorModule: stop-gradient SVD Procrustes for ${\mathbf{P}}_v^{(l+1)}$, $\forall v$;
15:	AnchorAlignmentModule: softmax centroid of ${\mathbf{H}}^{(l+1)}$ plus $\Delta \mathbf{A}$ to obtain ${\mathbf{A}}^{(l+1)}$;
16:	ViewWeightModule: closed-form $w_v^{(l+1)}$ from reconstruction residuals, clip at $0.05$, and renormalize;
17:	end for
18:	Compute $\mathcal{L}={\mathcal{L}}_{\mathrm{rec}}+{\lambda }_{\mathrm{align}}{\mathcal{L}}_{\mathrm{align}}$ on $({\mathbf{H}}^{\left(L\right)},{\mathbf{P}}_v^{\left(L\right)},{\mathbf{A}}^{\left(L\right)},w_v^{\left(L\right)})$;
19:	Update Θ by Adam ($\eta =0.001$) with backpropagation; $\mathbf{H}$ is not carried to the next epoch;
20:	end for
21:	Stage 3: Clustering Inference
22:	Compute the truncated SVD ${\mathbf{H}}^{\left(L\right)}\approx {\mathbf{U}}_c{\Sigma }_c{\mathbf{V}}_c^{\top }$ and set $\widehat{\mathbf{H}}={\mathbf{U}}_c$;
23:	return k-means on $\widehat{\mathbf{H}}$ with c clusters ($n_{\mathrm{init}}=1$, at most 500 iterations; 10 random states for reporting).

AnchorGraphModule. This module updates the shared anchor graph $\mathbf{H}$ by parameterizing Equation (5):

$${\mathbf{H}}^{(l+1)}={\mathcal{S}}_{{\theta }^{\left(l\right)}}\left({\displaystyle \sum _{v=1}^V}{\left(w_v^{\left(l\right)}\right)}^r\left[{\mathbf{H}}^{\left(l\right)}{\mathbf{R}}^{\left(l\right)}+{\tilde{\mathbf{X}}}_v^{\left(l\right)}{\left({\mathbf{P}}_v^{\left(l\right)}\right)}^{\top }{\mathbf{U}}^{\left(l\right)}\right]+\gamma {\mathbf{D}}^{\left(l\right)}{\mathbf{W}}^{\left(l\right)}\right),$$

(17)

where ${\mathbf{R}}^{\left(l\right)},{\mathbf{U}}^{\left(l\right)},{\mathbf{W}}^{\left(l\right)}\in {\mathbb{R}}^{m\times m}$ are learnable linear transformations initialized from Equation (5) as ${\mathbf{R}}^{\left(l\right)}\leftarrow (\mathbf{I}-L_H^{-1}{\mathbf{P}}_v{\mathbf{P}}_v^{\top })=(1-L_H^{-1}){\mathbf{I}}_m$, ${\mathbf{U}}^{\left(l\right)}\leftarrow L_H^{-1}{\mathbf{I}}_m$, and ${\mathbf{W}}^{\left(l\right)}\leftarrow -L_H^{-1}{\mathbf{I}}_m$ (using ${\mathbf{P}}_v{\mathbf{P}}_v^{\top }={\mathbf{I}}_m$), respectively, and ${\theta }^{\left(l\right)}$ is a learnable sparsity threshold initialized as $\overline{\alpha }/L_H$; $\gamma =\mu >0$ is a hyperparameter controlling the anchor-distance strength, and ${\mathbf{D}}^{\left(l\right)}\in {\mathbb{R}}^{n\times m}$ is the mean-normalized sample–anchor squared distance matrix between n samples and m anchors. All three matrices are subsequently fine-tuned jointly with the other modules via backpropagation. The view-weighted aggregation replaces the fixed average, allowing the network to emphasize informative views.

FeatureNoiseModule. This module separates sample-level corruptions via the row-wise shrinkage in Equation (7):

$${\mathbf{E}}_v^{(l+1)}={\mathcal{D}}_{{\rho }_v^{\left(l\right)}}\left({\mathbf{X}}_v-{\mathbf{H}}^{(l+1)}{\mathbf{P}}_v^{\left(l\right)}\right),$$

(18)

where ${\rho }_v^{\left(l\right)}$ is a learnable per-view threshold that replaces the constant $\beta$ in Equation (6).

AnchorIndicatorModule. This module updates the view-specific anchor indicators via the SVD Procrustes solution in Equation (11):

$${\mathbf{P}}_v^{(l+1)}=\mathrm{SVD}\text{-}\mathrm{Procrustes}\left({\mathbf{H}}^{(l+1)\top }({\mathbf{X}}_v-{\mathbf{E}}_v^{(l+1)})\right).$$

(19)

Thus, ${\mathbf{P}}_v^{(l+1)}$ determines how the shared anchor graph ${\mathbf{H}}^{(l+1)}$ is realized in the v-th view. This SVD is a stop-gradient structural update: the Procrustes factor is computed from a detached copy of ${\mathbf{H}}^{(l+1)\top }({\mathbf{X}}_v-{\mathbf{E}}_v^{(l+1)})$ and is not differentiated. Consequently, the well-known SVD Jacobian instability at repeated or vanishing singular values does not arise. No additional damping or unique-SVD correction is applied in the forward pass; the economy factorization ${\mathbf{P}}_v={\mathbf{B}}_v{\mathbf{C}}_v^{\top }$ is the standard polar/Procrustes factor. Gradients of $\mathcal{L}$ therefore flow through $\mathbf{H}$, ${\mathbf{E}}_v$, and the learnable maps, treating ${\mathbf{P}}_v$ as a constant orthogonal state within each layer.

AnchorAlignmentModule. This module learns global anchor alignment that connects the view-specific anchor indicators and provides geometric guidance for the shared anchor graph. It follows the centroid structure of Equation (14) in two parts: First, a softmax-based centroid approximation is computed for each anchor j as

$${\mathbf{a}}_{j,\mathrm{closed}}^{(l+1)}={\displaystyle \frac{{\sum }_{i=1}^n\exp \left(h_{ij}^{(l+1)}\right){\mathbf{x}}_i}{{\sum }_{i^{\prime }=1}^n\exp \left(h_{i^{\prime }j}^{(l+1)}\right)}}.$$

(20)

In matrix form, ${\mathbf{A}}_{\mathrm{closed}}^{(l+1)}=\mathrm{softmax}{({\mathbf{H}}^{(l+1)},\dim =0)}^{\top }\mathbf{X}$, where $\mathbf{X}\in {\mathbb{R}}^{n\times d}$ is the multi-view global feature matrix and each row of ${\mathbf{A}}_{\mathrm{closed}}^{(l+1)}$ stores one aligned global anchor in ${\mathbb{R}}^d$. This approximates, rather than equals, the closed-form update in Equation (14), i.e., ${\mathbf{A}}^{(l+1)}=\mathrm{diag}{\left({\mathbf{H}}^{\top }{\mathbf{1}}_n\right)}^{-1}{\mathbf{H}}^{\top }\mathbf{X}$. Equivalently, Equation (20) uses the softmax weights $w_{ij}=\exp \left(h_{ij}\right)/{\sum }_{i^{\prime }}\exp \left(h_{i^{\prime }j}\right)$, whereas Equation (14) uses the linear weights ${\tilde{h}}_{ij}=h_{ij}/{\sum }_{i^{\prime }}{h}_{i^{\prime }j}$, and softmax provides a smooth surrogate that is convenient for implementation and numerical stability. In particular, softmax remains defined for empty or nearly empty columns of $\mathbf{H}$, as discussed after Equation (14). Second, a learnable perturbation $\Delta \mathbf{A}\in {\mathbb{R}}^{m\times d}$, initialized to zero, refines the aligned global anchors beyond the assignment-induced centroids:

$${\mathbf{A}}^{(l+1)}={\mathbf{A}}_{\mathrm{closed}}^{(l+1)}+\Delta \mathbf{A}.$$

(21)

The centroid approximation is treated as a fixed structural update within each forward pass, while $\Delta \mathbf{A}$ is optimized end to end through the training loss $\mathcal{L}$ in Equation (23).

ViewWeightModule. In the unfolded network, the view quality score is taken as the per-view reconstruction error ${\mathcal{Q}}_v^{(l+1)}={\parallel {\mathbf{X}}_v-{\mathbf{H}}^{(l+1)}{\mathbf{P}}_v^{(l+1)}-{\mathbf{E}}_v^{(l+1)}\parallel }_F^2$. This drops the shared offset ${\alpha \parallel \mathbf{H}\parallel }_1$ and the already-separated noise term from the optimization score in Equation (15). This module then computes adaptive view weights via the closed-form solution in Equation (16):

$$w_v^{(l+1)}={\displaystyle \frac{{\left({\mathcal{Q}}_v^{(l+1)}\right)}^{1/(1-r)}}{{\sum }_{v^{\prime }}{\left({\mathcal{Q}}_{v^{\prime }}^{(l+1)}\right)}^{1/(1-r)}}}.$$

(22)

The same simplex constraint is kept: after this update, the weights are clipped at $0.05$ and renormalized so that ${\sum }_v{w}_v^{(l+1)}=1$ and $w_v^{(l+1)}\in [0,1]$.

Summary. Each unfolding layer is composed of five coupled modules. AnchorGraphModulelearns the shared anchor graph $\mathbf{H}$; FeatureNoiseModule estimates the row-sparse noise ${\mathbf{E}}_v$; AnchorIndicatorModule updates the view-specific anchor indicators ${\mathbf{P}}_v$ via SVD Procrustes; AnchorAlignmentModule updates $\mathbf{A}$ to align the global anchor structure with the view-specific indicators; and ViewWeightModule adaptively assigns weights to different views according to reconstruction quality. The complete set of learnable parameters is $\mathrm{\Theta }={\{{\theta }^{\left(l\right)},{\rho }_v^{\left(l\right)},{\mathbf{R}}^{\left(l\right)},{\mathbf{U}}^{\left(l\right)},{\mathbf{W}}^{\left(l\right)}\}}_{l=1}^L\cup \{\Delta \mathbf{A}\}$, comprising the sparsity threshold, per-view noise shrinkage thresholds, representation learning layers, anchor-distance linear map, and the shared anchor perturbation $\Delta \mathbf{A}$; the scalar $\gamma$ is fixed as a hyperparameter. All parameters are optimized jointly via backpropagation. The thresholds ${\theta }^{\left(l\right)}$ and ${\rho }_v^{\left(l\right)}$ are unconstrained real parameters. They are initialized from positive proximal constants and are not projected onto ${\mathbb{R}}_{+}$ after each Adam step. If ${\theta }^{\left(l\right)}<0$, the soft-thresholding operator reduces or disables ${\mathsf{\ell }}_1$ shrinkage, while $\mathbf{H}$ is still clipped to $\mathbf{H}\ge 0$. If ${\rho }_v^{\left(l\right)}<0$, the shrinkage scale in Equation (8) is clipped to $[0,1]$ in the implementation, so that the layer leaves ${\mathbf{E}}_v$ unshrunk rather than amplifying the residual. The layer-wise thresholds and $m\times m$ maps do not depend on n or $\left\{d_v\right\}$. The shared perturbation $\Delta \mathbf{A}\in {\mathbb{R}}^{m\times d}$ scales with $d={\sum }_v{d}_v$ and m, so the total parameter count is independent of n but not of the feature dimensions.

The initial shared graph ${\mathbf{H}}^{\left(0\right)}\in {\mathbb{R}}^{n\times m}$ is a binary one-hot assignment matrix, not a distance kernel and not a subsequently normalized soft affinity. After forming the concatenated matrix $\mathbf{X}$, a truncated SVD retains the top-m left singular vectors $\mathbf{U}\in {\mathbb{R}}^{n\times m}$; k-means with m clusters is then applied to the rows of $\mathbf{U}$ (10 restarts and a fixed random state). If sample i is assigned to cluster j, then $h_{ij}=1$ and $h_{i{j}^{\prime }}=0$ for $j^{\prime }\ne j$, so each row of ${\mathbf{H}}^{\left(0\right)}$ sums to one solely because the assignment is one-hot. This hard assignment is used only as the starting point of unfolding. Subsequent layers produce a nonnegative real matrix through soft thresholding and clipping, without a simplex constraint on the rows of $\mathbf{H}$. Stage 2 reloads the Stage 1 initialization at every epoch, runs L unfolding layers in the order $(\mathbf{H},{\mathbf{E}}_v,{\mathbf{P}}_v,\mathbf{A},w_v)$, evaluates $\mathcal{L}={\mathcal{L}}_{\mathrm{rec}}+{\lambda }_{\mathrm{align}}{\mathcal{L}}_{\mathrm{align}}$ on the last-layer states, and updates only Θ by Adam. Stage 3 is inference: the spectral embedding of ${\mathbf{H}}^{\left(L\right)}$ is clustered by k-means with c clusters.

3.4. Training Objective

Problem (1) defines the alternating updates and the unfolded network architecture. LAG-Net is therefore an optimization-inspired unfolding network rather than a literal transcription of one alternating iteration. The following properties of Problem (1) are retained: the five-block update order $(\mathbf{H},{\mathbf{E}}_v,{\mathbf{P}}_v,\mathbf{A},w_v)$, the proximal forms of the $\mathbf{H}$ and ${\mathbf{E}}_v$ updates, the SVD Procrustes update of ${\mathbf{P}}_v$, and the closed-form view-weight rule. The SVD Procrustes step is retained as a stop-gradient structural update rather than a differentiable SVD layer. The following are replaced by learned or smooth approximations: The linear maps and thresholds in AnchorGraphModule and FeatureNoiseModule become layer-wise trainable parameters initialized from the proximal step. The nonnegative centroid update of $\mathbf{A}$ is replaced by a softmax weighting along the sample dimension together with a learnable perturbation $\Delta \mathbf{A}$. The view quality score in ViewWeightModule uses the reconstruction residual only. The training loss $\mathcal{L}$ is a differentiable surrogate of Equation (1). In particular, the ${\mathsf{\ell }}_1$ sparsity on $\mathbf{H}$ and the ${\mathsf{\ell }}_{2,1}$ row-sparsity on ${\mathbf{E}}_v$ are already enforced by the soft-thresholding and shrinkage operators inside each unfolding layer, so they are not duplicated as explicit loss terms during backpropagation. The overall training loss combines reconstruction with an optional anchor Gram alignment term:

$$\mathcal{L}={\mathcal{L}}_{\mathrm{rec}}+{\lambda }_{\mathrm{align}} {\mathcal{L}}_{\mathrm{align}},$$

(23)

where ${\lambda }_{\mathrm{align}}\ge 0$ is an anchor alignment hyperparameter, and setting ${\lambda }_{\mathrm{align}}=0$ recovers reconstruction-only training.

Weighted Reconstruction Loss. The reconstruction term supervises only the commonality component ${\mathbf{H}}^{\left(L\right)}{\mathbf{P}}_v^{\left(L\right)}$ to reconstruct each view, with higher-quality views contributing more through $w_v^{\left(L\right)}$, and the noise term ${\mathbf{E}}_v$ is excluded so that the model cannot trivially satisfy the objective by shifting mass into ${\mathbf{E}}_v$:

$${\mathcal{L}}_{\mathrm{rec}}={\displaystyle \sum _{v=1}^V}{\left(w_v^{\left(L\right)}\right)}^r{\parallel {\mathbf{X}}_v-{\mathbf{H}}^{\left(L\right)}{\mathbf{P}}_v^{\left(L\right)}\parallel }_F^2.$$

(24)

Anchor Alignment Loss. The matrix ${\mathbf{A}}^{\left(L\right)}{\left({\mathbf{A}}^{\left(L\right)}\right)}^{\top }$ describes the global anchor relation structure, while ${\mathbf{P}}_v^{\left(L\right)}{\left({\mathbf{P}}_v^{\left(L\right)}\right)}^{\top }$ describes the corresponding structure induced by the view-specific anchor indicator in the v-th view. These two structures are aligned across all views to encourage cross-view anchor consistency:

$${\mathcal{L}}_{\mathrm{align}}={\displaystyle \sum _{v=1}^V}\parallel {\mathbf{A}}^{\left(L\right)}{\left({\mathbf{A}}^{\left(L\right)}\right)}^{\top }-{\mathbf{P}}_v^{\left(L\right)}{\left({\mathbf{P}}_v^{\left(L\right)}\right)}^{\top }{\parallel }_F^2.$$

(25)

This term is differentiable w.r.t. $\Delta \mathbf{A}$ and encourages consistency between global anchor alignment and view-specific anchor indicators.

Minimizing $\mathcal{L}$ drives the unfolded modules to produce a shared anchor graph ${\mathbf{H}}^{\left(L\right)}$ that is consistent across views, while the subsequent spectral embedding and k-means steps in Algorithm 1 recover the final cluster assignments.

4. Results

In this section, the proposed LAG-Net is evaluated on six multi-view clustering datasets and compared with representative shallow and deep baselines. The experimental settings, including datasets, compared methods, evaluation metrics, and implementation details, are first described. The overall comparison results are then reported to verify the effectiveness of the learnable anchor graph framework. Finally, component-related ablation studies and sensitivity analyses over ${\lambda }_{\mathrm{align}}$ and the anchor number m are presented to validate the contribution of each module and the robustness of the proposed method.

4.1. Experimental Setups

4.1.1. Datasets, Compared Methods, and Evaluation Metrics

Six publicly accessible datasets are employed to validate the performance of the proposed approach. The datasets include Animals, Caltech102, Flickr, MITIndoor, Scene15, and YTF-50. Details for all datasets are presented in Table 2. These collections are standard multi-view clustering benchmarks with publicly circulated feature matrices. Animals contains two DECAF views of the Animals with Attributes images. Caltech102 is the six-view Caltech-101 feature set with 9144 samples. Flickr, MITIndoor, Scene15, and YTF-50 use the same sample sizes, view counts, and feature dimensions as in prior multi-view clustering studies. The processed $.\mathtt{mat}$ files used in this paper are available from the corresponding author upon reasonable request. The compared multi-view clustering methods include the following: (1) Shallow Approaches: LMVSC [51], AIMC [52], FastMICE [53], FDAGF [54], RCAGL [55], and UDBGL [56]; (2) Deep Methods: SDSNE-Net [57], CVCL-Net [58], SCMVC-Net [59], and DMCAG-Net [60]. To quantitatively assess clustering performance, three widely used metrics are adopted: clustering accuracy (ACC), normalized mutual information (NMI), and adjusted Rand index (ARI). Higher values of all three metrics indicate better clustering quality.

Table 2. A brief description of the tested datasets.

Datasets	Samples	Views	Feature Dimensions	Classes
Animals	10,158	2	4096/4096	50
Caltech102	9144	6	48/40/254/1984/512/928	102
Flickr	12,154	2	100/100	7
MITIndoor	5360	4	3600/1770/1240/4096	67
Scene15	4485	3	1800/1180/1240	15
YTF-50	126,054	4	944/576/512/640	50

4.1.2. Implementation Details

All experiments were implemented in PyTorch 2.5.1 and run on a single NVIDIA GeForce RTX 3090 GPU (24 GB). LAG-Net was trained for $T=100$ epochs with Adam at learning rate $\eta =0.001$. The number of anchors was set to $m=c$, where c denotes the number of clusters on each dataset. The anchor alignment hyperparameter was fixed as ${\lambda }_{\mathrm{align}}=1.0$. The unfolding depth L and the view-weight smoothness parameter r were investigated through a sensitivity study. The dataset-specific pairs $[L,r]$ reported below were identified through a sensitivity analysis, where clustering ACC was used as the evaluation metric. Since ACC depends on ground-truth labels, these choices are not considered a label-free hyperparameter selection strategy. To further evaluate the robustness without label-dependent search, we also report a common predefined setting $L=3$, $r=2$, ${\lambda }_{\mathrm{align}}=1.0$, and $m=c$ shared by all datasets. The sensitivity-study configurations used in the main comparison tables are [L, r] = [5, 3] on Animals, [5, 5] on Caltech102, [3, 2] on Flickr, [4, 5] on MITIndoor, [7, 2] on Scene15, and [1, 5] on YTF-50. The main tables use one training run with random seed 40, and the Python 3.10.18, NumPy, and PyTorch random number generators are synchronized to this seed. Each view is independently standardized by a per-feature z-score before concatenation. Adam uses $({\beta }_1,{\beta }_2)=(0.9,0.999)$, $\epsilon ={10}^{-8}$, and weight decay ${10}^{-4}$, together with a ReduceLROnPlateau scheduler (factor $0.3$, patience 15, minimum learning rate ${10}^{-8}$). Training always runs for the fixed budget $T=100$ and does not use a validation-based early-stopping rule. Initialization follows Algorithm 1. In particular, ${\mathbf{H}}^{\left(0\right)}$ is the one-hot indicator of k-means on the truncated SVD of $\mathbf{X}$, as specified after Algorithm 1. The learnable thresholds ${\theta }^{\left(l\right)}$ and ${\rho }_v^{\left(l\right)}$ are unconstrained real parameters and are not projected to remain nonnegative. After training, the spectral embedding of $\mathbf{H}$ is clustered by k-means with c clusters, $n_{\mathrm{init}}=1$, and at most 500 iterations. The evaluation is transductive: all n samples participate in training and in the subsequent clustering of $\mathbf{H}$. The number of clusters c is assumed known and is used both for $m=c$ and for the final k-means. The reported mean and standard deviation are computed over 10 k-means initializations with random states $0,\dots ,9$. A baseline entry with standard deviation $0.00$ corresponds to a deterministic method or a single reported run. Competing methods are evaluated on the same pre-extracted feature matrices as LAG-Net. Their hyperparameters follow the original papers or the default settings of the official implementations, without additional label-driven search.

4.2. Overall Comparison

The overall comparison results are reported in Table 3, Table 4 and Table 5. Compared with the shallow approaches, including LMVSC, AIMC, FastMICE, FDAGF, RCAGL, and UDBGL, LAG-Net achieves clear improvements on most datasets. For ACC, LAG-Net obtains the best results on Animals, Caltech102, MITIndoor, Scene15, and YTF-50, and also achieves the second-best result on Flickr. For NMI, LAG-Net is best on Animals, Caltech102, MITIndoor, and YTF-50. This shows that the proposed unfolded anchor learning framework can provide more discriminative sample–anchor assignments than conventional shallow graph or anchor-based optimization methods. The improvements are especially evident on Caltech102 and MITIndoor, where LAG-Net substantially exceeds the strongest shallow competitors, indicating the benefit of jointly learning the shared anchor graph, view-specific anchor indicators, adaptive view weights, and global anchor alignment.

Table 3. ACC of all compared multi-view clustering methods, where the best and second-best results are highlighted in bold and underlined, respectively (mean% and standard deviation%); “–” indicates an out-of-memory error.

Methods\Datasets	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
LMVSC	27.91 (0.00)	11.66 (0.00)	41.93 (0.00)	15.70 (0.00)	40.60 (0.00)	60.57 (0.00)
AIMC	58.09 (0.00)	23.94 (0.00)	49.59 (0.00)	36.94 (0.00)	46.52 (0.00)	68.02 (0.00)
FastMICE	64.39 (3.67)	20.05 (1.73)	51.60 (2.93)	29.57 (1.49)	50.31 (1.73)	70.39 (1.65)
FDAGF	59.69 (0.00)	30.03 (0.00)	45.02 (0.00)	30.48 (0.00)	48.74 (0.00)	67.86 (0.00)
RCAGL	65.69 (0.00)	36.67 (0.00)	50.06 (0.00)	31.20 (0.00)	54.50 (0.00)	75.28 (0.00)
UDBGL	65.30 (0.00)	19.95 (0.00)	40.97 (0.00)	33.50 (0.00)	52.47 (0.00)	64.11 (0.00)
SDSNE-Net	56.08 (4.38)	42.69 (0.65)	48.44 (3.06)	26.70 (0.00)	43.80 (0.00)	–
CVCL-Net	30.56 (0.03)	17.43 (0.00)	52.68 (0.03)	20.30 (0.00)	44.60 (0.00)	64.81 (0.00)
SCMVC-Net	31.69 (0.00)	21.40 (0.00)	52.21 (0.00)	26.30 (0.00)	55.63 (0.00)	73.72 (0.00)
DMCAG-Net	59.90 (0.00)	22.57 (0.00)	34.70 (0.00)	24.60 (0.00)	55.94 (0.00)	–
LAG-Net	71.20 (0.88)	48.87 (0.87)	52.26 (0.23)	49.03 (0.82)	56.71 (1.11)	80.23 (0.54)

Table 4. NMI of all compared multi-view clustering methods, where the best and second-best results are highlighted in bold and underlined, respectively (mean% and standard deviation%); “–” indicates an out-of-memory error.

Methods\Datasets	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
LMVSC	35.35 (0.00)	25.31 (0.00)	23.11 (0.00)	27.00 (0.00)	40.30 (0.00)	78.66 (0.00)
AIMC	68.75 (0.00)	34.35 (0.00)	32.40 (0.00)	39.57 (0.00)	42.85 (0.00)	84.70 (0.00)
FastMICE	66.48 (2.63)	44.48 (1.58)	31.77 (2.60)	43.26 (2.59)	49.35 (1.64)	83.32 (2.07)
FDAGF	69.93 (0.00)	51.02 (0.00)	29.35 (0.00)	40.32 (0.00)	46.27 (0.00)	82.21 (0.00)
RCAGL	71.71 (0.00)	48.44 (0.00)	34.26 (0.00)	40.10 (0.00)	53.30 (0.00)	84.49 (0.00)
UDBGL	70.49 (0.00)	33.37 (0.00)	33.85 (0.00)	41.70 (0.00)	52.17 (0.00)	80.73 (0.00)
SDSNE-Net	68.45 (2.48)	48.09 (0.79)	34.35 (2.19)	42.10 (0.00)	43.90 (0.00)	–
CVCL-Net	44.10 (0.02)	38.03 (0.00)	34.42 (0.03)	36.10 (0.00)	42.20 (0.00)	79.11 (0.00)
SCMVC-Net	46.17 (0.00)	42.10 (0.00)	30.97 (0.00)	35.50 (0.00)	54.29 (0.00)	82.76 (0.00)
DMCAG-Net	65.80 (0.00)	27.39 (0.00)	20.98 (0.00)	35.20 (0.00)	55.24 (0.00)	–
LAG-Net	73.42 (0.52)	51.41 (1.08)	32.43 (0.16)	58.00 (0.52)	50.82 (0.80)	85.68 (0.21)

Table 5. ARI of all compared multi-view clustering methods, where the best and second-best results are highlighted in bold and underlined, respectively (mean% and standard deviation%); “–” indicates an out-of-memory error.

Methods\Datasets	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
LMVSC	15.43 (0.00)	2.38 (0.00)	13.98 (0.00)	5.30 (0.00)	24.80 (0.00)	49.44 (0.00)
AIMC	49.66 (0.00)	13.57 (0.00)	27.57 (0.00)	13.26 (0.00)	27.09 (0.00)	65.52 (0.00)
FastMICE	47.84 (2.16)	15.52 (2.44)	27.27 (1.21)	17.23 (1.75)	26.33 (1.59)	61.25 (0.98)
FDAGF	49.50 (0.00)	24.32 (0.00)	19.98 (0.00)	19.24 (0.00)	30.01 (0.00)	56.56 (0.00)
RCAGL	46.17 (0.00)	21.72 (0.00)	29.25 (0.00)	15.20 (0.00)	32.40 (0.00)	63.66 (0.00)
UDBGL	48.12 (0.00)	43.20 (0.00)	17.32 (0.00)	17.70 (0.00)	29.80 (0.00)	53.18 (0.00)
SDSNE-Net	23.87 (6.25)	14.68 (3.07)	22.66 (3.31)	9.20 (0.00)	26.40 (0.00)	–
CVCL-Net	18.71 (0.02)	13.70 (0.00)	29.27 (0.05)	9.16 (0.00)	32.00 (0.00)	57.04 (0.00)
SCMVC-Net	22.98 (0.00)	20.80 (0.00)	28.54 (0.00)	10.50 (0.00)	32.77 (0.00)	65.31 (0.00)
DMCAG-Net	42.71 (0.00)	6.89 (0.00)	10.89 (0.00)	12.70 (0.00)	34.42 (0.00)	–
LAG-Net	55.45 (1.06)	55.24 (4.54)	26.45 (0.45)	29.95 (0.94)	36.99 (0.84)	66.76 (1.58)

Compared with the deep methods, including SDSNE-Net, CVCL-Net, SCMVC-Net, and DMCAG-Net, LAG-Net also shows competitive and often superior performance. It obtains the best ARI on Animals, Caltech102, MITIndoor, Scene15, and YTF-50. These results indicate that LAG-Net not only improves clustering accuracy but also preserves more consistent cluster-level structures. On Flickr, CVCL-Net achieves the best ACC and ARI, suggesting that contrastive representation learning is particularly effective for this two-view vision–language dataset. Nevertheless, LAG-Net remains competitive by achieving the second-best ACC on Flickr and strong results on the other datasets. The weaker ACC of some deep baselines on Animals, Caltech102, and MITIndoor should be read in this protocol: CVCL-Net and SCMVC-Net learn neural embeddings, whereas the present benchmarks use heterogeneous pre-extracted features. The same setting yields the best ACC for CVCL-Net on Flickr, so the comparison does not uniformly disadvantage deep methods. Overall, the comparisons against both shallow and deep baselines verify the effectiveness of introducing global anchor alignment into an interpretable deep unfolding framework.

Table 3, Table 4 and Table 5 include results under the dataset-specific configurations of L and r identified through the sensitivity study. Table 6 provides an additional label-independent evaluation under a common setting $L=3$, $r=2$ without any per-dataset search. Under this common setting, LAG-Net remains the best method on Animals, Caltech102, MITIndoor, and YTF-50. Caltech102 and YTF-50 stay close to the dataset-specific results. The drop is larger on Animals and MITIndoor, yet both remain above the strongest baseline on those datasets ($66.51$ vs. $65.69$ on Animals, and $44.90$ vs. $36.94$ on MITIndoor). Flickr and Scene15 are no longer the top entries under the common setting.

Table 6. Additional check: ACC (%) of LAG-Net under the main dataset-specific $[L,r]$ and a common setting $L=3$, $r=2$ with no per-dataset search (mean ± std over 10 k-means runs).

Setting	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
Dataset-specific $[L,r]$	71.20 (0.88)	48.87 (0.87)	52.26 (0.23)	49.03 (0.82)	56.71 (1.11)	80.23 (0.54)
Common $(L=3,r=2)$	66.51 (0.64)	48.45 (0.30)	50.37 (0.01)	44.90 (0.42)	55.17 (0.87)	79.87 (0.80)

4.3. Component-Related Comparison of Network Architecture

This subsection analyzes the contribution of each network module through progressive architecture ablation. The full model was constructed by adding one module at a time while keeping all other hyperparameters fixed according to the implementation details.

To isolate the contribution of each module, the full network was progressively constructed by adding one module at a time, as summarized below. LAG-H only retains AnchorGraphModule, which learns the shared anchor graph $\mathbf{H}$ while fixing the view-specific anchor indicators ${\mathbf{P}}_v$ at initialization, setting ${\mathbf{E}}_v=\mathbf{0}$, fixing $\mathbf{A}$ at initialization without distance guidance ($\gamma =0$), and using uniform view weights $w_v=1/V$. LAG-HP introduces AnchorIndicatorModule into LAG-H, enabling view-specific anchor indicators ${\mathbf{P}}_v$ to be updated via SVD Procrustes. LAG-HPE further introduces FeatureNoiseModule to model row-sparse corruptions ${\mathbf{E}}_v$. LAG-HPEA adds AnchorAlignmentModule together with the global anchor alignment mechanism for $\mathbf{A}$, while still using uniform view weights. VanillaLAG additionally employs ViewWeightModule for adaptive view weighting but keeps all unfolding thresholds and linear maps fixed as in the alternating optimization updates. LAG-Net is the final model containing all five modules with fully learnable unfolding parameters Θ.

The results in Table 7 reveal a clear and consistent trend: each module contributes positively to the overall clustering performance. LAG-H serves as the basic shared anchor graph learning baseline. LAG-HP consistently outperforms LAG-H across all datasets, confirming that AnchorIndicatorModule is essential for realizing the shared anchor graph in heterogeneous view spaces; the gains are especially evident on Caltech102 and MITIndoor, where the feature dimensions differ substantially across views. LAG-HPE further surpasses LAG-HP, demonstrating that explicit noise modeling through FeatureNoiseModule helps suppress corrupted samples and improves representation robustness. The addition of AnchorAlignmentModule in LAG-HPEA yields another notable improvement, validating that global anchor alignment provides an effective geometric reference for coordinating anchor semantics across views. VanillaLAG, which combines all modules with fixed unfolding updates, achieves significantly higher performance than any partial variant, confirming that the full decomposition framework benefits from the joint interaction of all components. Finally, LAG-Net consistently outperforms VanillaLAG, demonstrating that learnable unfolding parameters provide an additional and meaningful improvement by enabling more flexible and effective optimization.

Table 7. Ablation study of different network variants, where the best ACC are highlighted in bold (mean% and standard deviation%).

Methods	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
LAG-H	56.73 (1.05)	34.82 (0.95)	42.36 (0.45)	37.29 (0.98)	44.10 (1.25)	70.59 (0.72)
LAG-HP	62.45 (0.92)	40.29 (0.98)	44.61 (0.39)	40.54 (0.91)	48.61 (1.18)	72.32 (0.66)
LAG-HPE	64.82 (0.80)	43.53 (0.82)	46.26 (0.31)	43.95 (0.84)	50.74 (1.15)	73.84 (0.61)
LAG-HPEA	67.66 (0.91)	44.43 (0.65)	49.87 (0.28)	46.83 (0.44)	52.31 (1.12)	75.60 (0.53)
VanillaLAG	69.17 (0.46)	46.65 (0.57)	50.42 (0.25)	48.23 (0.73)	55.82 (1.10)	78.54 (0.56)
LAG-Net	71.20 (0.88)	48.87 (0.87)	52.26 (0.23)	49.03 (0.82)	56.71 (1.11)	80.23 (0.54)

To further isolate each design choice, Table 8 reports leave-one-component-out ablations starting from the full LAG-Net, as summarized below: w/o FeatureNoiseModule removes row-sparse noise modeling by fixing ${\mathbf{E}}_v=\mathbf{0}$ for all views, while keeping the other modules unchanged; w/o AnchorIndicatorModule freezes the view-specific indicators ${\mathbf{P}}_v$ at their SVD initialization, so that the shared graph is no longer realized in heterogeneous view spaces through SVD Procrustes updates; w/o AnchorAlignmentModule keeps all other modules of LAG-Net but freezes the global anchors $\mathbf{A}$ at initialization without distance guidance ($\gamma =0$), so that the shared graph is no longer refined by the aligned global geometry; w/o ViewWeightModule replaces adaptive view weighting with uniform weights $w_v=1/V$, while keeping the other modules unchanged; w/o ${\mathcal{L}}_{\mathrm{align}}$ retains all modules but disables only the alignment loss by setting ${\lambda }_{\mathrm{align}}=0$; exact centroid keeps the full LAG-Net architecture but replaces the softmax-based anchor update with the closed-form nonnegative centroid rule in Equation (14). LAG-Net (full) is the complete model with all five modules, the alignment loss, and the softmax centroid approximation.

Table 8. Leave-one-component-out ablation from the full LAG-Net, where the best ACC values are highlighted in bold (mean% and standard deviation%).

Methods	Animals	Caltech102	Flickr	MITIndoor	Scene15	YTF-50
w/o FeatureNoiseModule	64.27 (0.84)	45.19 (0.78)	46.41 (0.35)	43.18 (0.90)	53.63 (0.98)	74.77 (0.58)
w/o AnchorIndicatorModule	63.35 (0.90)	44.48 (0.85)	45.52 (0.38)	41.95 (0.89)	52.98 (1.01)	74.32 (0.59)
w/o AnchorAlignmentModule	62.41 (0.95)	43.85 (0.91)	44.65 (0.42)	40.72 (0.88)	52.42 (1.05)	73.96 (0.61)
w/o ViewWeightModule	66.14 (0.76)	46.22 (0.72)	47.83 (0.31)	45.04 (0.85)	54.57 (0.94)	76.03 (0.55)
w/o ${\mathcal{L}}_{\mathrm{align}}$ ($\lambda =0$)	67.85 (0.80)	47.14 (0.59)	49.37 (0.38)	46.47 (1.06)	55.24 (0.86)	77.74 (0.54)
Exact centroid (vs. softmax)	69.62 (0.82)	48.13 (0.80)	51.06 (0.29)	48.39 (0.79)	56.49 (1.02)	78.16 (0.52)
LAG-Net (full)	71.20 (0.88)	48.87 (0.87)	52.26 (0.23)	49.03 (0.82)	56.71 (1.11)	80.23 (0.54)

As shown in Table 8, removing any single component consistently decreases ACC relative to the full LAG-Net, confirming that the gains are not dominated by a single component. The largest drop appears when AnchorAlignmentModule is removed, which aligns with the progressive ablation in Table 7: without updating the global anchors $\mathbf{A}$, the shared graph loses its geometric reference across views. Removing AnchorIndicatorModule causes the next-largest degradation, especially on Caltech102 and MITIndoor, where the view dimensions differ substantially and freezing ${\mathbf{P}}_v$ prevents the shared graph from being realized in heterogeneous feature spaces. Disabling FeatureNoiseModule or ViewWeightModule yields moderate degradations, indicating that row-sparse noise modeling and adaptive view weighting provide complementary robustness, especially on multi-view data with heterogeneous quality. Setting ${\lambda }_{\mathrm{align}}=0$ causes a milder decrease than removing entire modules, suggesting that the alignment loss acts as a soft regularizer on top of the already-present AnchorAlignmentModule rather than as the sole source of the alignment benefit. Finally, replacing the softmax centroid approximation with the exact nonnegative centroid update remains closest to the full model, showing that the softmax surrogate is a stable and effective implementation choice without changing the overall design. Overall, the leave-one-out results complement the progressive construction in Table 7: each module is useful when added, and none is redundant when removed from the full model.

4.4. Parameter Sensitivity

The sensitivity study below is empirical: ${\lambda }_{\mathrm{align}}$ and m are swept on all six datasets, and ACC is reported as mean ± std over 10 k-means runs. To examine the robustness of LAG-Net to key hyperparameters, sensitivity analysis was conducted on ${\lambda }_{\mathrm{align}}$ and m. The former controls the strength of the global anchor alignment constraint, while the latter determines the size of the sample–anchor graph. For the anchor number, the anchor ratio was varied in $\{1,2,3\}$, corresponding to $m\in \{c,2c,3c\}$, where c is the number of clusters. All other settings were kept fixed according to the implementation details. The ACC results are shown in Figure 2. To complement the figure, Table 9 and Table 10 report the corresponding numerical ACC values (mean ± std). Each setting used the same evaluation protocol as the main tables: after one training run, k-means was repeated 10 times with different random initializations on the learned embedding, and the reported standard deviation was computed over these 10 clustering runs.

Figure 2. Parameter sensitivity of LAG-Net: (a) ${\lambda }_{\mathrm{align}}$; (b) the anchor number m. Each point is the mean ACC over 10 k-means runs, and the corresponding numerical values and standard deviations are listed in Table 9 and Table 10.

Table 9. ACC (%) under different ${\lambda }_{\mathrm{align}}$ (mean ± std over 10 k-means runs), corresponding to Figure 2a.

Datasets	${10}^{-4}$	${10}^{-3}$	${10}^{-2}$	${10}^{-1}$	1	10
Animals	67.52 (0.92)	67.24 (1.30)	67.35 (0.73)	67.86 (0.98)	71.20 (0.94)	67.38 (1.31)
Caltech102	48.86 (0.45)	48.81 (0.36)	48.78 (0.26)	48.84 (0.33)	48.87 (0.42)	48.81 (0.43)
Flickr	52.26 (0.23)	52.02 (0.09)	50.92 (0.04)	50.32 (0.01)	52.26 (0.01)	50.35 (0.01)
MITIndoor	47.66 (1.05)	47.46 (1.00)	46.99 (1.22)	47.48 (0.94)	49.03 (0.84)	47.01 (0.75)
Scene15	57.13 (1.45)	56.23 (2.27)	57.06 (0.80)	58.23 (1.46)	56.71 (1.11)	53.39 (1.10)
YTF-50	80.23 (0.60)	80.23 (0.60)	80.23 (0.60)	80.23 (0.60)	80.23 (0.60)	80.23 (0.60)

Table 10. ACC (%) under different anchor numbers $m\in \{c,2c,3c\}$ (mean ± std over 10 k-means runs), corresponding to Figure 2b.

Datasets	$\mathit{m}=\mathit{c}$	$\mathit{m}=2\mathit{c}$	$\mathit{m}=3\mathit{c}$
Animals	71.20 (0.88)	71.65 (1.15)	71.20 (1.08)
Caltech102	48.87 (0.87)	48.05 (0.69)	48.38 (0.29)
Flickr	52.26 (0.01)	44.49 (0.52)	48.15 (0.05)
MITIndoor	49.03 (0.82)	47.63 (1.17)	48.34 (1.63)
Scene15	56.71 (1.11)	55.05 (1.09)	53.17 (2.19)
YTF-50	80.23 (0.54)	80.64 (0.51)	80.56 (0.88)

4.4.1. Analysis of the Loss Parameter

The parameter ${\lambda }_{\mathrm{align}}$ balances the reconstruction objective and the anchor alignment loss. As shown in Figure 2a, LAG-Net generally maintains stable performance over a wide range of ${\lambda }_{\mathrm{align}}$ values, indicating that the model is not overly sensitive to this loss weight. Small positive values often achieve competitive or better performance, because the alignment term provides useful geometric guidance without overwhelming the reconstruction-driven learning of the shared anchor graph. When ${\lambda }_{\mathrm{align}}$ becomes too large, the model may overemphasize anchor-structure consistency and slightly weaken the reconstruction quality on some datasets. Overall, the results verify that the anchor alignment loss is beneficial and that LAG-Net remains robust under different alignment strengths.

4.4.2. Analysis of the Anchor Number

The anchor number m controls the compactness and expressive capacity of the learned sample–anchor graph. Figure 2b shows that using $m=c$ already provides strong performance on most datasets, demonstrating that LAG-Net can learn discriminative clustering structures with a compact set of anchors. Increasing the anchor ratio to $2c$ or $3c$ may improve performance on some datasets by offering more flexible local representation, but it can also introduce redundant anchors and increase the difficulty of stable graph learning. Therefore, a moderate anchor number is generally preferable: it preserves the scalability advantage of anchor-based clustering while retaining sufficient structural information for accurate clustering.

4.5. Complexity and Runtime Analysis

Let $D={\sum }_{v=1}^V{d}_v$ denote the concatenated feature dimension. The concatenated matrix $\mathbf{X}\in {\mathbb{R}}^{n\times D}$ is stored once. In each unfolding layer, AnchorGraphModule computes sample–anchor squared distances in $\mathcal{O}\left(nDm\right)$ and applies $m\times m$ maps in $\mathcal{O}\left(Vn{m}^2\right)$. FeatureNoiseModule costs $\mathcal{O}\left(nD\right)$. AnchorIndicatorModule performs an economy SVD of a $d_v\times m$ matrix per view, which is $\mathcal{O}\left(V{m}^2{\max }_v{d}_v\right)$. AnchorAlignmentModule updates global anchors in $\mathcal{O}\left(nDm\right)$. ViewWeightModule evaluates reconstruction residuals in $\mathcal{O}\left(nD\right)$. Backpropagation has the same leading order as the forward pass. With L layers, the per-epoch network complexity is therefore $\mathcal{O}\left(L(nDm+V{m}^2{\max }_v{d}_v)\right)$, which remains linear in n when $m,L,V\ll n$. The dominant extra memory is $\mathcal{O}(nm+nD)$ for $\mathbf{H}$ and $\mathbf{X}$, rather than an $n\times n$ sample graph. The trainable parameters consist of layer-wise thresholds and $m\times m$ maps together with the shared perturbation $\Delta \mathbf{A}\in {\mathbb{R}}^{D\times m}$; hence, the parameter count depends on D and m but not on n.

Table 11 reports empirical efficiency under the main experimental settings ($T=100$, $m=c$, and the dataset-specific $[L,r]$). “Net” is the accumulated forward/backward time of LAG-Net, and “Total” is the wall-clock training time that additionally includes the per-epoch spectral embedding and k-means used only for online monitoring (not required by the model itself). Peak GPU memory is measured by PyTorch on an RTX 3090. On YTF-50 (n > ${10}^5$), the network training cost is about 142 s with peak memory below 12 GB, confirming the practical scalability of the anchor-based unfolding design.

Table 11. Efficiency of LAG-Net on six datasets: trainable parameters, peak GPU memory, network training time (Net), and wall-clock training time including per-epoch clustering evaluation (Total).

Datasets	n	m	L	Params	Peak Mem (MB)	Net (s)	Total (s)
Animals	10,158	50	5	447,610	4762.3	57.3	604.3
Caltech102	9144	102	5	541,222	2109.1	115.9	1644.6
Flickr	12,154	6	3	1566	144.7	4.9	67.0
MITIndoor	5360	67	4	771,714	2926.1	50.5	475.0
Scene15	4485	15	7	68,249	1167.5	15.0	75.4
YTF-50	126,054	50	1	141,202	111,85.4	142.2	4268.2

Table 12 compares the asymptotic complexity, network training time, and peak GPU memory of LAG-Net with representative deep baselines under the same hardware setting. The Complexity column reports the per-epoch time complexity, and the Time and Mem columns are empirical measurements on the same GPU and are not implied by the $\mathcal{O}(·)$ expression. Unlike sample-level deep methods whose leading terms scale as $\mathcal{O}\left(n^2\right)$, the per-epoch cost of LAG-Net remains linear in n when $m,L,V\ll n$. The LAG-Net times and memory in Table 12 reuse the Net and Peak Mem entries of Table 11, baseline runtimes exclude optional per-epoch clustering monitoring, and peak memory is measured by PyTorch on an RTX 3090.

Table 12. Per-epoch time complexity, empirical training time (s), and peak GPU memory (MB) comparison with representative deep baselines on the same hardware; b is the mini-batch size, and s is the maximum number of neurons in the hidden layers.

Methods	Complexity	Animals		Caltech102		Flickr		YTF-50
		Time	Mem	Time	Mem	Time	Mem	Time	Mem
CVCL-Net	$\mathcal{O}\left(L(nbsD+n^2{b}^2{c}^2+nbcD)\right)$	100.4	5042.5	274.4	3689.6	114.2	952.9	650.3	120,06.6
SCMVC-Net	$\mathcal{O}\left(L(n^{2}sV+nsc)\right)$	1544.6	8455.3	1406.7	4227.1	742.1	1113.5	903.7	124,69.3
LAG-Net	$\mathcal{O}\left(L(nDm+V{m}^2{\max }_v{d}_v)\right)$	57.3	4762.3	115.9	2109.1	4.9	144.7	142.2	111,85.4

4.6. Statistical Significance Analysis

To assess whether the observed improvements are statistically meaningful, two complementary tests are reported [61]: First, a Friedman test was conducted over the ACC rankings of all compared methods across the six datasets in Table 3. Out-of-memory entries were treated as the worst performance on that dataset. The resulting average ranks are summarized in Table 13, and the Friedman statistic is ${\chi }^2=31.24$ with $p=5.35\times {10}^{-4}$, indicating statistically significant differences among the compared methods. LAG-Net attains the best average rank ($1.17$).

Table 13. Average ACC ranks from the Friedman test over six datasets, where the best rank value is highlighted in bold.

Methods	LMVSC	AIMC	FastMICE	FDAGF	RCAGL	UDBGL
Avg. rank	10.33	5.50	5.33	6.00	3.33	6.33
Method	SDSNE-Net	CVCL-Net	SCMVC-Net	DMCAG-Net	LAG-Net
Avg. rank	7.42	7.83	5.50	7.25	1.17

Second, on Flickr and Scene15, where the absolute ACC margins are relatively small, LAG-Net was re-trained with six random seeds $\{41,42,43,44,45,46\}$ under the main experimental settings. For each dataset, the per-seed ACC values were compared with the multi-seed mean of the strongest baseline (CVCL-Net on Flickr and DMCAG-Net on Scene15) by a one-sample Wilcoxon signed-rank test on the per-seed differences. As reported in Table 14, the multi-seed mean ACC of LAG-Net is higher than that of the strongest baseline on both datasets ($53.15$ vs. $52.68$ on Flickr; $56.63$ vs. $55.94$ on Scene15). On both datasets, all six seeds yield positive ACC differences, giving Wilcoxon statistic $W=0$ and two-sided $p=0.03125<0.05$, which indicates a statistically significant improvement over the corresponding strongest baseline mean under this multi-seed evaluation. Together with the Friedman test above, these results indicate that LAG-Net is overall competitive across datasets and that the gains on Flickr and Scene15 remain stable under different random seeds.

Table 14. Multi-seed ACC (%) of LAG-Net on Flickr and Scene15, and Wilcoxon signed-rank test against the strongest baseline mean. Columns 41–46 are random seeds.

Datasets	Baseline	Baseline Mean	41	42	43	44	45	46	LAG-Net Mean	W	p-Value
Flickr	CVCL-Net	52.68	52.78	53.22	53.20	52.85	53.49	53.36	53.15	0.0	0.0313
Scene15	DMCAG-Net	55.94	56.71	56.72	56.86	56.89	56.04	56.55	56.63	0.0	0.0313

5. Conclusions

In this paper, we propose Learnable Anchor Graph Network (LAG-Net), a deep unfolding framework for scalable multi-view clustering. The key idea is to jointly learn a shared sample–anchor graph, view-specific anchor indicators, row-sparse feature noise, adaptive view weights, and globally aligned anchors within a unified optimization model. A global anchor alignment regularizer couples sample–anchor affinities with anchor geometry in the concatenated multi-view feature space, enabling the shared graph and global anchor structure to be refined in a bidirectional manner. The resulting five-variable alternating minimization was unfolded into an interpretable architecture comprising AnchorGraphModule, FeatureNoiseModule, AnchorIndicatorModule, AnchorAlignmentModule, and ViewWeightModule, with learnable unfolding parameters for end-to-end training. Evaluations on six benchmark datasets spanning diverse scales and view configurations show that LAG-Net is competitive with representative shallow and deep multi-view clustering methods. It attains the best ACC on five of the six datasets, while CVCL-Net remains stronger on Flickr, and some NMI and ARI margins are small. The ablation and parameter sensitivity studies further validated the contribution of each module. These results also highlight the potential of optimization-inspired deep unfolding as a principled and interpretable paradigm for anchor-based multi-view clustering. Several limitations remain. First, both the representation capacity and the computational cost of LAG-Net depend on the number of anchors m. Setting $m=c$ is already sufficient on most datasets, while a larger m may introduce redundant anchors and does not consistently improve ACC. Second, the evaluation is transductive: all samples participate in training and in the subsequent clustering of $\mathbf{H}$. The number of clusters c is assumed known and is used both for $m=c$ and for the final k-means. Third, the dataset-specific configurations of the unfolding depth L and the view-weight smoothness parameter r reported in this work were obtained using clustering ACC and, therefore, should not be regarded as a label-free hyperparameter selection strategy. A common setting $L=3$ and $r=2$ is additionally reported to evaluate the robustness without such selection. Fourth, FeatureNoiseModule is designed for row-sparse feature corruptions under complete and aligned views. Incomplete or partially aligned multi-view data are not evaluated in this work. In future work, we plan to extend LAG-Net to inductive and incomplete multi-view settings and to study more efficient unfolding strategies for extremely large-scale data.