Multi-View Alignment and Denoising via Center-Guided Spectral Diffusion

1 Introduction

Multi-view data is prevalent in a wide range of real-world applications, including multi-camera vision systems and crosslingual text modeling [Xie et al., 2020; Santosh et al., 2020; Liu et al., 2024]. While different views offer complementary observations of the same underlying structure from diverse perspectives, challenges arise due to differences in representation, noise interference, and structural misalignment across views [Fang et al., 2024; Lu et al., 2025;

∗Corresponding author.

Figure 1: (a) Traditional methods introduce noise during the alignment process and then amplify it during the fusion stage. (b) The impact of adding noise to a single view and then fusion the view using various baseline model. Here, GEGCN learns the consistent representations through neural network; CON denotes the concatenation method. The red dashed line indicates the performance of the deleted noisy view on the baseline model.

Liu et al., 2025a]. A central problem in multi-view learning is how to extract consistent and discriminative representations from such heterogeneous and partially redundant information [Tian et al., 2023; Peng et al., 2024]. In recent years, researchers have attempted to model similarity matrices to uncover potential relationships among samples [Chen et al., 2023c; Du et al., 2024; Wu et al., 2025]. While this graph-based approach has achieved impressive results, the presence of extensive noise has exacerbated the heterogeneity between views, making denoising particularly important. Mainstream methods typically treat view denoising and alignment as two separate sequential processes. In this separationbased paradigm, each view is first denoised, and then aligned by mapping heterogeneous modalities to a shared subspace via a neural network [Geng et al., 2021; Xu et al., 2023; Lu et al., 2024a]. Research shows that denoising and alignment operations in multi-view domains help models obtain more generalizable representations [Zhang et al., 2020; Lian et al., 2026].

This separation-based paradigm has achieved tremendous success. However, we have identified a limitation: Anomaly features in low-quality views may distort cross-view alignment, which introduces additional noise into the alignment and fusion process, leading to denoising failure. Recently research has shown that noisy samples tend to have larger gradient norms during training, causing the alignment process to Residual focus excessively on these noisy samples [Huang et al., 2021; Peng et al., 2022], which may even steer the view alignment toward the noise subspace. As illustrated in Figure 1(a), the distorted embeddings of low-quality views propagate to other views during alignment through parameter-sharing mechanisms or interaction between views. The contaminated interactions force the representations of other views to shift toward noise, thereby disrupting the original semantic structure. In end-to-end training, the polluted representations serve as input for the next training iteration, further exacerbating information distortion. We conduct experiments to simulate the impact of different noise intensities on view alignment, as illustrated in Figure 1(b). Specifically, noise is only added to the first view, and we then evaluate the performance of different alignment methods. The experimental results show that as the noise intensity in a single view increases, traditional methods continuously contaminate the remaining views during the alignment process. The performance may even deteriorate to the point where abandoning the view will yield better results.

To address the above challenges, we propose Center Guided Spectral Diffusion (CGSD). Unlike previous methods such as direct mapping, we use a generative process to replace alignment to achieve synergistic optimisation of denoising and alignment. Specifically, we first model an unconditional diffusion stochastic differential equations system in both the feature space and the low-rank spectral space, which is used to iteratively update a consensus center anchor within multi-view. This consensus center anchor is then used as a input to a conditional diffusion model. Since anchors are consensus representations distilled from multi-view, conditional diffusion will be guided to generate aligned samples. After the model captures the underlying data distribution, in order to balance the truncation error and the number of network evaluations, we employ an Heun's second-order method as the solver for the differential equations and utilize Bayesian inference during the sampling phase to guide the generation of aligned samples. The contributions of can be summarized as follows:

• Generative models replace alignment: The CGSD framework is proposed to effectively mitigate noise amplification in multi-view data fusion by replacing traditional alignment with generative process.
• Center-guided mechanism is designed: The multiview consensus anchor points are extracted through an unconditional diffusion process and then applied to conditional diffusion, guiding the alignment of multi-view.
• Extensive experiments on real-world datasets: The experimental results show that the proposed model achieves state-of-the-art results on multiple datasets.

2 RelatedWork

In this section, we will briefly introduce multi-view learning and score-based generative diffusion models.

2.1 Graph-basedMulti-view Learning

The main objective of multi-view learning is to utilise complementary information from multiple perspectives or multisource data to improve the performance and applicability of machine learning models. In recent years, with the advancement of graph learning methods, multi-view learning tasks have achieved significant breakthroughs [Lian et al., 2024; Liu et al., 2025b; Wang et al., 2025]. [Chen et al., 2023b] proposed to jointly learn feature and graph fusion for multiview data through a two-stage, which integrated a feature fusion network and a learnable graph convolutional network with a novel differentiable shrinkage activation function to capture discriminative node relationships. [Lu et al., 2024b] proposed to address multi-view semi-supervised classification by introducing a generative essential graph convolutional network that unified multi-graph consistency extraction, graph refinement, and task-specific optimization, leveraging a learnable threshold shrinkage function to obtain effective graph embeddings. [Chen et al., 2023a; Lin et al., 2025] focused on the optimisation and denoising of multi-view graph structures. Research shows that, compared with traditional methods, graph-based multi-view learning can achieve better results.

2.2 Generation DiffusionModel

Diffusion models are widely used due to their powerful ability to learn data distributions [Ho et al., 2020; Dhariwal and Nichol, 2021; Chen et al., 2024; Liu et al., 2023]. In recent years, relevant advances in the field of graph generation have been continuously explored, with related work mainly focusing on graph-level tasks. [Jo et al., 2022] used a continuoustime diffusion framework, enabling effective graph generation that satisfied structural dependencies and permutation invariance. However, running full-rank diffusion on the adjacency matrix causes fatal errors in the graph structure connections. [Luo et al., 2023] applied low-rank diffusion SDEs in the graph spectrum space to efficiently learn graph topology generation, achieving excellent theoretical guarantees and state-of-the-art performance. In terms of nodes-level tasks, [Li et al., 2024] enhanced latent space learning with discriminative content and a content-preservation mechanism, significantly improving anomaly detection performance while maintaining low time and space complexity. Although diffusion models have achieved remarkable success in many fields, their extension to multi-view learning remains challenging due to the inherent information redundancy and noise from complementary multi-source inputs.

3 Preliminary

In this section, we first introduce some basic rules for using symbols in Table 1. Since diffusion models have limited applications in multi-view scenes, we also provide a comprehensive introduction to the basic theory of diffusion models.

3.1 Score-based Generative DiffusionModel

Diffusion models establish a unique bidirectional mapping mechanism between a complex data distribution D and a priori distribution S through a Stochastic Differential Equation (SDE) framework. Unlike traditional generative models (e.g., VAE and GAN) that focus only on the unidirectional mapping from S to D, the diffusion model considers the bidirectional relationship between D and S at the same time. After sufficient training, the model can reconstruct the data from normally distributed random Gaussian noise.

[Karras et al., 2022] extended the SDE of [Song et al., 2021] to a more general form:

where p(·) is the probability density function of x, ∇x log p(x) is the score function of x, dx+ and dx- represent the forward and reverse processes of the diffusion model, β(t) denotes the relative rate at which the existing noise is replaced by the new noise, and ˙σ(t) is the first-order derivative of σ(t). When β(t) = ˙σ(t)/σ(t), the SDE of [Song et al., 2021] is recovered. Then the SDE for forward diffusion can be formalized as:

where f (·) is the drift function, g(·) is the diffusion function, and dωt is the differential of theWiener process, representing Gaussian random noise. The reverse time process SDEs can be derived as:

In order to learn the unknown score function to match the noise in the inverse process, the following matching scores need to be minimized:

where ϵθ is the neural network that learns the probability density gradient score matching. Ideally, it will guide the data towards the real data distribution D.

4 Methodology

This section details the methodology proposed in this study, and the architecture of the model is illustrated in Figure 2.

4.1 Multi-viewDenoising Spectral Diffusion

Denote a set of views X as {X1 ,X2 , · · · ,XV }. Each view can be represented as a graph Gv = (V (v) , E (v) ,A(v) ,X(v) ), where V (v) is the set of nodes, E (v) is the set of edges, A(v) ∈ RN×N and X(v) ∈ RN×dv , N is the total number of nodes and dv indicates the feature dimension. We aim for the diffusion model to learn the underlying distribution and recover more discriminative data during the sampling process. Specifically, we train two score-based neural networks, including spectral space diffusion ϵϕ(·) and feature space diffusion sθ(·), to generate high-fidelity denoised samples by running a reversed time SDEs.

Applying full-rank diffusion directly on sparse adjacency matrices can lead to significant errors, spectral-domain diffusion offers improved fidelity in reconstruction [Luo et al., 2023]. However, unlike this work, our diffusion operation relies on eigenvectors rather than eigenvalues to obtain purer and more discriminative information. Specifically, let the spectral decomposition of A be given by ΦΛΦT, where Φ is the orthogonal of eigenvectors and Λ is the diagonal matrix of eigenvalues. Noise removal using truncated eigenvectors is an effective technique grounded in graph frequency domain analysis. This approach offers several advantages, including a solid theoretical foundation, effective noise suppression, and the ability to capture global structural information [Bruna et al., 2014]. In our method, the diffusion process emphasizes the low-frequency components of the eigenvectors and achieves noise reduction by reconstructing the first k components of the truncated eigenvectors. For clarity, we denote these truncated vectors as U. The forward spectral diffusion is then described by the following system of SDEs:

whereXvt andUvt represent the evolutionary states of the feature vectors and eigenvectors, v is the view index, f X (·, t) : RN×dv ↦→ RN×dv and f U (·, t) : RN×N ↦→ RN×N denote the drift terms, σX(t) and σU(t) denote the corresponding diffusion coefficients. The stochastic processes ωX t and ωU t are standard Wiener processes.

Forward diffusion perturbs the original data with progressively stronger Gaussian noise under a prescribed drift f X (·, t). Now we define the reverse time SDEs as follows:

where pt(Xv ) and pt(Uv ) denote the marginal probability density functions of Xv and Uv over time, ¯ω is reversetime Wiener process, and the score matching functions ∇X log pt(Xv ) and ∇U log pt(Uv ) are used to train the neural network to learn the gradient of the data distribution.

4.2 Learning Center Anchor via Spectral Diffusion

As mentioned in previous sections, existing methods amplify noise during alignment, leading to failed denoising or even negative effects. To address this issue, we design a module that dynamically learns data centers to align multi-view representations and mitigate noise interference in views, ensuring that the model accurately captures the intrinsic relationships between them.

We begin by following the work of [Arazo et al., 2019], which highlights that noise samples often deviate from the true data distribution. Specifically, we compute the similarity between the original representation and its reconstructed counterpart, which can be formalized as follows:

where τ is the temperature coefficient that controls the smoothness of the weights, and si represents the cosine similarity between the original representation X and the current reconstructed representation ˆX. After calculating the weights using Equation (9), we use it to reduce the impact of noisy samples on the data center. Specifically, we initialize the value as the representation mean to anchor the initial data center (that is, c(v) = 1N ∑Ni=1 X(v) i ) and iteratively refine it through the training process of the unconditional diffusion:

where c(v) represents the representation center of view v, and cfinal represents the global center of multi-view data. Iterative updates of the anchor enable the diffusion model to align representations into a consistent space during the reverse denoising process. In practice, we substitute X and U into the above equation to obtain the corresponding representations cXfinal and cUfinal . In subsequent steps, these two consensus center anchors are used as conditional inputs to the diffusion model to guide its generation process.

4.3 Center-GuidedMulti-viewAlignment

Inspired by the Classifier-Free Guidance [Ho and Salimans, 2021], we use the consensus center anchor obtained from unconditional diffusion model as conditional inputs to the conditional diffusion model. Based on Bayes' theorem, we construct an adjustable guidance mechanism by integrating gradient information from both conditional and unconditional probabilities during the sampling process:

where λ ≥ 0 is a hyperparameter that modulates the influence of the conditional guidance, cXfinal is the joint anchor of the representation of the multi-view features, and cUfinal denotes the joint anchor of the multi-view spectral vector.

The traditional Euler method has a local truncation error of O(h2 ) relative to the step size h when solving ordinary differential equations, while high-order Runge-Kutta methods require multiple evaluations of the neural network sθ and ϵϕ [S¨uli and Mayers, 2003; Karras et al., 2022]. In contrast, Heun's second-order method [Butcher, 2016] provides a favourable trade-off between truncation error and the number of neural network function evaluations [Jolicoeur-Martineau et al., 2021]. Following the guidance framework defined by Equations (12) and (13), the integration process can be approximated using the trapezoidal rule:

where X∗t-1 and U∗t-1 are intermediate predicted values given by the first-order Euler formula. For the specific derivation process of Equations (14) and (15), please refer to Appendix. By predicting the noise in the data through the above system, we can both recover the underlying data distribution and guide the data toward alignment.

4.4 Training and Inference

Training Stage. During the training stage, the consensus center anchor is first obtained by training the unconditional diffusion model using loss (4). Subsequently, the conditional diffusion model is then iteratively optimized according to the following loss functions:

Inference Stage. During the inference Stage, the data is reconstructed based on Equations (16) and (17) to obtain the aligned and denoised single-view outputs, denoted as ˆX and ˆU. In addition, once ˆU is generated through the spectral diffusion process, it is reintroduced into the decomposition equation to reconstruct ˆA as ˆUΛˆUT. For a more detailed algorithm flowchart, please refer to Appendix.

4.5 Theoretical Analysis

Proposition 1. Let A ∈ RN×N be the original data matrix, and U ∈ RN×k be the matrix consisting of the top-k eigenvectors of A, where k ≪ N. For two diffusion processes, the upper bound of the generation error for the truncated eigenvector diffusion is strictly tighter than that of the full matrix diffusion, satisfying:

Remark: Proposition 1 demonstrates that running diffusion on truncated eigenvectors yields tighter error bounds. Furthermore, as the sample size increases and k becomes negligible relative to N, the advantage in error bounds is further amplified. Consequently, our method exhibits significant advantages on large-scale datasets. The proof can be found in Appendix.

Complexity. The overall computational complexity of the diffusion phase is O((4R + 1) · V · N · D2 ). Upon simplification, the complexity exhibits a linear relation. A detailed analysis of the complexity is provided in Appendix.

5 Experiments

In this section, we first introduce the dataset and experimental setup, and then evaluate the performance of the proposed model. In addition, we conduct a series of experiments to further demonstrate the effectiveness of the model.

5.1 Datasets and ComparedMethods

Datasets. We evaluate the proposed method on eight different real-world datasets and three large-scale datasets, including 100leaves, Animals, ALOI, Flower17, GRAZ02, Scene15, MNIST, NUS-WIDE, NUSWIDEOBJ, NoisyMNIST, and YTF. Among these, NUSWIDEOBJ, NoisyMNIST, and YTF are classified as large-scale datasets. Further details are provided in Appendix.

Compared Methods. We compare the proposed method with state-of-the-art baseline models, including Co-GCN [Li et al., 2020], DSRL [Wang et al., 2021], ECMGD [Lu et al., 2024a], RCML [Xu et al., 2024], LGCN-FF [Chen et al., 2023b], and GEGCN [Lu et al., 2024b]. Here LGCNFF and GEGCN are graph-learning models, and a detailed description of the methodology can be found in Appendix.

5.2 Experimental Setting

To mitigate randomness, each experiment repeats five times, and both the mean and variance are reported. For classification tasks, we follow the standard data partitioning protocols and conduct experiments using a fixed learning rate of 0.001 over 1000 training epochs. The experiments employ two authoritative evaluation metrics: Accuracy and F1-Score. More detailed experimental settings can be found in Appendix.

5.3 Classification onMulti-viewDatasets

Performance. Graph convolutional networks are widely used for node classification tasks [Kipf and Welling, 2017; Li et al., 2018; Xu et al., 2019]. By feeding the aligned and denoised views obtained through CGSD into the GCN, we evaluate our node classification performance. The experimental results presented in Table 2 demonstrate that the proposed algorithm consistently outperforms other baseline methods across most datasets. Among them, LGCN-FF and GEGCN are denoising alignment algorithms. Specifically, it achieves notable accuracy improvements of 4.01%, 10.81%, and 2.61% on the 100leaves, Flower17, and Scene15 datasets, respectively. However, the GRAZ02 dataset contains too few samples to fully demonstrate the algorithm's advantages. Furthermore, due to the linear time complexity of CGSD

diffusion process, the method is well-suited for large-scale datasets. As shown in Table 3, CGSD maintains competitive performance on large-scale multi-view datasets, achieving state-of-the-art performance on all three datasets.

Visualization. Figure 3 shows the t-SNE visualization results of the classification performance of different contrast methods on the Scene15 dataset. Observing the chart, compared to other models, we can see that CGSD significantly improves the inter-class separability within this dataset, indicating that our approach extracts more discriminative features. For visualization of all compared algorithms, please refer to Appendix.

5.4 Visualisation of Denoising Results

Figure 4 visualizes the original adjacency matrix, alongside the results of two advanced graph learning methods and our proposed denoising method. Notably, our method produces the most distinct diagonal block structure, indicating that the learned similarity matrix focuses on intra-class relationships with a marked reduction in heterogeneous connections. This structure offers a significant advantage for learning node embeddings via graph convolutional networks, as reflected in our superior performance on node classification tasks. The effectiveness of our method stems from the use of a diffusion model, which integrates informative signals frommulti-view while simultaneously denoising truncated eigenvector.

5.5 Time Efficiency Analysis

We conduct a comprehensive evaluation of the computational efficiency and performance of various algorithms on multiview data, as shown in Figure 5. The results demonstrate that our proposed method not only achieves higher accuracy but also requires significantly less training time compared to most competing approaches. This improvement is largely due to our linear-time complexity diffusion process and advanced view fusion.

5.6 Ablation Study

To validate the effectiveness of CGSD, we conduct ablation studies, the results of which are illustrated in Figure 6. We independently remove the alignment operations within the feature and spectral spaces to isolate the impact of these alignment mechanisms. It can be observed that the ablated variants exhibit significant performance differences across the majority of datasets. This further indicates that the data sampled following our diffusion process possesses superior discriminability compared to the original data distribution. Detailed results of the ablation studies for the remaining datasets are provided in Appendix.

5.7 Parameter Sensitivity

We conduct experiments based on Equations (12) and (13) using various values of λ, with the detailed results presented in Figure 7. Specifically, we report node classification performance for λ = 0 (representing purely unconditional diffusion) and for λ values ranging from 0.1 to 1.0, capturing

(c) GEGCN

(d) Ours

LGCN-FF ACC

F1

2.43 ± 1.27

YTF

87.85 ±1.87 98.77 ±0.01

87.59 ±1.76 98.76 ±0.02

OOM

OOM

OOM

OOM

88.40 ±0.07 78.62 ±0.52

88.05 ±0.08 76.58 ±0.47

OOM

85.56 ± 0.49

OOM

GEGCN ACC

F1

OOM

93.63 ± 0.57

OOM

OOM

93.52 ± 0.54

OOM

Ours ACC

38.22 ± 0.21

97.80 ±0.36 99.86 ±0.00

F1

23.62 ± 0.16

97.78 ±0.37 99.79 ±0.01

Table 3: Classification performance on large-scale datasets. The

best results are highlighted in green, while the second-best are high-

lighted in brown, where 'OOM' denotes Out-of-Memory.

Animals: Time and Accuracy

Time(s)

(b) Animals

Figure 5: A comparsion of the runtime (s) and accuracy (%) of the algorithms on the ALOI and Animals datasets.

different levels of guidance strength. The results show that both datasets achieve optimal performance when λ is approximately 0.4, suggesting that moderate guidance effectively facilitates the alignment process by enabling the model to extract complementary information from multi-view. In contrast, excessive guidance leads to a decline in performance. Detailed results of the parameter sensitivity for the remaining datasets are shown in Appendix.

(a) ALOI

(b) 100leaves

Figure 6: Performance comparison between different model variants

on the ALOI and 100leaves datasets.

6 Conclusion and FutureWork

This paper proposes a framework called CGSD, whose main innovation is to replace the alignment process with a generative approach. Unlike existing multi-view learning methods, we learn the underlying distribution of samples to generate aligned samples that conform to the distribution of multiview data. This method effectively alleviates the problem of noise amplification in current alignment methods. Experiments on multiple real-world datasets validate the superiority of CGSD. In the future, we plan to further refine CGSD through several promising directions, such as developing adaptive guidance mechanisms for generation and investigating more stable sampling strategies, thereby extending it to increasingly complex real-world scenarios.

Accuracy(%)

Accuracy(%)

Accuracy(%)

Acknowledgements

This work is in part supported by the National Natural Science Foundation of China under Grants U25A20527 and 62276065, and the Fujian Provincial Natural Science Foundation of China under Grant 2024J01510026.