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Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise

Xiucai Ding, Chao Shen, Hau-Tieng Wu

TL;DR

GRAB-MDM addresses robust multiview fusion under high-dimensional noise by introducing view-dependent bandwidths within a diffusion-map framework. It constructs a block kernel with a global diffusion operator across $K$ views and performs a two-stage bandwidth selection with per-view scales $h_\ell$ and a global factor $c$, yielding a joint embedding that leverages cross-view geometry. Theoretical analysis shows the limiting operator is a mixture of Laplace-Beltrami operators with view-specific lower-order terms; robustness in high-dimensional regimes is established, extending diffusion-map guarantees to more than two views. Empirically, GRAB-MDM consistently outperforms fixed-bandwidth baselines and a suite of multiview methods in spectral clustering and manifold-learning tasks, demonstrating improved embedding quality and clustering stability in noisy, high-dimensional settings.

Abstract

Multiview datasets are common in scientific and engineering applications, yet existing fusion methods offer limited theoretical guarantees, particularly in the presence of heterogeneous and high-dimensional noise. We propose Generalized Robust Adaptive-Bandwidth Multiview Diffusion Maps (GRAB-MDM), a new kernel-based diffusion geometry framework for integrating multiple noisy data sources. The key innovation of GRAB-MDM is a {view}-dependent bandwidth selection strategy that adapts to the geometry and noise level of each view, enabling a stable and principled construction of multiview diffusion operators. Under a common-manifold model, we establish asymptotic convergence results and show that the adaptive bandwidths lead to provably robust recovery of the shared intrinsic structure, even when noise levels and sensor dimensions differ across views. Numerical experiments demonstrate that GRAB-MDM significantly improves robustness and embedding quality compared with fixed-bandwidth and equal-bandwidth baselines, and usually outperform existing algorithms. The proposed framework offers a practical and theoretically grounded solution for multiview sensor fusion in high-dimensional noisy environments.

Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise

TL;DR

GRAB-MDM addresses robust multiview fusion under high-dimensional noise by introducing view-dependent bandwidths within a diffusion-map framework. It constructs a block kernel with a global diffusion operator across views and performs a two-stage bandwidth selection with per-view scales and a global factor , yielding a joint embedding that leverages cross-view geometry. Theoretical analysis shows the limiting operator is a mixture of Laplace-Beltrami operators with view-specific lower-order terms; robustness in high-dimensional regimes is established, extending diffusion-map guarantees to more than two views. Empirically, GRAB-MDM consistently outperforms fixed-bandwidth baselines and a suite of multiview methods in spectral clustering and manifold-learning tasks, demonstrating improved embedding quality and clustering stability in noisy, high-dimensional settings.

Abstract

Multiview datasets are common in scientific and engineering applications, yet existing fusion methods offer limited theoretical guarantees, particularly in the presence of heterogeneous and high-dimensional noise. We propose Generalized Robust Adaptive-Bandwidth Multiview Diffusion Maps (GRAB-MDM), a new kernel-based diffusion geometry framework for integrating multiple noisy data sources. The key innovation of GRAB-MDM is a {view}-dependent bandwidth selection strategy that adapts to the geometry and noise level of each view, enabling a stable and principled construction of multiview diffusion operators. Under a common-manifold model, we establish asymptotic convergence results and show that the adaptive bandwidths lead to provably robust recovery of the shared intrinsic structure, even when noise levels and sensor dimensions differ across views. Numerical experiments demonstrate that GRAB-MDM significantly improves robustness and embedding quality compared with fixed-bandwidth and equal-bandwidth baselines, and usually outperform existing algorithms. The proposed framework offers a practical and theoretically grounded solution for multiview sensor fusion in high-dimensional noisy environments.
Paper Structure (31 sections, 11 theorems, 197 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 31 sections, 11 theorems, 197 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The proposed algorithm and bandwidth selection procedure
  3. The proposed GRAB-MDM algorithm
  4. The bandwidth selection procedure in GRAB-MDM
  5. The common manifold model and limiting operators
  6. The model and basic definitions
  7. The limiting operators
  8. Asymptotic analysis
  9. Connection with the Laplace-Beltrami operator: Bias analysis
  10. Variance analysis
  11. Robustness for high-dimensional and noisy datasets
  12. Robustness of the proposed GRAB-MDM
  13. Effectiveness of the bandwidth selection algorithm
  14. Numerical results
  15. Multiview spectral clustering
  16. ...and 16 more sections

Key Result

Theorem 4.2

Suppose Assumptions assu_model and assum_main hold. Fix $1 \leq i \leq \mathsf{K}$. Denote $\mathbf{f}=(f_1,\ldots,f_{\mathsf{K}}) \in C^4(\iota_1(\mathcal{M}))\times \ldots \times C^4(\iota_{\mathsf{K}}(\mathcal{M}))$. For $i=1,\ldots, \mathsf{K}$, denote $\bar{f}\in C^4(\iota_i(\mathcal{M})))$ as When $\epsilon_1,\ldots,\epsilon_{\mathsf{K}}$ are sufficiently small and satisfy $\epsilon_i \asym

Figures (4)

  • Figure 1: Illustration of the latent common manifold model. The data generating process starts with a common (latent) manifold $\mathcal{M}.$ For each point $z \in \mathcal{M},$ it can be simultaneously embedded into $\mathsf{K}$ different submanifolds $\iota_\ell(\mathcal{M}), 1 \leq \ell \leq \mathsf{K},$ via $\mathsf{K}$ different embeddings. The observed point clouds are collections of data points sampled jointly according to (\ref{['ew_densitydefinition']}) and (\ref{['eq_commonmode2']}) from those embedded submanifolds. In other words, for $1 \leq i \neq j \leq \mathsf{K}$ and $1 \leq \alpha \leq n,$$\mathbf{x}_\alpha^j=\vartheta_{ji}(\mathbf{x}_\alpha^i).$
  • Figure 2: Comparison of the robustness of various methods. We use Setup (1), and the results are based on 1,000 repetitions.
  • Figure 3: Comparison of the robustness of our proposed method with and without sketching. The detailed implementation is described in Appendix \ref{['sec_practicalnoisereduction']}, and the reported results are averaged over 1,000 repetitions.
  • Figure 4: Eigen-ratio and eigenvector plots. We present a representative plot of the eigen-ratios and the second eigenvector of $\mathcal{A}$ in (\ref{['eq_finaloperator']}). The eigen-ratio plot indicates that $m=1$ is the appropriate choice, and accordingly only the second eigenvector and its average across three views are displayed. This also explains why our algorithm can efficiently identify the three distinct classes and why taking the average is beneficial. The results are based on Setup (2) in Table \ref{['tab_comparison_1']}.

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 4.2
  • Remark 4.3
  • ...and 18 more