Diffusion-based Semi-supervised Spectral Algorithm for Regression on Manifolds (2410.14539v1)

Abstract: We introduce a novel diffusion-based spectral algorithm to tackle regression analysis on high-dimensional data, particularly data embedded within lower-dimensional manifolds. Traditional spectral algorithms often fall short in such contexts, primarily due to the reliance on predetermined kernel functions, which inadequately address the complex structures inherent in manifold-based data. By employing graph Laplacian approximation, our method uses the local estimation property of heat kernel, offering an adaptive, data-driven approach to overcome this obstacle. Another distinct advantage of our algorithm lies in its semi-supervised learning framework, enabling it to fully use the additional unlabeled data. This ability enhances the performance by allowing the algorithm to dig the spectrum and curvature of the data manifold, providing a more comprehensive understanding of the dataset. Moreover, our algorithm performs in an entirely data-driven manner, operating directly within the intrinsic manifold structure of the data, without requiring any predefined manifold information. We provide a convergence analysis of our algorithm. Our findings reveal that the algorithm achieves a convergence rate that depends solely on the intrinsic dimension of the underlying manifold, thereby avoiding the curse of dimensionality associated with the higher ambient dimension.

• The paper presents a novel diffusion-based algorithm that employs graph Laplacian approximations to estimate manifold heat kernels for adaptive regression.
• It integrates labeled and unlabeled data to enhance performance in applications with scarce labels, such as medical imaging and speech recognition.
• Convergence analysis shows that the algorithm's efficiency depends solely on the manifold’s intrinsic dimension, mitigating the curse of high ambient dimensions.

Diffusion-based Semi-supervised Spectral Algorithm for Regression on Manifolds

The paper presents a novel approach to regression analysis on manifolds, focusing on handling high-dimensional data embedded within lower-dimensional structures. Traditional spectral methods typically rely on predetermined kernel functions, which often fail to adequately capture the complex geometric properties inherent to manifold data. This work introduces a diffusion-based spectral algorithm that leverages graph Laplacian approximations and the local properties of the heat kernel to provide an adaptive, data-driven regression framework.

Key Contributions

1. Graph Laplacian Approximation: The proposed method employs the graph Laplacian to estimate the manifold's heat kernel. This provides a flexible, computationally feasible alternative to directly calculating manifold-based kernels, which can be challenging in practice.
2. Semi-supervised Learning Framework: By integrating both labeled and unlabeled data, the algorithm enhances performance in environments where obtaining labeled data is costly or requires specific expertise. This aspect is crucial for many real-world applications, such as medical imaging and speech recognition.
3. Convergence Analysis: The paper provides a convergence analysis demonstrating that the algorithm achieves a rate dependent solely on the manifold's intrinsic dimension. This avoids the curse of dimensionality often associated with high ambient dimensions.

Numerical Results

The algorithm shows promising numerical results across various tests, including simulations on synthetic manifold data. These results illustrate the algorithm's potential to effectively capture the underlying manifold structure and accurately perform regression tasks.

Theoretical Implications

Theoretical contributions include extending the applicability of graph Laplacian approximations in estimating heat kernels directly on sampled data points. The ability to operate fully within the manifold's intrinsic structure without dimensionality reduction underscores the method's potential to address the complexities presented by high-dimensional data.

Practical Implications

The practical implications are significant, especially in fields where labeled data is scarce, and obtaining it is expensive. The semi-supervised approach allows practitioners to better utilize available resources by leveraging abundant unlabeled data.

Future Work

Future explorations could focus on refining the algorithm's dependency on selected hyperparameters, such as the diffusion time and truncation number. Additionally, extending the approach to further improve robustness and efficiency across diverse manifold topologies and in non-synthetic, real-world data scenarios would be valuable.

In summary, this paper offers a significant contribution to the field of manifold learning by providing a novel diffusion-based approach to spectral regression. This method addresses key challenges in existing algorithms, providing a promising direction for both theoretical exploration and practical applications.