Abstract
Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the “true” geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem.
Highlights
Since objects vary gradually in the real world, the manifold assumption indicates that the data points depict the state of an object should distribute on a smooth low-dimensional manifold embedded in high-dimensional observation space [1]
Our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem
A local structure term is added in the Tikhonov model to make the noise points diffuse on the tangent space of the manifold
Summary
Since objects vary gradually in the real world, the manifold assumption indicates that the data points depict the state of an object should distribute on a smooth low-dimensional manifold embedded in high-dimensional observation space [1]. (2) Our method improves the Tikhonov model to make the variational diffusion on the tangent space of the manifold for a high-dimensional nonrigid point cloud dataset. The key factors that control the geometric distribution of the dataset are maintained and the characteristics of individual points are removed as noise. Many machine learning methods proposed the noise-resistant model for outliers but did not discuss denoising as an independent problem [7, 20]. We propose a denoising method for the dataset. is method improves the Tikhonov method by adding a local structure term. e optimal solution is obtained by minimizing the objective function through a variational diffusion approach
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