Robust Spherical Laplacian Embedding

Area of Science:

• Machine Learning
• Data Science
• Dimensionality Reduction

Background:

• Traditional Laplacian embedding (LE) struggles with nonlinear structures in high-dimensional data.
• Existing distance metrics in LE lack discriminative power for real-world datasets like text and images.
• Cosine similarity is sensitive to outliers and noise, limiting its effectiveness.

Purpose of the Study:

• To develop a robust Laplacian embedding method (RS-LE) for improved data representation.
• To address the limitations of existing distance functions in embedding sparse and noisy data.
• To enhance the discriminative capabilities of low-dimensional projections for complex datasets.

Main Methods:

• Proposing a novel metric unifying Euclidean distance and cosine similarity in a spherical space using a robust $\ell _{p,p}$ -norm.
• Introducing an efficient proximal alternating linearized minimization (ALM) algorithm to solve the non-convex, non-smooth optimization problem.
• Developing robust spherical Laplacian embedding (RS-LE).

Main Results:

• The proposed RS-LE method provides more discriminative representations compared to existing techniques.
• The proximal ALM algorithm ensures global and objective convergence for the optimization problem.
• Experimental validation demonstrates the effectiveness of RS-LE on real-world datasets.

Conclusions:

• RS-LE offers a powerful and robust approach to dimensionality reduction for complex, high-dimensional data.
• The novel metric and efficient algorithm overcome limitations of traditional LE methods.
• This work advances data analysis techniques for applications in text and image processing.