Low-Rank Riemannian Optimization for Graph-Based Clustering Applications.

Area of Science:

• Machine Learning
• Optimization Theory
• Data Science

Background:

• Machine learning applications are increasingly prevalent due to abundant data.
• Data preprocessing via clustering is crucial for robust statistical analysis.
• Optimization problems on positive semidefinite stochastic matrices are computationally intensive.

Purpose of the Study:

• To develop a low-complexity Riemannian optimization framework for solving problems on positive semidefinite stochastic matrices.
• To enable efficient graph-based clustering applications.
• To investigate the geometric properties of Riemannian manifolds for optimization.

Main Methods:

• Factorization of the optimization variable X=YY^T to reduce complexity.
• Derivation of conditions for satisfactory solutions based on the number of Y columns.
• Investigation of embedded and quotient geometries, including tangent space, Riemannian gradients, and Hessians.
• Development of a retraction operator for efficient first and second-order optimization methods.

Main Results:

• The proposed algorithms exhibit a clear complexity advantage over state-of-the-art Euclidean and Riemannian approaches.
• The framework is effective for graph clustering applications.
• The geometric analysis provides a foundation for efficient optimization methods.

Conclusions:

• The developed Riemannian optimization framework offers a computationally efficient solution for graph clustering.
• The low-complexity approach enhances the feasibility of advanced machine learning techniques.
• This work contributes to the advancement of optimization methods on Riemannian manifolds.