Hierarchical simplicial manifold learning.

Area of Science:

• Data science
• Computational topology
• Machine learning

Background:

• Learning global structures from complex data is crucial across scientific fields.
• Current methods often combine local data representations with global structure assembly.
• Integrating algebraic/computational topology with machine learning is a key challenge.

Purpose of the Study:

• To propose a novel hierarchical simplicial manifold learning algorithm.
• To construct simplicial complexes and decode their topological properties.
• To demonstrate the algorithm's applicability, convergence, and efficiency.

Main Methods:

• A hierarchical simplicial manifold learning algorithm is proposed.
• The algorithm utilizes nested clustering and topological reduction.
• It constructs simplicial complexes from sampled data.

Main Results:

• The learned simplicial complex preserves the topology of the original data manifold.
• The algorithm demonstrates convergence and computational efficiency.
• Applicability is shown on both synthetic and real-world datasets.

Conclusions:

• The proposed algorithm effectively learns global topological structures from complex data.
• It offers a robust method for constructing and analyzing simplicial complexes.
• This approach integrates manifold learning with topological data analysis.