A Mapper Algorithm with Implicit Intervals and Its Optimization.

Area of Science:

• Topological Data Analysis
• Machine Learning
• Computational Biology

Background:

• The Mapper algorithm visualizes high-dimensional data but requires manual parameter tuning and overlooks data uncertainty.
• Existing variants often still need manual parameter adjustments and operate within deterministic frameworks.

Purpose of the Study:

• To develop a novel framework for automated parameter optimization in data visualization.
• To address limitations of the standard Mapper algorithm, including manual tuning and deterministic processing.
• To enhance the ability to capture complex data structures and identify subgroups in biomedical data.

Main Methods:

• Introduced a soft Mapper framework with implicit interval representation via a hidden assignment matrix.
• Utilized a Gaussian mixture model for flexible and implicit interval construction.
• Developed a stochastic gradient descent (SGD) algorithm with a topological loss function for parameter optimization.

Main Results:

• Demonstrated effectiveness in capturing underlying topological structures through simulation and application studies.
• Successfully identified a distinct subgroup of Alzheimer's Disease in an RNA expression dataset.
• The Mapper graph mode was introduced as a robust point estimation for the output graph.

Conclusions:

• The soft Mapper framework offers automated parameter optimization, overcoming limitations of standard Mapper algorithms.
• The method effectively visualizes complex data and has potential for identifying disease subgroups in biomedical research.
• The developed framework provides a more robust and flexible approach to topological data analysis.