Topology Applied to Machine Learning: From Global to Local.

Area of Science:

• Computational topology
• Data science
• Machine learning

Background:

• Persistent homology (PH) initially focused on global data shape, disregarding short topological features as noise.
• Recent advancements utilize PH for local data geometry analysis, crucial for machine learning.

Purpose of the Study:

• To explain the evolution of applied topology since the advent of persistent homology.
• To highlight the importance of local geometric features in data analysis.
• To survey applications and methods for integrating PH into machine learning.

Main Methods:

• Reviewing early PH applications emphasizing global shape (e.g., three-circle model, cyclo-octane molecule).
• Examining recent PH uses for local geometry and vectorization techniques (persistence landscapes, persistence images).
• Surveying diverse applications across shape recognition, agent-based modeling, materials science, archaeology, and biology.

Main Results:

• Short persistent homology bars, often dismissed as noise, are vital for machine learning tasks.
• PH, through methods like persistence landscapes and images, effectively combines local and global data geometry.
• PH connects to geometric features such as curvature and fractal dimension.

Conclusions:

• Applied topology has shifted from global to local data analysis using persistent homology.
• Short topological features are as significant as long ones for machine learning.
• Integrating PH into machine learning offers powerful tools for diverse scientific fields.