Knot data analysis using multiscale Gauss link integral.

Area of Science:

• Data Science
• Applied Mathematics
• Computational Biology

Background:

• Topological data analysis (TDA) is a powerful approach in data science, but knot theory applications are limited by lack of localization and quantization.
• Existing methods struggle to capture both global topological properties and local data structures effectively.

Purpose of the Study:

• To address the limitations of traditional TDA in practical applications by introducing Knot Data Analysis (KDA).
• To develop a multiscale geometric topology approach for analyzing complex datasets.

Main Methods:

• Introduced Knot Data Analysis (KDA), integrating curve segmentation and multiscale analysis into the Gauss link integral.
• Developed the multiscale Gauss link integral (mGLI) to recover global topological properties and capture local structures.
• Integrated mGLI with machine learning and deep learning models.

Main Results:

• The multiscale Gauss link integral (mGLI) effectively recovers global knot properties and local data connectivities at appropriate scales.
• KDA, particularly when combined with deep learning, significantly outperformed state-of-the-art methods on 13 complex biological datasets.
• Demonstrated superior performance in protein flexibility analysis, protein-ligand interactions, ion channel blockade screening, and toxicity assessment.

Conclusions:

• Knot Data Analysis (KDA) provides a robust framework for applying knot theory to complex data, overcoming previous limitations.
• The multiscale Gauss link integral (mGLI) offers a novel approach for multiscale geometric topology analysis.
• KDA opens a new research frontier in 'knot deep learning' with broad implications for data science and computational biology.