Interpretability and Representability of Commutative Algebra, Algebraic Topology, and Topological Spectral Theory for...

Area of Science:

• Artificial Intelligence
• Topological Data Analysis
• Mathematical AI

Background:

• Mathematical Artificial Intelligence (AI) is rapidly advancing.
• Persistent homology (PH) is a successful AI approach for extracting multiscale structural features, crucial for explainable AI and world models.
• Interpretability and representability are key challenges in AI.

Purpose of the Study:

• To investigate the interpretability and representability of three foundational mathematical AI methods: PH, persistent Laplacians (PL), and persistent commutative algebra (PCA).
• To examine the geometric, topological, and algebraic properties of these methods when applied to diverse datasets.
• To compare the strengths and weaknesses of PH, PL, and PCA for data analysis.

Main Methods:

• Application of persistent homology (PH) using persistence barcodes to identify topological invariants.
• Utilizing persistent Laplacians (PL) derived from topological spectral theory to quantify spectral information, geometric stiffness, and connectivity.
• Employing persistent commutative algebra (PCA) rooted in Stanley-Reisner theory to extract algebraic invariants like Betti numbers and f/h-vectors.

Main Results:

• PH effectively captures topological features like loops and voids with computational efficiency.
• PL enhances geometric sensitivity by incorporating spectral information.
• PCA provides rich algebraic interpretability through combinatorial and algebraic invariants.
• A comparative analysis highlights the distinct advantages of each method across different data types (geometric shapes, fullerene structures, biomolecular systems).

Conclusions:

• PH, PL, and PCA form a hierarchy of mathematical representations for data analysis.
• Each method offers unique perspectives: PH for topology, PL for geometry, and PCA for algebra.
• Together, these methods contribute to developing more explainable and generalizable AI for real-world applications.