Interpretability and Representability of Commutative Algebra, Algebraic Topology, and Topological Spectral Theory for Real-World Data

TL;DR

The paper investigates the interpretability and representability of three mathematical AI frameworks—PH, PL, and PCA—across multiscale datasets, from geometric shapes to biomolecular structures. It develops and employs persistent Stanley–Reisner theory, persistent Hochster-based graded Betti numbers, and persistent f-/h-vectors to capture topological, geometric, and algebraic evolution in parallel. Through benchmark and real-data examples (octagon, octahedron, fullerene, and protein/DNA complexes), it demonstrates that PH offers efficient, intuitive summaries, PL adds geometric sensitivity via spectral data, and PCA provides rich algebraic interpretability, together yielding a robust hierarchy for explainable AI. This integrated perspective advances scalable, interpretable representations for real-world data and highlights trade-offs between computational cost and information richness across the PH–PL–PCA spectrum.

Abstract

Recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis (TDA) via persistent homology (PH) that provides explainable AI (xAI) by extracting multiscale structural features from complex datasets. This work investigates the interpretability and representability of three foundational mathematical AI methods, PH, persistent Laplacians (PL) derived from spectral theory, and persistent commutative algebra (PCA) rooted in Stanley-Reisner theory. We apply these methods to a set of data, including geometric shapes, synthetic complexes, fullerene structures, and biomolecular systems to examine their geometric, topological and algebraic properties. PH captures topological invariants such as connected components, loops, and voids through persistence barcodes. PL extends PH by incorporating spectral information, quantifying topological invariants, geometric stiffness and connectivity via harmonic and non-harmonic spectra. PCA introduces algebraic invariants such as graded Betti numbers, facet persistence, and f/h-vectors, offering combinatorial, topological, geometric, and algebraic perspectives on data over scales. Comparative analysis reveals that while PH offers computational efficiency and intuitive visualization, PL provides enhanced geometric sensitivity, and PCA delivers rich algebraic interpretability. Together, these methods form a hierarchy of mathematical representations, enabling explainable and generalizable AI for real-world data.

Figures (16)

• Figure 1: Illustration of PH and PL on the regular octagon. Top: Vietoris--Rips filtration process. Bottom: PL and PH for $0$-, $1$-, and $2$-dimensional features.
• Figure 2: Illustrations of persistent commutative algebra analysis on the regular octagon using a Rips complex-based filtration process.
• Figure 3: Illustration of PH and PL on the regular octahedron. Top: Vietoris--Rips filtration process. Bottom: PL and PH for $0$-, $1$-, and $2$-dimensional features.
• Figure 4: Illustrations of persistent commutative algebra analysis on the octahedron using a Rips complex-based filtration process.
• Figure 5: Illustration of PH and PL on the fullerene structure C$_{20}$. Top: Vietoris--Rips filtration process. Bottom: PL and PH for $0$-, $1$-, and $2$-dimensional features.