GBNL: Graded Betti number learning of complex biological data

While persistent homology (PH) is widely used for data shape analysis, persistent commutative algebra (PCA) has seen limited adoption in machine learning (ML) and data science. Unlike PH, which delivers topological invariants in the form of Betti numbers, PCA provides both algebraic invariants and graded Betti numbers. In this work, we introduce a first-of-its-kind application of commutative algebra graded Betti numbers in ML and data science. Specifically, we present Graded Betti Number Learning (GBNL) for protein-nucleic acid binding prediction. Protein-DNA/RNA interactions are fundamental to cellular processes such as replication, transcription, translation, and gene regulation, however their understanding and prediction remain challenging. GBNL represents each nucleic acid sequence as a family of $ k $-mer-specific sets and derives persistent graded Betti invariants from PCA, generating multiscale topological representations of local nucleotide organization. To incorporate cross-molecule context, these graded Betti representations are paired with transformer-based protein embeddings, linking nucleotide-level signals with global protein patterns. The proposed framework effectively detects single-site mutations and distinguish complete mutation patterns. Operating on primary sequences with minimal preprocessing, GBNL bridges commutative algebra, reduced algebraic topology, combinatorics, and ML, establishing a new paradigm for comparative sequence analysis. Numerical studies using three datasets highlight the success of GBNL in protein-nucleic acid binding prediction.