Lifting Cocycles: From Heuristic to Theory

Abstract: The circular coordinates algorithm, a key tool in topological data analysis, relies on a theoretically unvalidated lifting step to convert cocycles from a prime field to integer coefficients. We provide a rigorous analysis of this procedure, establishing a criterion for its success. We also introduce a novel algebraic method to reduce any lifted cocycle to a cocycle with winding number 1, ensuring feature correctness. These principles are extended to homology cycles, solidifying the theoretical foundation of this widely used feature extraction technique.