Data-Driven Rank Inference with Certified Sylvester Reconstruction for Robust Binary Waring Decomposition

Binary Waring decomposition seeks to express a homogeneous binary form as a minimal sum of powers of linear forms. In the binary setting, Sylvester’s theorem gives a classical algebraic route for rank determination and parameter recovery through structured Hankel/catalecticant matrices. Although this procedure is exact and interpretable in ideal arithmetic, practical rank identification may become unstable when the input coefficients are contaminated by noise or when the underlying roots are close to degenerate configurations. This paper develops a data-driven rank inference framework coupled with certified Sylvester reconstruction for robust binary Waring decomposition. The proposed method first converts the coefficient sequence into a Hankel-aware graph that captures recurrence-induced dependencies among polynomial coefficients. A graph neural network is then used to infer plausible rank candidates from this structured representation. Instead of accepting a single prediction directly, the framework performs explicit Sylvester reconstruction and algebraic residual verification for candidate ranks. To further improve decision reliability, a lightweight meta-verification module integrates reconstruction residuals, model confidence scores, and stability-related indicators to select the most credible rank. Experiments on large-scale synthetic binary forms demonstrate that the proposed approach improves rank identification accuracy and verified reconstruction success under low-to-moderate noise, while maintaining the transparency and auditability of classical symbolic–numeric computation. These results suggest that data-driven rank inference can serve as an effective front-end for algebraically certified reconstruction, especially in numerically ambiguous regimes where fixed threshold-based Sylvester implementations are fragile.