Schur–Padé error analysis of the reverse recurrence

Derive a Schur–Padé error analysis for the reverse recurrence used to compute the backward pass of the decomposition-free polynomial matrix-logarithm normalizer.

Background

The proposed backward pass uses a reverse three-term recurrence over cached polynomial basis matrices and is validated numerically through finite-difference gradient checks and comparison with automatic differentiation. However, the paper does not provide a Schur–Padé-based error analysis of this reverse recurrence.

Such an analysis would address the numerical fidelity and stability of the recurrence relative to a high-precision Schur–Padé reference, complementing the empirical gradient and reconstruction-error evaluations.

References

We leave these, with dense prediction and a Schur--Padé error analysis of the reverse recurrence, to future work.

— Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling  (2608.19021 - Rahman et al., 19 Aug 2026) in Section Conclusion, paragraph “Limitations”