Extending MF-LogDet beyond polynomial objectives

Extend the matrix-free log-determinant (MF-LogDet) affine normal computation framework beyond sparse polynomial objectives to broader structured function classes, specifically trigonometric polynomials, symmetric polynomials, and low-rank structured models.

Background

The proposed approach exploits the algebraic closure and sparsity of polynomial objectives to enable efficient matrix-free evaluation of gradients, Hessian–vector products, and directional third-order contractions.

The authors explicitly state that extending this computational framework to additional classes with exploitable structure—such as trigonometric or symmetric polynomials and low-rank models—remains an open direction for future study.

References

Several directions remain open for future study, including the design of effective preconditioners for tangent linear systems, multi-CPU and GPU implementations of the matrix-free kernels, and extensions beyond the polynomial setting to broader structured function classes such as trigonometric polynomials, symmetric polynomials, and low-rank structured models.

— Affine Normal Directions via Log-Determinant Geometry: Scalable Computation under Sparse Polynomial Structure  (2604.01163 - Niu et al., 1 Apr 2026) in Section 7 (Conclusion)