Preconditioners for tangent linear systems in MF-LogDet

Design effective preconditioners for the tangent linear systems (H_T + λ I) u = b that arise in the matrix-free log-determinant (MF-LogDet) affine normal computation of affine normal directions, so as to accelerate Krylov subspace solves within the tangent space of the level set.

Background

The MF-LogDet framework computes affine normal directions by solving tangent-space linear systems involving the projected Hessian H_T and, when needed, a regularization shift. The overall complexity and runtime depend strongly on the efficiency of these Krylov solves.

While the paper outlines basic preconditioning ideas, it explicitly identifies the design of effective preconditioners tailored to the tangent operator H_T as an open direction, anticipating that such preconditioners would reduce inner iteration counts and improve scalability.

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)