Extend PALM convergence theory to general nonconvex hyperpriors

Establish convergence guarantees for the proximal alternating linearized minimization (PALM) algorithm when applied to the empirical Bayes framework objective with a hyperprior H(γ) that is neither convex nor concave, thereby accommodating general nonconvex hyperpriors.

Background

The paper introduces a PALM-based algorithm to minimize the empirical Bayes objective that combines a data-fit term, a log-determinant term, and a hyperprior. The authors prove convergence of PALM under the assumption that the hyperprior H is either convex or concave, covering important cases such as half-Laplace and certain generalized Gaussian hyperpriors.

However, many practically useful hyperpriors, especially heavy-tailed or more intricate nonconvex forms, do not satisfy global convexity or concavity. Extending the convergence theory to general nonconvex hyperpriors would broaden the applicability of the proposed method and solidify its theoretical foundations.

References

Several directions remain open for future work. First, it is important to extend the convergence theory of PALM beyond the convex or concave setting, to accommodate more general nonconvex hyperpriors.

— Sparsity via Hyperpriors: A Theoretical and Algorithmic Study under Empirical Bayes Framework (2511.06235 - Li et al., 9 Nov 2025) in Section 6 (Conclusions)