Convergence of the alternating alignment-regression algorithm

Establish whether the alternating algorithm for jointly estimating elastic alignments and the spherical regression function has iterates that approach a minimum of the penalized risk, despite the risk being nonconvex in the alignments.

Background

The proposed elastic kernel ridge regression procedure alternates between aligning each observed SRVF to its current fitted value and updating the regression function. The penalized empirical risk is nonconvex with respect to the alignment parameters. Although each alternating step does not increase the risk, the paper does not establish that the resulting sequence of iterates converges to a minimizer.

References

The penalized risk is not convex in the alignments and, even though the alternating algorithm does not increase it at any step, it is not guaranteed that the iterates approach a minimum.

— Elastic kernel Ridge regression, with applications in phonetics  (2610.08386 - Matteo et al., 6 Oct 2026) in Section Discussion