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Develop effective numerical optimization methods for Stiefel codes

Develop and analyze optimization strategies that efficiently find high-quality or provably optimal codes in Stiefel manifolds StF(d,r) under the chordal distance, improving on existing heuristic approaches.

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Background

Previous work has used numerical optimization to construct Stiefel codes, but without guarantees of global optimality. The present paper provides exact constructions achieving known bounds, highlighting the need for systematic numerical methods that can find or certify optimality more broadly.

The open question seeks principled optimization strategies tailored to the geometry and symmetries of Stiefel manifolds that can reliably produce putatively optimal configurations.

References

In this paper, we constructed several optimal codes in the Stiefel manifold with chordal distance, but many open problems remain. From a numerical standpoint, what are good optimization strategies for finding putatively optimal codes in the Stiefel manifold?

— Optimal codes in the Stiefel manifold (2407.01813 - Jasper et al., 1 Jul 2024) in Section 5 (Discussion)