Existence of gradient flows for infinite-dimensional deformation groups

Establish existence of the gradient flow arising from the indirect shape-matching framework when it is applied to tomography using infinite-dimensional deformation groups.

Background

The paper derives gradient flows for indirect shape matching on finite-dimensional Lie groups and applies them to reconstruct protein backbones from cryo-electron microscopy data. The authors note that the same geometric framework could be extended to tomography, where the relevant deformation groups may be infinite-dimensional rather than finite-dimensional.

For infinite-dimensional groups, existence of solutions to the gradient-flow equation is not guaranteed by the finite-dimensional theory used in the paper. Establishing such existence is therefore identified as a concrete direction for future work.

References

For instance, tomography is a possible application, and requires the use of infinite-dimensional groups. In this setting, questions such as existence of the flow are non-trivial to answer, and are something we intend to explore in future work.

— Recovering protein conformations from single-particle cryo-EM data via indirect shape matching gradient flows  (2608.17759 - Jansson et al., 18 Aug 2026) in Conclusion and outlook