Uniqueness of solutions to scalar variational principles for Kac–Rice complexities

Establish whether the scalar variational problems obtained by simplifying the Kac–Rice variational formulas for the annealed complexities of local minima, sub-extensive-index saddles, and generic critical points admit unique solutions, or rigorously characterize parameter regimes in which multiple solutions (e.g., multiple fixed points) occur.

Background

The paper reduces functional variational formulas for complexities to scalar variational principles to enable efficient numerical solution. While numerical evidence suggests uniqueness for local minima and sub-extensive-index saddles, the authors observed multiple fixed points for the total complexity of all critical points in a limited parameter region, indicative of a first-order phase transition.

The authors explicitly state they cannot guarantee uniqueness of solutions in general, motivating a rigorous investigation of uniqueness or multiplicity conditions for these scalar variational problems.

References

Finally, while we can not guarantee mathematically that these scalar variational principles do not admit several solutions, we numerically never found more than one solution for local minima and saddles of sub-extensive index. This is however not the case for the complexity of all critical points, for which we exhibited the presence of two fixed points in a (very limited) region of parameters, associated to a first-order phase transition, see Appendix~\ref{subsec_app:phase_transition_total_complexity}.

— Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval  (2602.17779 - Maillard et al., 19 Feb 2026) in Section 1.3 (Summary of contributions)