Geodesic Switches: Mechanisms & Models

• Geodesic switches are transition phenomena where geodesic trajectories change regimes, defined by local deterministic or probabilistic rules.
• They underpin mechanisms in discrete models, neural networks, and geometric triangulations, enabling efficient mode connectivity and manifold reconfiguration.
• These switches reveal deep links between geometry, topology, and stochastic processes, with practical implications in optimization and quantum control.

Geodesic switches are transition phenomena or update rules manifesting when geodesic trajectories or flows meet, interact, or change regime in geometric, combinatorial, and stochastic settings. The concept arises in multiple domains, including discrete particle flows on abstract manifolds, neural network mode connectivity, combinatorial triangulations, computational geometry, quantum control, and random percolation models. These switches encode the deterministic or probabilistic mechanisms by which geodesics "jump," "collide," or undergo regime changes (periodic ↔ drift ↔ divergent) under the influence of underlying geometric, algebraic, or environmental constraints.

1. Geodesic Switches in Discrete Geometric Dynamics

In the evolution of multiple particles on discrete manifolds, geodesic switches correspond to local deterministic, reversible update rules triggered by particle encounters on shared facets. In the framework of interacting geodesics on discrete manifolds (Knill, 30 May 2025), the phase space is a frame bundle $P$ over an abstract simplicial complex $G$, with particles represented as signed, totally ordered maximal simplices. Every particle moves independently along discrete geodesic paths until multiple particles coincide on the same facet, at which point a geodesic switch is executed:

• Switch Mechanism: The local rule applies only at multisite encounters, relying solely on local degree (net positive/negative particles), ensuring that interactions are abelian and locally causal.
• Mathematical Model: The evolution operator

$T = BA$

(with $A$, $B$ involutions) updates particle signs and positions with explicit action on frame permutations and sign reversal, governed by local multiplicities.

• Physical Significance: The switch generalizes classical momentum-exchange collisions and enables creation or destruction of composite geodesic entities ("molecules") and reversible space deformations.
• Space-Matter Duality: The global evolution of divisors realizes a bijective, volume-preserving time-dependent deformation of the manifold itself—a dynamic blend of "matter" (particle configuration) and "space" (frame bundle rearrangement).

2. Geodesic Mode Connectivity and Switching in Neural Networks

In the context of neural network loss landscapes, geodesic switches underlie the transition between solutions (modes) through low-loss paths, notably reframed via information geometry (Tan et al., 2023):

• Loss Landscape Geometry: Networks' parameter spaces induce Riemannian manifolds of probability distributions, equipped with Fisher-Rao metric,

$$g_{ij}(\bm{\theta}) = \mathbb{E}_{p(\bm{x}, \bm{\hat{y}}; \bm{\theta})} \left[ \frac{\partial \log p}{\partial \theta_i} \frac{\partial \log p}{\partial \theta_j} \right]$$

• Geodesic Path Construction: Given two trained network weights $\bm{\theta}_a, \bm{\theta}_b$, a discretized geodesic is found by minimizing the sum of Jensen-Shannon divergences between subsequent model distributions:

$$\mathcal{L}(\{\bm{\theta}_i\}_{i=1}^N) = \sum_{i=1}^{N-1} \mathrm{JSD}(p_i || p_{i+1})$$

• Switching Interpretation: The geodesic switch is the process of morphing between modes along the manifold of distributions, effectively moving from one minimum to another without crossing high-loss regions, unlike linear interpolation in parameter space.
• Empirical Evidence: Narrow ResNet models on CIFAR-10 attain mode connectivity only under geodesic optimization, demonstrating that geodesic switches enable traversability of normally disconnected loss minima.

3. Switches in Geometric Triangulations and Bistellar Moves

Geodesic switches are formalized in geometric topology as bistellar moves or edge flips that reorganize triangulations of Riemannian manifolds while preserving geodesic embedding (Kalelkar et al., 2019):

• Geometric Triangulation: Each simplex is totally geodesic. Edge flips (2D) and higher-dimensional bistellar moves yield the universal "geodesic switch" operation.
• Connectivity: In dimensions 2 and 3, the flip graph of geometric triangulations is connected under geodesic switches, mirroring combinatorial triangulation connectivity but with geodesic constraints. Derived subdivisions are required in higher dimensions to enable connectivity.
• Implication: Any invariant under bistellar moves is an invariant of the manifold for constant curvature spaces, as all triangulations are related by geodesic switches after sufficient subdivision.

Setting	Switch Mechanism	Connectivity/Outcome
Discrete Manifold (particles)	Local deterministic rule ($T = BA$)	Reversible interactions and space deformation
Neural net loss landscape	Geodesic path through Fisher-Rao/JSD	Traversal between minima (mode connectivity)
Geometric triangulation (2-3D)	Edge flip/bistellar move	Universal connectivity under switches

4. Regime Transitions on Periodic Surfaces

On surfaces such as the Necker cube (Hooper et al., 2023), geodesic switches correspond to arithmetically-induced dynamical transitions in long-term trajectory behavior:

• Regime Switches: The initial direction $m = p/q$ (with odd/odd, odd/even, or irrational values) completely dictates whether the geodesic is closed, drift-periodic, or divergent. A switch occurs at arithmetic boundaries in the direction space.
• Symmetry/Switch Boundaries: The half-translation cover and affine symmetries classify and facilitate these transitions, with Dehn twist cylinders arising in directions supporting closed geodesics.
• Global Dynamical Properties: Geodesic switches produce a sharp, non-gradual transition in the dynamical behavior of trajectories, partitioning phase space into invariant classes.

5. Intersection and Switching Algorithms in Computational Geometry

Geodesic switches are algorithmically essential in computational geometry for managing intersection transitions and ensuring robust enumeration of distinct geodesic crossings (Karney, 2023):

• Intersection Algorithms: Efficiently locate the closest or all intersections between geodesics on an ellipsoid using iterative triangle solves and $L_1$ metrics in displacement space.
• Switching Principle: Minimal spacing between intersections ($|\mathbf{T} - \mathbf{T}'| \ge 2t_1$) ensures discrete, non-overlapping transitions (switches) between intersection events. Handling of coincident geodesics involves flagging and canonicalization, effectively switching between distinct intersection modalities.

6. Geodesic Switches in Quantum Control Optimization

In quantum gate synthesis, geodesic switches are procedural steps enabling escape from local minima or adaptation to restrictive Hamiltonian constraints in multi-qubit gate implementation (Lewis et al., 11 Jan 2024):

• Geodesic Algorithm: The optimization follows geodesics on the $\mathrm{SU}(2^n)$ manifold, updating Hamiltonian parameters via projection onto effective geodesic direction in Lie algebra.
• Switching Operation: When the trajectory stalls or constraints impede direct geodesic following, Gram-Schmidt restarts (orthogonal exploration) constitute algorithmic geodesic switches, systematically searching new directions for global solution attainment and fidelity optimization.

Domain	Switch Methodology	Algorithmic/Physical Role
Ellipsoid intersections	Minimal metric separation, iterative solve	Discrete intersection selection
Quantum gates	Geodesic projection + orthogonal restart	Escape from minima, optimal synthesis

7. Stochastic Geodesic Switching in Random Growth Models

In stochastic models such as dynamical Brownian last passage percolation, geodesic switches quantify noise-induced macroscopic changes in optimal paths (Bhatia, 31 Oct 2025):

• Dynamical BLPP Model: Geodesics $\Gamma_{(0,0)}^{(n,n),r}$ change structure under random environment resampling; each "switch" is a coarse-grained reconfiguration away from endpoints as time evolves.
• Quantitative Law: The expected number of switches away from endpoints is bounded by $n^{5/3+o(1)}(t-s)$, saturating the KPZ spatial fluctuation scaling.
• Exceptional Times: The set of times admitting bi-infinite geodesics has Hausdorff dimension at most $1/2$, while times allowing directional bigeodesics form a set of dimension zero—demonstrating the profound sparsity of non-typical switching events under persistent randomness.

Conclusion

Geodesic switches encapsulate diverse structural and dynamic transitions in geometric and stochastic system evolution, ranging from deterministic cellular automaton rules on discrete manifolds, regime transitions in global dynamical flows, and combinatorial topology transformations, to noise-driven reconfigurations in integrable probability and optimized quantum control. Their common feature is the local-to-global impact of interactions or boundary crossings on geodesic trajectories, governed by precise mathematical rules and sharply-classified mechanisms. These phenomena form foundational elements in understanding manifold connectivity, neural network interpolation, quantum gate synthesis, and the geometry of randomness, providing rigorous frameworks for both theoretical analysis and algorithmic implementation across disciplines.