GEORCE: A Fast New Control Algorithm for Computing Geodesics

Frederik M\"obius Rygaard; S{\o}ren Hauberg

REVIEW 4 major objections 4 minor 66 references

GEORCE: A Fast New Control Algorithm for Computing Geodesics

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that GEORCE, a control-theoretic reformulation of the discretized geodesic energy, converges globally and locally quadratically while using no Hessians, and reports much faster and more accurate geodesics than ADAM, BFGS…

desk verdict Novel fixed-metric control solver for geodesics with strong empirical results, but the quadratic local convergence proof has a real gap and should not be taken as established. read the letter →

arxiv 2505.05961 v3 pith:SWYBQYKQ submitted 2025-05-09 math.DG stat.CO

Frederik Möbius Rygaard , S{o}ren Hauberg This is my paper · ORCID

classification math.DGstat.CO MSC 53C2249Q9965D15

keywords geodesicsRiemannianmanifoldsFinsleroptimalcontroldiscreteenergyminimizationglobalconvergencequadraticinformationgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

GEORCE is a new algorithm for computing geodesics—locally shortest paths—on Riemannian and Finsler manifolds. The paper's central claim is that discretizing the geodesic energy as a discrete optimal control problem makes each iteration an explicit linear solve, because the cost becomes convex in the velocity variables once the metric matrices are frozen. The authors prove global convergence to a local minimum and local quadratic convergence near it, matching Newton's rate without forming a Hessian. They report that GEORCE is much faster and finds shorter curves than ADAM, BFGS, and sparse Newton methods on spheres, ellipsoids, hyperbolic spaces, statistical manifolds, and learned generative-model manifolds.

What carries the argument

The central mechanism is the discrete control reformulation: the curve is represented by states $x_t$ and velocity controls $u_t$ linked by $x_{t+1}=x_t+u_t$, with endpoints fixed. Each GEORCE iteration freezes the positive-definite metric matrices $G_t=G(x_t^{(i)})$ and the derivative vectors at the current iterate, so the Pontryagin necessary conditions become a linear system whose closed-form solution (Proposition 2) is a sum of inverses $G_t^{-1}$. The soft line search (Armijo backtracking with decay $\rho=0.5$) converts each descent direction into a globally convergent step. The cost per iteration is $O(Td^3)$, linear in the number of grid points $T$ and cubic in the manifold dimension $d$, and only first-order derivatives of the metric are needed.

What would settle it

Run GEORCE with an exact line search on a strongly convex metric near a known geodesic—for example, a small perturbation of a great-circle geodesic on $S^2$—and record the accepted steps $\alpha^*$ as the iterates approach the optimum. If $\alpha^*$ is not 1 in an entire neighborhood of the optimum, or if the ratio $\|z^{(i+1)}-z^*\|/\|z^{(i)}-z^*\|^2$ fails to stay bounded, then the assumptions of Proposition 8 do not hold and the quadratic-rate claim would not hold on that example.

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Extended reading notes

Core claim

The discovery is that the geodesic boundary-value problem, discretized as minimising $\sum_{t=0}^{T-1} (x_{t+1}-x_t)^\top G(x_t)(x_{t+1}-x_t)$, can be rewritten as a discrete control problem in which the objective is strictly convex in the controls once the metric is frozen. The time-discrete Pontryagin maximum principle yields first-order necessary conditions, and fixing $G_t=G(x_t^{(i)})$ and the metric-derivative terms at the current iterate turns those conditions into a closed-form update built from a sum of inverse metric matrices. The resulting scheme, GEORCE, is proven to converge from any feasible straight-line initialization to a local minimum and, under local strict convexity and acceptance of the full step, to converge quadratically near the minimum. The same construction, with one additional velocity-derivative term $\zeta_t$, extends to Finsler manifolds.

Load-bearing premise

The load-bearing premise is that near the optimum the line search always accepts the full GEORCE step ($\alpha^* = 1$); the paper does not prove this, and converting the proven quadratic decrease of the energy into quadratic convergence of the iterates needs an unstated quadratic-growth condition on the energy.

Editorial extensions

If this is right

A geodesic boundary-value problem can be solved by a sequence of closed-form linear solves, each costing $O(Td^3)$, with no Christoffel symbols, Hessians, or second-order metric information.
Because the same update with one extra term handles Finsler metrics, asymmetric shortest-path problems such as navigation under a force field inherit the same convergence guarantees and practical speed.
On the benchmark manifolds, GEORCE reaches the gradient tolerance in a handful of iterations and returns shorter computed lengths than ADAM, BFGS, and sparse Newton at both $T=50$ and $T=100$.
On Fisher-Rao statistical manifolds and on variational-autoencoder latent spaces, the few-iteration behavior makes geodesic computation practical at higher dimension and grid size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural reading is that GEORCE is a metric-aware quasi-Newton method: each iteration solves the exact quadratic model obtained by freezing $G$, which explains Newton-like local speed when the frozen model approximates the true energy well.
Replacing the exact inverse metric sum with a cheaper surrogate (low-rank, diagonal-plus-sampling, or iterative solve) could reduce the $O(d^3)$ per-iteration cost; the paper names inverse evaluation as the main limitation but does not pursue such surrogates.
Because the first control $u_0$ estimates the logarithmic map, GEORCE could plug directly into downstream tasks such as Fréchet means, geodesic regression, or parallel transport without extra machinery.
The Randers-type benchmarks suggest an immediate test in time-optimal path planning with asymmetric costs, where reversing the force field should reverse the optimal geodesic and expose the Finsler distance asymmetry.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: GEORCE is a genuinely new way to compute geodesics on Riemannian and Finsler manifolds by reformulating the discrete energy as a control problem and freezing the metric each iteration. The empirical gains are large and consistent across many manifolds, including VAE latent spaces, with code released. The paper deserves a serious referee. But the headline theoretical claim—quadratic local convergence—is not proven as written. Proposition 8 assumes α*=1 locally (i.e., no line search) without argument, and even granting that, it derives a bound on the energy decrease O(||z_i−z*||^2) and then jumps to a bound on the iterate distance. That jump needs a quadratic-growth lower bound on the energy near the minimum; local strict convexity does not supply it (z^4 on R is the standard counterexample). The proof gap is load-bearing because the abstract and conclusion advertise quadratic convergence. The Finsler extension inherits the same issue via "completely similar." This is a fixable problem—either prove α*→1 and quadratic growth under stronger assumptions, or soften the claim to superlinear/linear with a sharp rate. The empirical quadratic-looking behavior in Fig. 3 is suggestive but not a proof. What the paper does well: the control reformulation is clean, the fixed-metric linear solve is new relative to Noakes–Zhang, the complexity analysis is honest, and the benchmarks are broad and reproducible. The global convergence argument (Prop. 5) is essentially a descent argument and seems plausible, though the notation is messy and the line search details could be clearer. One more soft spot: the empirical accuracy metric is the same discrete energy the algorithms minimize, so "more accurate" is not an independent check. That is standard for optimization papers, but worth flagging. Overall, the algorithm is likely to be useful to people computing geodesics in information geometry, medical imaging, and generative modeling. I would cite it if I worked in that area, and I would send it to peer review despite the proof gap, with the expectation that the authors either repair the local convergence claim or restate it accurately.

Referee Report

4 major / 4 minor

Summary. The paper introduces GEORCE, an algorithm for computing geodesics on Riemannian and Finsler manifolds by rewriting the discretized geodesic energy (Eq. 5) as a discrete optimal control problem (Eq. 6), freezing the metric and its derivative at the current iterate, solving the resulting linear system explicitly (Eq. 11), and combining this with a line search. The authors claim global convergence and local quadratic convergence for the discretized energy, an extension to Finsler manifolds, and substantial empirical speedups and lower final energies relative to ADAM, BFGS, and sparse Newton methods across spheres, ellipsoids, tori, hyperbolic space, SPD matrices, Fisher-Rao statistical manifolds, and VAE-learned manifolds. The paper includes pseudocode for the Riemannian and Finsler versions, derivations for a sparse Newton baseline, extensive tables for T=50 and T=100, and a public code repository.

Significance. If the convergence claims are made rigorous, GEORCE would be a genuinely useful contribution: it is a first-order method with an O(Td^3) per-iteration cost that scales linearly in the number of grid points, and the benchmarks cover a broad and relevant set of manifolds, including information-geometric and learned-manifold settings. The paper is also commendable for shipping open-source code, using no fitted parameters in the algorithm itself, and comparing against a wide range of optimization baselines on CPU and GPU. However, the theoretical results currently outrun the proofs: the quadratic-convergence theorem has a substantive logical gap, the Finsler proof is deferred to a sketch, and the global-convergence proof conflates convergence of the energy with convergence of the iterates. The empirical evidence is suggestive but cannot substitute for those proofs, so the central theoretical claims are not yet established as stated.

major comments (4)

[Section 3.2, Proposition 8 and Eqs. (15)-(16)] The proof of local quadratic convergence is not complete. Eq. (15) and the surrounding argument establish, at most, a bound on the energy difference E(z^{(i+1)}) - E(z*) of order O(||z^{(i)} - z*||^2). To convert such an energy bound into the iterate bound ||z^{(i+1)} - z*|| <= c||z^{(i)} - z*||^2 in Eq. (16), one needs a quadratic lower bound E(z) - E(z*) >= m||z - z*||^2 near z*. Assumption 6 only postulates local strict convexity, which does not imply quadratic growth; for example, f(z) = z^4 is locally strictly convex at z* = 0 but has no quadratic lower bound there. The final step of the proof uses Lipschitz continuity to write E(z^{(i+1)}) - E(z*) = O(||z^{(i+1)} - z*||), which is an upper bound on the energy gap, not the required lower comparison. Without an additional strong-convexity or quadratic-growth hypothesis, the claimed Q-quadratic convergence does not follow.
[Section 3.2, Proposition 8 and Algorithm 1] The proposition assumes "locally alpha* = 1, i.e. no line search," but this is not proved for either the exact line search or the Armijo/backtracking procedure described in Algorithm 1. If, in a neighborhood of z*, the accepted step size is bounded away from 1, the update is a damped iteration and one would generally expect only linear convergence. The paragraph after the proof gives a heuristic justification, not a mathematical argument. This hypothesis is load-bearing for the quadratic-convergence claim and must either be proved or be made an explicit assumption with the theorem restated accordingly.
[Section 3.2, Proposition 5] The global-convergence proof shows that the scalar energy sequence E^{(i)} is decreasing and bounded below, hence convergent, but it does not establish convergence of the iterates z^{(i)}. The sentence "denote the convergence point (x-hat, u-hat)" presupposes the desired conclusion. Without compactness of the feasible set or a separate argument that every cluster point is stationary and that the energy limit is attained, an energy-convergent sequence need not have a well-defined limit point. The subsequent paragraph about jumping between two equal-energy local minima is not a substitute for a standard global-convergence argument. Thus the claim that GEORCE "will converge to a (local) minimum" is underproved as written.
[Appendix A.2] The Finsler extension is deferred with the statement that the proof is "completely similar" to the Riemannian case. This is not sufficient because the Finsler version inherits both gaps identified above: the unproved assumption that alpha* = 1 locally and the missing quadratic-growth condition. In addition, because the fundamental tensor G(x,u) now depends on the control variable, the Taylor expansion and the identity in Eq. (A5) need to be verified carefully with the velocity-dependent metric; the sketch does not carry the proof. The abstract's claim of quadratic local convergence for Finsler manifolds is therefore not currently supported.

minor comments (4)

[Appendix B.2, Algorithms 3 and 4] Both algorithms contain malformed conditional statements ("if then ...") and the switching condition between the gradient step and the Newton step is not fully specified; the pseudocode should be corrected and clarified.
[Section 2, paragraph on Levi-Civita connection] The text states that existence and uniqueness of the Levi-Civita connection are "guaranteed by completeness of the metric." The Levi-Civita connection exists for every smooth Riemannian metric without a completeness assumption; this sentence should be corrected.
[Section 5 and Tables 2-7] The empirical accuracy metric is the same discrete energy functional in Eq. (5) that all compared algorithms minimize. Phrases such as "more accurate" should be framed as "lower discrete energy and shorter computed length" rather than accuracy against an independent ground truth, unless a reference solution is available for the specific manifold.
[Figure 2] The caption and text mention dimensions n = 2,3,5,10,20, but the horizontal axis of the figure is labeled only up to 10; please align the description with the plotted dimensions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: GEORCE's derivation is self-contained, and its convergence claims are conditional theorems rather than reductions to fitted inputs or self-citations.

full rationale

The derivation of GEORCE is a genuine construction: the discretized energy is reformulated as a control problem, the optimality conditions are stated, and an iterative scheme is obtained by fixing the metric and its derivative. No parameter is fitted to data and then renamed as a prediction. Proposition 5 is a standard descent-with-line-search global convergence argument, and Proposition 8 assumes local strict convexity and alpha* = 1 and derives a quadratic rate; its conclusion is not identical to its hypotheses. The Finsler extension is deferred as 'completely similar', which is a proof gap rather than a circular reduction. The empirical benchmarks compare methods on the same discretized energy objective, which is standard optimization benchmarking and does not make the comparison circular; no fitted value is relabeled as an independent ground-truth result. Self-citations are used for background and for learned-manifold context, but they are not load-bearing for the main convergence or derivation claims. Therefore no specific circular step can be exhibited from the paper's equations or citations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The algorithm introduces no fitted parameters or invented entities. It rests on standard differential geometry assumptions plus a specific regularity condition for the quadratic convergence proof.

assumptions (5)

domain assumption The manifold is given in a local chart and the entire candidate curve remains within that chart.
Stated in Section 3: 'We assume access to a local chart of the manifold and assume that the entire candidate curve of the geodesic can be represented within the same chart.' Without this, the linear-system update may not be defined.
domain assumption End points are not conjugate points and the discretization is fine enough that critical points of the discrete energy are local or global minima.
Stated in Section 3 bullet list; needed for the limit of GEORCE to be a true local minimum of the energy.
standard math The metric matrix G(x_t) is symmetric positive definite at every point used.
Definition of a Riemannian metric; used to invert G_t in Proposition 2.
ad hoc to paper Close to the solution, the line search step alpha* equals 1 and E is locally strictly convex.
Assumption 6 and Proposition 8; this is load-bearing for the quadratic convergence claim and is not proven for the line search.
standard math The discretized energy functional is smooth, so first-order Taylor expansion yields O(||Delta z||^2) terms.
Used throughout the convergence proofs; smoothness of the metric implies smoothness of E.

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Cite this review

Pith. "Pith review of GEORCE: A Fast New Control Algorithm for Computing Geodesics." pith.science (2026). https://pith.science/paper/SWYBQYKQ

@misc{pith2026250505961,
  author       = {Pith},
  title        = {Pith review of: GEORCE: A Fast New Control Algorithm for Computing Geodesics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWYBQYKQ}},
  note         = {Machine review of arXiv:2505.05961}
}

read the original abstract

Computing geodesics for Riemannian manifolds is a difficult task that often relies on numerical approximations. However, these approximations tend to be either numerically unstable, have slow convergence, or scale poorly with manifold dimension and number of grid points. We introduce a new algorithm called GEORCE that computes geodesics in a local chart via a transformation into a discrete control problem. We show that GEORCE has global convergence and quadratic local convergence. In addition, we show that it extends to Finsler manifolds. For both Finslerian and Riemannian manifolds, we thoroughly benchmark GEORCE against several alternative optimization algorithms and show empirically that it has a much faster and more accurate performance for a variety of manifolds, including key manifolds from information theory and manifolds that are learned using generative models.

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