{"id":"d7368011-5d76-4a2d-8c13-b234f96bcb8d","arxiv_id":"2505.05961","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A control-based iterative solver computes geodesics on Riemannian and Finsler manifolds with global convergence and empirically fast, accurate performance.","lead":"Researchers introduce GEORCE, a new algorithm that computes geodesics, the shortest paths on curved spaces, by reframing the problem as a discrete control problem. It could make geodesic computation faster and more reliable for machine learning, medical imaging, and robotics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 8 does not establish the claimed Q-quadratic convergence: it assumes α*=1 without proof and needs an unstated quadratic-growth condition to pass from energy decrease to iterate distance; the central theoretical claim is unproven.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: Proposition 8 assumes α* = 1 and only proves quadratic shrinkage of the energy, not of the iterate distance, unless an unstated quadratic-growth condition is added. My reading of the proof confirms this. The global convergence proof (Proposition 5) also has gaps — energy decrease alone does not guarantee iterate convergence without compactness or a Zoutendijk-type condition, and Lemma 4's notation is inconsistent in places — but the local quadratic convergence claim is the more distinctive part of the central claim and is where the argument is least secure. The paper does provide real support elsewhere: a released implementation, broad benchmarks on many manifolds in both Riemannian and Finsler settings, and reproducible experimental details. Those empirical results are consistent with fast practical convergence, but they do not substitute for the missing proof of the stated rate. Because the reader already returned CONDITIONAL, my concern does not change the verdict; it sharpens the condition: the paper should either prove the α* = 1 premise and add an explicit quadratic-growth assumption to Proposition 8, or weaken the claimed convergence rate to what is actually proven.","tokens_in":42426,"tokens_out":6929,"duration_ms":74957,"concrete_test":"Independently re-derive the final step of Proposition 8: from E(z^{(i+1)}) - E(z*) ≤ C||z^{(i)} - z*||^2 and local Lipschitzness of E, derive ||z^{(i+1)} - z*|| ≤ c||z^{(i)} - z*||^2. Record every additional assumption needed, in particular a quadratic-growth or strong-convexity condition on E near z*. If the derivation cannot be completed using only Assumption 6 as stated, then Proposition 8's conclusion is not established. Optionally, also instrument Algorithm 1 on S^2 and E(50) near convergence and record the accepted α values; if α* < 1 occurs within a neighborhood of the solution, the proposition's premise fails in the very regime it claims to cover.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that GEORCE has quadratic local convergence rests on Proposition 8, and that proof has two unsecured links. First, the proposition assumes 'locally α* = 1, i.e. no line search' (Section 3.2). This is a substantive hypothesis, not a consequence of the previous arguments: if the accepted Armijo step is bounded away from 1 in a neighborhood of z*, the update is damped and one would generically expect linear, not quadratic, convergence. The paper provides no argument that the exact or backtracking line search accepts the full GEORCE step locally, only a heuristic sentence after the proof. Second, even granting α* = 1, the proof shows at most an energy decrease bound. From the displayed equations around Eq. 15, it derives E(z^{(i+1)}) - E(z*) ≤ O(||z^{(i)} - z*||^2). To conclude ||z^{(i+1)} - z*|| ≤ c||z^{(i)} - z*||^2 from this requires a quadratic lower bound E(z) - E(z*) ≥ m||z - z*||^2 near z*. Assumption 6 states only 'locally strictly convex', which does not imply quadratic growth: E(z) = z^4 on R is locally strictly convex at 0 but has no quadratic lower bound there. The final step of the proof implicitly treats the local Lipschitz upper bound on E as an equivalence with distance, which is invalid without strong convexity. The Finsler extension in Appendix A.2 inherits the same gap by being 'completely similar' and also omits any discussion of α* = 1. The empirical quadratic-looking behavior in Fig. 3 does not repair the proof gap, since those experiments may lie in regimes where G(x) is nearly constant and the step is effectively Newton's method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":42769,"tokens_out":5869,"duration_ms":68618,"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":[{"comment":"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":"Section 3.2, Proposition 8 and Eqs. (15)-(16)"},{"comment":"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":"Section 3.2, Proposition 8 and Algorithm 1"},{"comment":"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.","section":"Section 3.2, Proposition 5"},{"comment":"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.","section":"Appendix A.2"}],"minor_comments":[{"comment":"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":"Appendix B.2, Algorithms 3 and 4"},{"comment":"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":"Section 2, paragraph on Levi-Civita connection"},{"comment":"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.","section":"Section 5 and Tables 2-7"},{"comment":"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.","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The paper's numerical contribution appears solid and the code availability is a plus, but the two central theoretical claims (global convergence and quadratic local convergence) are not yet proven in the submitted manuscript. I would encourage a revision that either supplies the missing arguments and assumptions or carefully restates the claims at the level that is actually proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.","headline":"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.","tokens_in":43312,"tokens_out":1159,"would_cite":true,"duration_ms":14942,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C22","49Q99","65D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["geodesics","Riemannian manifolds","Finsler manifolds","optimal control","discrete energy minimization","global convergence","quadratic convergence","information geometry"],"falsifier":"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.","tokens_in":42209,"feed_emoji":"📐","tokens_out":9708,"duration_ms":92589,"temperature":0.7,"pith_summary":"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.","feed_headline":"New geodesic algorithm matches Newton's speed minus the Hessians","feed_subtitle":"A control-problem reformulation computes geodesics far faster than ADAM, BFGS, and sparse Newton in benchmarks.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the time-discrete Pontryagin maximum principle used to derive the first-order necessary conditions that GEORCE solves in closed form.","marker":"[47]"},{"why":"Provides the Armijo backtracking condition used by the soft line search, the mechanism that forces the energy to decrease in each GEORCE iteration.","marker":"[50]"},{"why":"States the standard convergence facts (gradient descent is global but linear, Newton is local but quadratic) against which GEORCE's convergence claims are positioned.","marker":"[36]"},{"why":"Defines the ADAM optimizer used as a primary baseline in the benchmark tables.","marker":"[40]"},{"why":"Defines the BFGS quasi-Newton method used as the main second baseline in the benchmarks.","marker":"[32–35]"},{"why":"Supplies the Randers/Finsler metric construction (microswimmer under a force field) used for all Finsler manifold benchmarks.","marker":"[51]"},{"why":"Provides the variational autoencoder whose decoder pull-back metric defines the learned-manifold benchmark.","marker":"[52]"},{"why":"Provides the closed-form Fisher-Rao metric matrices used for the information-geometry benchmarks.","marker":"[30]"},{"why":"Establishes the discretized energy minimization formulation on latent spaces that GEORCE is compared against.","marker":"[11, 12]"}],"fun_headline_variants":["GEORCE: control trick yields quadratic geodesic convergence","Geodesic algorithm beats ADAM, BFGS with global convergence","GEORCE extends to Finsler manifolds with no Hessian overhead","Fast geodesic computation via discrete control reformulation","GEORCE: global and quadratic convergence for geodesics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["GEORCE: control trick yields quadratic geodesic convergence","Geodesic algorithm beats ADAM, BFGS with global convergence","GEORCE extends to Finsler manifolds with no Hessian overhead","Fast geodesic computation via discrete control reformulation","GEORCE: global and quadratic convergence for geodesics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1323,"prompt_tokens":875,"completion_tokens":448,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":363}},"tokens_in":491,"tokens_out":448,"duration_ms":4140,"temperature":1.0,"reasoning_tokens":363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:52:28.071587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"URL https://books.google.dk/ books?id=fCh2SAtWIdwC","cited_arxiv_id":null,"evidence_quote":"Supplies the time-discrete Pontryagin maximum principle used to derive the first-order necessary conditions that GEORCE solves in closed form."},{"cited_title":"Minimization of functions having Lipschitz continuous first partial derivatives.Pacific Journal of Mathematics16, 1–3 (1966)","cited_arxiv_id":null,"evidence_quote":"Provides the Armijo backtracking condition used by the soft line search, the mechanism that forces the energy to decrease in each GEORCE iteration."},{"cited_title":"& Ye, Y.Linear and Nonlinear ProgrammingInternational Series in Operations Research & Management Science (Springer US, 2008)","cited_arxiv_id":null,"evidence_quote":"States the standard convergence facts (gradient descent is global but linear, Newton is local but quadratic) against which GEORCE's convergence claims are positioned."},{"cited_title":"& Golestanian, R","cited_arxiv_id":null,"evidence_quote":"Supplies the Randers/Finsler metric construction (microswimmer under a force field) used for all Finsler manifold benchmarks."},{"cited_title":"K., Meneghetti, F","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form Fisher-Rao metric matrices used for the information-geometry benchmarks."}],"review_version":1}