{"id":"d4c9aa6c-39fe-4767-a0bf-46ff3930d19d","arxiv_id":"2508.19790","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"APT* combines adaptive batch sizing with elliptical nearest-neighbor regions defined by a Coulomb-like force to speed up sampling-based path planning.","lead":"APT* is a sampling-based motion planner that adapts its batch size and shapes its search region based on the current solution cost. The paper reports faster initial solution times and lower final path costs than several existing planners in R4 to R16 and on a real robot.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Batch size mmin=1 makes Eq. 2's radius r=0, so the graph stops adding edges; the asymptotic optimality claim is unsupported and likely false.","rationale":"The reader's weakest assumption targeted the elliptical region, but that region is a superset of the standard r-ball (since d1>r and d2..n=r), so connectivity is not reduced. The true failure is the adaptive batch-size schedule: it can (and in the limit will) set B=1, and Eq. 2 yields r=0 because log(1)=0. With no neighbors, new samples cannot join the graph, so convergence to the optimum cannot continue. This is a concrete algebraic consequence of the paper's own equations, not merely a missing proof. The proposed test (track edges per sample as B approaches 1) would settle the issue empirically; however, the analytical computation r(1)=0 is already conclusive. Since the reader's verdict REJECT is consistent with this concern, no change to the verdict is needed.","tokens_in":13875,"tokens_out":9382,"duration_ms":92444,"concrete_test":"Run APT* in an obstacle-free hypercube in R^4 with a long time limit (e.g., 100s) and log Badapt and the number of edges added per new sample over time. If Badapt reaches 1 (mmin), Eq. 2 predicts r=0; verify that the edges-per-sample rate drops to zero and the solution cost plateaus. Alternatively, analytically compute r(mmin) from Eq. 2: since mmin=1, r=0, which alone falsifies the asymptotic optimality guarantee.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that APT* is asymptotically optimal is undermined by the adaptive batch-size module itself. Algorithm 2 sets mmin=1, and Eq. 10 drives B(Theta_d) to mmin as Theta_d->0 (Eq. 11), which occurs as the informed-set ratio Gi shrinks with cost improvement (Eq. 13). For B=1, the connection radius in Eq. 2 becomes r(1)=2*eta*(1+1/n)*(lambda(X_fhat)/lambda(UB,n) * log(1)/1)^{1/n} = 0 because log(1)=0. With r=0, the elliptical neighbor region in Eq. 9 has zero volume, so any new sample has no neighbors, and no edges are added or rewired. The graph therefore stops growing after this point, and the solution cost cannot converge to c*. Standard sufficient conditions for asymptotic optimality require the connection radius to scale as gamma*(log N/N)^{1/n} for cumulative sample count N; using the shrinking batch size B instead of N violates this condition once B is small. The paper provides no theorem showing this schedule preserves AO, and the formula directly contradicts it. Note that the elliptical region is a superset of the ball of radius r(B) (since d1>r and d2..n=r), so the anisotropic shape itself does not hurt connectivity; the failure is the zero radius. Thus 'Asymptotically Optimal' is not merely unproven but likely false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"APT* is presented as an asymptotically optimal, anytime single-query motion planner extending FDIT*. It introduces two coupled modules: an adaptive batch-size module (Algorithm 2) that sets the number of samples per batch from the shrinking hypervolume of an informed set, and an elliptical r-nearest-neighbor module (Algorithm 1) in which vertices carry a \"charge\" determined by a tanh-based nonlinear function of the batch size, generating Coulomb-like forces that stretch the neighbor ellipsoid. The paper claims superior initial-solution time and convergence cost relative to RRT-Connect, Informed RRT*, BIT*, AIT*, EIT*, FIT*, and FDIT* in R4, R8, R14, and R16 benchmarks, plus two real-world manipulation tasks. The central advertised property is asymptotic optimality, repeated in the title, abstract, and conclusion.","tokens_in":14261,"tokens_out":4463,"duration_ms":53644,"significance":"If the theoretical claim were established and the empirical results were as reported, APT* would be a useful contribution to high-dimensional sampling-based planning: the adaptive batch scheduler tied to informed-set hypervolume and the force-directed elliptical neighbor region are plausible heuristics for accelerating convergence, and the benchmark suite is broad, including simulated and physical robot tasks. The paper also links to a repository with an OMPL implementation. However, the asymptotic-optimality claim is the headline contribution, and it is neither proved nor, on inspection, consistent with the algorithm's own formulas. The empirical evaluation also contains reporting inconsistencies. The useful heuristic content does not rescue the central claim as written.","major_comments":[{"comment":"The connection radius in Eq. (2) is r(B) = 2η(1+1/n)[λ(X_fhat)/λ(UB,n) · log(B)/B]^{1/n}. Algorithm 2 sets mmin=1, and Eq. (10) gives B(Θd)=floor(1+Θd(mmax−1)), so B=1 whenever Θd is sufficiently small. Eq. (11) makes Θd→0 as the informed ratio Gi→0 (Eq. (13)), which is exactly the intended optimization phase. For B=1, log(1)=0, hence r=0. In the elliptical region of Eq. (9), the semi-axes are d1=r(1+k||F_D||) and d2..n=r, so with r=0 the region has zero volume: no edges are added or rewired, and the graph cannot continue to converge to c*. Standard sufficient conditions for asymptotic optimality, such as those in [11] and [20], require the connection radius to grow with the cumulative sample count N; using the shrinking batch size B alone does not satisfy this. No theorem is supplied to show that the schedule preserves asymptotic optimality. Thus the title's central claim is not merely","section":"§III-A, Eq. (2); Algorithm 2; Eq. (10)"},{"comment":"The paper claims asymptotic optimality but contains no formal statement, proof, or even a proof sketch. The modifications to the nearest-neighbor graph are substantial: the neighbor set is an anisotropic ellipsoid aligned with a force vector (Eq. (9)), and the batch size—and hence the connection radius—is a function of the current solution cost through Eqs. (10)–(14). No argument is given that the resulting graph remains connected almost surely or that the rewiring preserves the conditions of Karaman and Frazzoli [11]. Citing RGG theory [4, 20] does not substitute for verifying that the proposed architecture satisfies those sufficient conditions. For a paper whose title and abstract advertise asymptotic optimality, this missing analysis is load-bearing.","section":"§III (entire), esp. §III-B and §III-C"}],"minor_comments":[{"comment":"The PDT report embedded in Fig. 6 states that 10 runs were executed for each planner, while the text and Table III say 100 runs. This factor-of-ten discrepancy should be corrected and the statistical claims (e.g., 99% confidence intervals) recomputed with the actual number of runs.","section":"Fig. 6 vs. §IV-A"},{"comment":"The Coulomb force formula in Eq. (3) uses r_i^{n−1} with unit vector, while Eqs. (4) and (5) use ||x−x_valid,i||^n times the displacement vector. These are dimensionally inconsistent as written; the intended normalization should be clarified.","section":"Eq. (4)–(6)"},{"comment":"The Taylor expansion formula is hard to parse: the notation B2i is overloaded (Bernoulli numbers vs. batch size B), the placement of parentheses in the summand is ambiguous, and the role of the expansion order α beyond a heuristic trade-off is not explained. A cleaner statement with a defined argument domain would improve reproducibility.","section":"Eq. (15)"},{"comment":"Algorithm 2 uses informal constructs such as \"pragma once\", \"calTuningParam\", and \"isCostBetter\" that are not defined in the pseudocode or the text. Algorithm 3 fixes qmin=0.1, qmax=1.9, ε=6, β=−0.5, and α is chosen from Table I; these are empirical choices made on one problem, and the sensitivity of the claimed performance to them is not reported.","section":"Algorithm 2 and Algorithm 3"},{"comment":"The selection of the tanh order α and the charge bounds is based on a single comparison in Table I. No held-out validation or ablation across the benchmark set is provided, so the claimed superiority of APT*-T should be viewed as tuned performance rather than a robust algorithmic property.","section":"§III-C, Table I"}],"recommendation":"reject","confidential_remarks":"The central advertised contribution is asymptotic optimality, but the adaptive batch-size mechanism drives the connection radius to zero (B=1 in Eq. (2)), which is a direct contradiction of the claim. This is not a gap that a local proof or a tightened analysis can fix: the algorithm as specified stops adding edges in the regime it is designed to reach. Even if the authors removed the asymptotic-optimality claim and repositioned the paper as a purely empirical heuristic study, the empirical reporting has a 10-vs-100 runs inconsistency and the hyperparameters are selected on the same benchmark. I therefore recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends the authors' own FDIT* with an adaptive batch-size module borrowed from FIT* and a tanh-based charge schedule. The empirical work is real: code is on GitHub, benchmarks span R4 to R16, a dual-WAM cage task, and a real DARKO robot experiment. On the evidence, APT* finds initial solutions faster than FIT* and FDIT* in most scenarios, with comparable or better final costs. That is a useful engineering contribution.\n\nThe problem is the title. \"Asymptotically Optimal\" is asserted, not proved. There is no theorem showing the modified neighbor region and adaptive batch schedule preserve the RGG connectivity and rewiring conditions needed for AO. The stress-test note nails the concrete failure mode: Eq. (2) gives r(B)=0 when B=1, and Algorithm 2 drives B toward mmin=1 as the informed-set ratio Gi shrinks. With r=0 the elliptical region has zero volume, no edges are added, and the graph stops improving. Even if B never hits exactly 1, using the current batch size instead of the cumulative sample count breaks the standard (log N/N)^{1/n} scaling required for AO. So the central claim is not merely unproven; it is likely false as stated.\n\nThere is also a smaller issue: the five prolation methods are compared on a single problem in Table I, and the tanh order α and charge bounds are tuned on that table. That is parameter selection on the test set, which weakens the comparison. The self-citation pattern is heavy but not disqualifying; the cited work is genuinely prior and relevant.\n\nWho is this for? Readers who want a faster heuristic in the FIT*/FDIT* family. If the AO claim were dropped and the paper reframed as an empirical heuristic extension, it would be a decent RA-L paper. As it stands, the mismatch between claim and evidence is too large.\n\nIf this crossed my desk, I would send it to reviewers — the empirical part deserves scrutiny — but I would expect them to reject or demand major revision, removing or seriously qualifying the AO claim.","headline":"APT* is a solid empirical extension of FIT*/FDIT* with good benchmark results, but the \"Asymptotically Optimal\" claim is unsupported and likely false because the batch-dependent connection radius collapses to zero.","tokens_in":14750,"tokens_out":3393,"would_cite":false,"duration_ms":39401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a planner which deforms nearest-neighbor search into a force-aligned ellipsoid and adapts batch size to the informed-set volume finds initial and converged solutions faster than prior single-query planners in dimensio","keywords":["motion planning","sampling-based planning","asymptotic optimality","elliptical r-nearest neighbors","adaptive batch size","Coulomb force","informed sampling","manipulation planning"],"falsifier":"On a problem with a known optimal cost, log the adapted batch size and the total number of samples as computation time increases. If Badapt ever reaches mmin=1 and remains there while time grows, the sample count stops growing and cost will plateau above the optimum; that observation would falsify the asymptotic-optimality claim. Equivalently, a formal check: compute whether the batch sizes B(Θd) produced by Eqs. 10–14 diverge to infinity over repeated cost improvements.","tokens_in":13799,"feed_emoji":"🤖","tokens_out":7537,"duration_ms":86113,"temperature":0.7,"pith_summary":"The paper proposes APT*, a sampling-based motion planner that changes the shape of the nearest-neighbor search region as planning progresses. Instead of always looking in an isotropic ball around each vertex, it treats sampled vertices as electric charges, computes a virtual Coulomb force from valid and invalid neighbors, and stretches the search region into an ellipsoid aligned with that force. It also replaces fixed batch sizes with batch sizes tied to the shrinking hypervolume of the informed set, so early search uses large batches and nearly round regions while later refinement uses small batches and elongated regions. The claim is that this feedback loop finds first paths sooner and converges to low-cost solutions faster than existing single-query anytime planners. The evidence is benchmark comparisons from R4 to R16 and two real dual-arm manipulation tasks.","feed_headline":"Force-shaped search finds robot paths faster in 4D to 16D","feed_subtitle":"APT* adapts elliptical neighbor regions and batch sizes, beating seven prior planners on cost, speed, and success.","key_machinery":"Two coupled modules carry the argument. First, elliptical r-nearest neighbors: Coulomb's law (Eq. 6) gives a force vector, and Eq. 9 defines a hyperellipsoid neighborhood centered at the vertex, with the major axis along the force and the remaining semi-axes equal to the original radius. Second, adaptive batch sizing: Eq. 10 sets samples per batch from a decay factor, Eq. 13 measures how much the informed-set hypervolume has contracted, and Eq. 15 converts batch size into a vertex charge via the Tanh/Taylor expansion. Higher charge means stronger force, hence a more elongated neighborhood. The claimed effect is that this pair of mechanisms balances exploration and refinement better than fixe","core_discovery":"APT* assigns every sampled vertex a charge. Valid neighbors attract, invalid (in-collision) neighbors repel, and the vector sum produces a net Coulomb force F. The usual isotropic nearest-neighbor ball is stretched along F, producing an elliptical region (Eq. 9) whose major axis points in the force direction. The charge—and therefore the eccentricity—is not fixed: it is scheduled by a Tanh/Taylor function of an adaptive batch size, which shrinks as the hypervolume of the informed set contracts with improving solution cost (Eqs. 10–14). The paper's claim is that this coupling lets APT* find a first feasible path quickly (large batch, small charge, nearly spherical region) and then refine it e","pith_inferences":["Because the deformed region always contains the standard r-ball, the connectivity side of asymptotic optimality may survive the ellipsoidal change; the unproven part is the batch schedule, which must supply infinitely many samples over time. A formal analysis of B(Θd)'s growth, or a counterexample where it stalls at mmin, would settle whether the 'asymptotically optimal' in the title is earned.","The Coulomb force is computed from valid and invalid neighbor locations, so it functions as an implicit local clearance heuristic. One could test whether the force direction aligns with the gradient of a distance-to-obstacle field; if so, the prolation is doing obstacle-aware exploration in a way the paper does not make explicit.","The Tanh/Taylor charge schedule is one smooth interpolation among many; a natural ablation would replace the charge function with a linear or exponential scheduler matched to the same informed-ratio signal and compare convergence, which would reveal how much of the gain comes from the specific nonlinearity rather than from the feedback loop itself.","The same feedback idea—shaping the search region by informed-set volume—could transfer to bidirectional planners or k-nearest variants, where the force direction from two growing trees may be even more informative; the paper does not explore this."],"forward_implications":["Planners can replace an isotropic r-nearest-neighbor ball with an ellipsoid whose major axis points along a virtual force and still improve anytime convergence; the ellipsoid contains the original ball, so the baseline connectivity radius is not reduced.","Batch size no longer needs a manually tuned schedule; the informed-set hypervolume ratio provides a continuous signal that shifts from large exploratory batches to small refinement batches.","The reported speedups in median initial-solution time grow with dimensionality (roughly 11–34% over FIT*/FDIT* across R4–R16, and 45.54% over FIT* in the R14 cage task), suggesting the prolation effect matters most when sampling is sparse.","Success rates on constrained real-world manipulation tasks improve as well: the paper reports 93.3% versus 76.7–80% for kitchen tool retrieval and 83.3% versus 60–66.7% for shelf insertion."],"supporting_citations":[{"why":"The FDIT* planner that APT* extends; supplies the Coulomb-force ellipsoidal neighbor concept and the baseline it must beat.","marker":"[13]"},{"why":"Supplies the informed-tree search structure and the r(B) nearest-neighbor radius formula that APT* adapts.","marker":"[17]"},{"why":"Introduces the adaptive batch-size idea in informed sampling, which APT* couples to the ellipsoidal neighbor region.","marker":"[18]"},{"why":"States the sufficient conditions for asymptotic optimality that the paper invokes when calling APT* asymptotically optimal.","marker":"[11]"},{"why":"Defines informed sampling and hyperellipsoid informed sets whose Lebesgue measure drives the batch-size module.","marker":"[15]"},{"why":"Provides the batch-based implicit RGG search paradigm that the adaptive batch module builds on.","marker":"[16]"},{"why":"Introduces the informed-set ellipsoidal heuristic used to focus sampling along the current solution.","marker":"[14]"},{"why":"Supplies the random geometric graph theory underlying the r-nearest-neighbor radius and asymptotic optimality claims.","marker":"[4]"}],"fun_headline_variants":["Coulomb-inspired forces accelerate robot path planning in high dimensions","APT* planner uses electric-charge-like forces to find better paths faster","Adaptive elliptical search speeds up motion planning from 4D to 16D","Robot planner bends neighbor search into ellipses via virtual charges"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the adaptive batch-size schedule still produces the growing sample density that asymptotic-optimality proofs require, and that the ellipsoidal neighbor selection preserves the graph connectivity and rewiring conditions; the paper asserts both but provides no theorem for them.","fun_headline_variants_meta":{"raw":{"variants":["Coulomb-inspired forces accelerate robot path planning in high dimensions","APT* planner uses electric-charge-like forces to find better paths faster","Adaptive elliptical search speeds up motion planning from 4D to 16D","Robot planner bends neighbor search into ellipses via virtual charges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00062,"raw_usage":{"total_tokens":2724,"prompt_tokens":765,"completion_tokens":1959,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":1884}},"tokens_in":509,"tokens_out":1959,"duration_ms":14226,"temperature":1.0,"reasoning_tokens":1884,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:27:14.594000+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a problem with a known optimal cost, log the adapted batch size and the total number of samples as computation time increases. If Badapt ever reaches mmin=1 and remains there while time grows, the sample count stops growing and cost will plateau above the optimum; that observation would falsify the asymptotic-optimality claim. Equivalently, a formal check: compute whether the batch sizes B(Θd) produced by Eqs. 10–14 diverge to infinity over repeated cost improvements.","supporting_citations":[],"review_version":1}