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Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds

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Pith's one-line read A matrix-valued heuristic from the Loewner order on positive definite matrices enables direct, rejection-free informed sampling under Riemannian metrics.

desk verdict The paper's main contribution is a Loewner-order matrix lower bound on the metric tensor that keeps directional structure and reduces Riemannian informed-set sampling to standard Euclidean hyperspheroid routines. read the letter →

arxiv 2606.02879 v1 pith:UQMTU6TD submitted 2026-06-01 cs.RO

Phone Thiha Kyaw , Jonathan Kelly This is my paper · ORCID
classification cs.RO
keywords informedsamplingRiemannianmanifoldsmotionplanningLoewnerorderadmissibleheuristicssampling-basedplannersrobotmanipulators
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

The paper establishes a way to keep informed sampling admissible when the underlying metric on configuration space varies with position and direction. It computes the tightest constant matrix lower bound that respects the Loewner order, then uses the Cholesky factor of that bound to transform the problem into ordinary Euclidean space. In that space the informed set becomes a standard prolate hyperspheroid that existing sampling routines can draw from without rejection. Readers working on robot motion planning care because the resulting search regions are smaller than those produced by scalar bounds, which in turn speeds up convergence of optimal planners on real manipulators.

What carries the argument

The matrix-valued admissible heuristic derived from the Loewner order on symmetric positive definite matrices, whose Cholesky factorization supplies the linear map to isotropic Euclidean space.

What would settle it

A sampled point whose true Riemannian distance to the goal exceeds the current best solution cost, or a configuration where the proposed matrix bound exceeds the actual metric tensor.

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Extended reading notes

Core claim

The Cholesky factorization of the tightest constant lower bound on the metric tensor, obtained via the Loewner order, defines a linear map to an isotropic Euclidean space in which the Riemannian informed set reduces to a standard prolate hyperspheroid, allowing direct use of existing Euclidean informed-sampling algorithms while remaining admissible.

Load-bearing premise

A single position-independent matrix lower bound obtained from the Loewner order stays admissible for the true Riemannian distance at every point and its Cholesky factor produces a linear map that correctly transforms the informed-set geometry.

Editorial extensions

If this is right

  • Informed sets remain admissible yet are strictly smaller than those obtained from Euclidean distance or scalar eigenvalue bounds.
  • Sampling inside the informed set requires no rejection step and reuses existing Euclidean algorithms.
  • Convergence of asymptotically optimal planners improves on 6-DoF to 14-DoF manipulators under multiple Riemannian metrics.
  • Directional structure of the metric is retained rather than collapsed to a scalar.

Reading between the lines

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

  • The same Loewner-order construction could be applied to time-varying or learned metrics during online replanning.
  • The linear-map reduction suggests that other manifold sampling tasks outside motion planning might benefit from analogous constant matrix bounds.
  • Parallel sampling in the transformed Euclidean space becomes straightforward once the map is computed.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper claims to introduce a matrix-valued admissible heuristic for informed sampling on Riemannian manifolds by using the Loewner order on SPD matrices to compute the tightest constant lower bound on the metric tensor while preserving its directional structure. The Cholesky factorization of this bound induces a linear map to an isotropic Euclidean space, reducing the Riemannian informed set to a standard prolate hyperspheroid that permits direct, rejection-free sampling via existing Euclidean algorithms. Experiments on 6-DoF UR5, 7-DoF Franka, and 14-DoF PR2 manipulators under three Riemannian metrics demonstrate consistently tighter informed sets and accelerated convergence relative to Euclidean and scalar-eigenvalue baselines across multiple asymptotically optimal planners.

Significance. If the central derivation holds, the result supplies an admissible, direction-preserving heuristic that avoids the conservatism of scalar bounds while remaining parameter-free and directly compatible with existing sampling routines. This is a targeted improvement for sampling-based planning on configuration-dependent metrics, a setting common in robotics, and the approach rests on standard Loewner-order properties without introducing ad-hoc parameters or self-referential fitting.

minor comments (2)
  1. The abstract states that the bound is the 'tightest constant lower bound' but does not indicate in which section the explicit construction or optimization procedure for obtaining this bound from a given g(x) is presented; a short algorithmic outline or pseudocode would clarify reproducibility.
  2. The experimental section reports improvement on three robots and three metrics, yet the precise definitions of those metrics (e.g., the functional form of g(x)) are not referenced in the abstract; adding a brief table or equation pointer would strengthen the claim of generality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee summary accurately captures the central contribution regarding the Loewner-order heuristic and its reduction to prolate hyperspheroid sampling. No major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The derivation relies on the standard mathematical property that any constant matrix M satisfying M ≼ g(x) for all x (in the Loewner order) produces an admissible heuristic d_M ≤ d_g for the Riemannian distance, which follows directly from the definition of path length and infimum over curves. The Cholesky factor of M then supplies an explicit linear isometry to Euclidean space, reducing the informed set to a prolate spheroid by algebraic construction without fitted parameters, self-citations, or ansatzes. No load-bearing step reduces to its own inputs or prior author work; the central claim is self-contained from first-principles properties of SPD matrices and the Loewner order.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claim rests on standard properties of the Loewner order for symmetric positive definite matrices and the assumption that the resulting constant matrix bound remains admissible under the Riemannian metric.

assumptions (1)
  • standard math The Loewner order defines a partial order on symmetric positive definite matrices that admits a greatest lower bound for any set of such matrices.
    Invoked to obtain the tightest constant lower bound on the metric tensor.
invented entities (1)
  • Matrix-valued admissible heuristic via Loewner order
    purpose: To produce a tighter informed set that retains the directional information of the Riemannian metric tensor.
    Newly introduced construction in the paper.

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Cite this review

Pith. "Pith review of Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds." pith.science (2026). https://pith.science/paper/UQMTU6TD

@misc{pith2026260602879,
  author       = {Pith},
  title        = {Pith review of: Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UQMTU6TD}},
  note         = {Machine review of arXiv:2606.02879}
}
read the original abstract

Informed sampling techniques accelerate sampling-based motion planners by focusing the search on promising regions of the state space, yet most existing methods rely on Euclidean heuristics that become inadmissible under configuration-dependent Riemannian metrics. While scalar eigenvalue bounds restore admissibility by uniformly scaling the Euclidean distance, they discard the directional structure of the metric, producing overly conservative informed sets. We propose a matrix-valued admissible heuristic that exploits the Loewner order on symmetric positive definite matrices to compute the tightest constant lower bound on the metric tensor while preserving its full directional structure. The Cholesky factorization of this bound defines a linear map to an isotropic Euclidean space in which the Riemannian informed set reduces to a standard prolate hyperspheroid, enabling direct, rejection-free sampling using existing algorithms. Experiments on manipulation tasks with a 6-DoF UR5, 7-DoF Franka, and 14-DoF PR2 under three distinct Riemannian metrics show that our heuristic produces consistently tighter informed sets than both the Euclidean and scalar eigenvalue bounds, accelerating convergence across multiple state-of-the-art asymptotically optimal planners.

Figures

Figures reproduced from arXiv: 2606.02879 by the authors.

Figure 1
Figure 1. Illustration of the proposed direct informed sampling approach for [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Search trees and informed sets for a 2-DoF planar manipulator planning from start ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Distribution of the heuristic-to-geodesic-distance ratio [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: Anytime planning performance across all nine robot–metric scenarios over 100 trials. Rows correspond to robots (top to bottom: UR5, Franka, PR2) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Works this paper leans on

45 extracted references · 2 canonical work pages

  1. [1]

    Randomized kinodynamic planning,

    S. M. LaValle and J. J. Kuffner Jr, “Randomized kinodynamic planning,” Int. J. Robot. Res., vol. 20, no. 5, pp. 378–400, 2001

    2001

  2. [2]

    Sampling-based algorithms for optimal motion planning,

    S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,”Int. J. Robot. Res., vol. 30, no. 7, pp. 846–894, 2011

    2011

  3. [3]

    Batch Informed Trees (BIT*): Informed asymptotically optimal anytime search,

    J. D. Gammell, T. D. Barfoot, and S. S. Srinivasa, “Batch Informed Trees (BIT*): Informed asymptotically optimal anytime search,”Int. J. Robot. Res., vol. 39, no. 5, pp. 543–567, 2020

    2020

  4. [4]

    Informed sampling for asymptotically optimal path planning,

    ——, “Informed sampling for asymptotically optimal path planning,” IEEE Trans. Robot., vol. 34, no. 4, pp. 966–984, 2018

    2018

  5. [5]

    Bullo and A

    F. Bullo and A. D. Lewis,Geometric control of mechanical systems: modeling, analysis, and design for simple mechanical control systems. Springer, 2019, vol. 49

    2019

  6. [6]

    Riemannian geometry as a unifying theory for robot motion learning and control,

    N. Jaquier and T. Asfour, “Riemannian geometry as a unifying theory for robot motion learning and control,” inProc. Int. Symp. Robot. Res. (ISRR). Springer, 2022, pp. 395–403

    2022

  7. [7]

    A Riemannian take on distance fields and geodesic flows in robotics,

    Y . Li, J. Qiu, and S. Calinon, “A Riemannian take on distance fields and geodesic flows in robotics,”Int. J. Robot. Res., p. 02783649261420233, 2024

    2024

  8. [8]

    Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds

    P. T. Kyaw and J. Kelly, “Geometry-aware sampling-based motion planning on Riemannian manifolds,”arXiv preprint arXiv:2602.00992, 2026

    work page Pith review arXiv 2026

Show all 45 references
  1. [9]

    Geometry-aware manipulability learning, tracking, and transfer,

    N. Jaquier, L. Rozo, D. G. Caldwell, and S. Calinon, “Geometry-aware manipulability learning, tracking, and transfer,”Int. J. Robot. Res., vol. 40, no. 2-3, pp. 624–650, 2021

    2021

  2. [10]

    Learning stable robotic skills on Riemannian manifolds,

    M. Saveriano, F. J. Abu-Dakka, and V . Kyrki, “Learning stable robotic skills on Riemannian manifolds,”Robot. Auton. Syst., vol. 169, p. 104510, 2023

    2023

  3. [11]

    A Rie- mannian metric for geometry-aware singularity avoidance by articulated robots,

    F. Mari´c, L. Petrovi ´c, M. Guberina, J. Kelly, and I. Petrovi ´c, “A Rie- mannian metric for geometry-aware singularity avoidance by articulated robots,”Robot. Auton. Syst., vol. 145, p. 103865, 2021

    2021

  4. [12]

    Geodesic methods for shape and surface processing,

    G. Peyr ´e and L. D. Cohen, “Geodesic methods for shape and surface processing,”Adv. Comput. Vis. Med. Image Process., pp. 29–56, 2009

    2009

  5. [13]

    Adaptively Informed Trees (AIT*) and Effort Informed Trees (EIT*): Asymmetric bidirectional sampling-based path planning,

    M. P. Strub and J. D. Gammell, “Adaptively Informed Trees (AIT*) and Effort Informed Trees (EIT*): Asymmetric bidirectional sampling-based path planning,”Int. J. Robot. Res., vol. 41, no. 4, pp. 390–417, 2022

    2022

  6. [14]

    Greedy heuristics for sampling-based motion planning in high-dimensional state spaces,

    P. T. Kyaw, A. V . Le, R. E. Mohan, and J. Kelly, “Greedy heuristics for sampling-based motion planning in high-dimensional state spaces,” arXiv preprint arXiv:2405.03411, 2024

    2024

  7. [15]

    Off the beaten track: Laterally weighted motion planning for local obstacle avoidance,

    J. Sehn, T. D. Barfoot, and J. Collier, “Off the beaten track: Laterally weighted motion planning for local obstacle avoidance,”IEEE Trans. Field Robot., vol. 1, pp. 249–275, 2024. Fig. 5: Anytime planning performance across all nine robot–metric scenarios over 100 trials. Row...

    2024

  8. [16]

    Gener- alizing informed sampling for asymptotically-optimal sampling-based kinodynamic planning via markov chain monte carlo,

    D. Yi, R. Thakker, C. Gulino, O. Salzman, and S. Srinivasa, “Gener- alizing informed sampling for asymptotically-optimal sampling-based kinodynamic planning via markov chain monte carlo,” inProc. IEEE Int. Conf. Robot. Autom. (ICRA), 2018, pp. 7063–7070

    2018

  9. [17]

    Hierarchical rejection sampling for informed kinodynamic planning in high-dimensional spaces,

    T. Kunz, A. Thomaz, and H. Christensen, “Hierarchical rejection sampling for informed kinodynamic planning in high-dimensional spaces,” inProc. IEEE Int. Conf. Robot. Autom. (ICRA), 2016, pp. 89–96

    2016

  10. [18]

    Asymptotically optimal sampling-based motion planning methods,

    J. D. Gammell and M. P. Strub, “Asymptotically optimal sampling-based motion planning methods,”Annu. Rev. Control Robot. Auton. Syst., vol. 4, no. 1, pp. 295–318, 2021

    2021

  11. [19]

    Verification and synthesis of admissible heuristics for kinodynamic motion planning,

    B. Paden, V . Varricchio, and E. Frazzoli, “Verification and synthesis of admissible heuristics for kinodynamic motion planning,”IEEE Robot. Autom. Lett., vol. 2, no. 2, pp. 648–655, 2017

    2017

  12. [20]

    Asymptotically optimal A* for kinodynamic planning,

    M. Przybylski, “Asymptotically optimal A* for kinodynamic planning,” IEEE Robot. Autom. Lett., vol. 9, no. 5, pp. 4353–4360, 2024

    2024

  13. [21]

    An admissible heuristic to improve convergence in kinodynamic planners using motion primitives,

    B. Sakcak, L. Bascetta, G. Ferretti, and M. Prandini, “An admissible heuristic to improve convergence in kinodynamic planners using motion primitives,”IEEE Control Syst. Lett., vol. 4, no. 1, pp. 175–180, 2019

    2019

  14. [22]

    Search-based motion planning for aggressive flight in SE(3),

    S. Liu, K. Mohta, N. Atanasov, and V . Kumar, “Search-based motion planning for aggressive flight in SE(3),”IEEE Robot. Autom. Lett., vol. 3, no. 3, pp. 2439–2446, 2018

    2018

  15. [23]

    Fast exact and approximate geodesics on meshes,

    V . Surazhsky, T. Surazhsky, D. Kirsanov, S. J. Gortler, and H. Hoppe, “Fast exact and approximate geodesics on meshes,”ACM Trans. Graph., vol. 24, no. 3, pp. 553–560, 2005

    2005

  16. [24]

    Landmark-based geodesic computation for heuristically driven path planning,

    G. Peyre and L. D. Cohen, “Landmark-based geodesic computation for heuristically driven path planning,” inProc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR), vol. 2, 2006, pp. 2229–2236

    2006

  17. [25]

    Landmark guided probabilistic roadmap queries,

    B. Paden, Y . Nager, and E. Frazzoli, “Landmark guided probabilistic roadmap queries,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), 2017, pp. 4828–4834

    2017

  18. [26]

    Riemannian fast-marching on cartesian grids, using V oronoi’s first reduction of quadratic forms,

    J.-M. Mirebeau, “Riemannian fast-marching on cartesian grids, using V oronoi’s first reduction of quadratic forms,”SIAM J. Numer. Anal., vol. 57, no. 6, pp. 2608–2655, 2019

    2019

  19. [27]

    Non-Euclidean motion planning with graphs of geodesically convex sets,

    T. Cohn, M. Petersen, M. Simchowitz, and R. Tedrake, “Non-Euclidean motion planning with graphs of geodesically convex sets,”Int. J. Robot. Res., vol. 44, no. 10-11, pp. 1840–1862, 2025

    2025

  20. [28]

    Probabilistic motion planning for non-Euclidean and multi-vehicle problems,

    A. Lukyanenko and D. Soudbakhsh, “Probabilistic motion planning for non-Euclidean and multi-vehicle problems,”Robot. Auton. Syst., vol. 168, p. 104487, 2023

    2023

  21. [29]

    S. Boyd, L. El Ghaoui, E. Feron, and V . Balakrishnan,Linear matrix inequalities in system and control theory. SIAM, 1994

    1994

  22. [30]

    Robust belief space planning under intermittent sensing via a maximum eigenvalue-based bound,

    S. D. Bopardikar, B. Englot, A. Speranzon, and J. van den Berg, “Robust belief space planning under intermittent sensing via a maximum eigenvalue-based bound,”Int. J. Robot. Res., vol. 35, no. 13, pp. 1609– 1626, 2016

    2016

  23. [31]

    Belief roadmap search: Advances in optimal and efficient planning under uncertainty,

    T. Shan and B. Englot, “Belief roadmap search: Advances in optimal and efficient planning under uncertainty,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), 2017, pp. 5318–5325

    2017

  24. [32]

    Robust online motion planning via contraction theory and convex optimization,

    S. Singh, A. Majumdar, J.-J. Slotine, and M. Pavone, “Robust online motion planning via contraction theory and convex optimization,” in Proc. IEEE Int. Conf. Robot. Autom. (ICRA), 2017, pp. 5883–5890

    2017

  25. [33]

    STOMP: Stochastic trajectory optimization for motion planning,

    M. Kalakrishnan, S. Chitta, E. Theodorou, P. Pastor, and S. Schaal, “STOMP: Stochastic trajectory optimization for motion planning,” in Proc. IEEE Int. Conf. Robot. Autom. (ICRA), 2011, pp. 4569–4574

    2011

  26. [34]

    CHOMP: Covariant hamiltonian optimization for motion planning,

    M. Zucker, N. Ratliff, A. D. Dragan, M. Pivtoraiko, M. Klingensmith, C. M. Dellin, J. A. Bagnell, and S. S. Srinivasa, “CHOMP: Covariant hamiltonian optimization for motion planning,”Int. J. Robot. Res., vol. 32, no. 9-10, pp. 1164–1193, 2013

    2013

  27. [35]

    Computing large convex regions of obstacle- free space through semidefinite programming,

    R. Deits and R. Tedrake, “Computing large convex regions of obstacle- free space through semidefinite programming,” inProc. Workshop Algorithmic Found. Robot. (WAFR). Springer, 2015, pp. 109–124

    2015

  28. [36]

    On semidefinite relaxations for matrix-weighted state-estimation problems in robotics,

    C. Holmes, F. D ¨umbgen, and T. Barfoot, “On semidefinite relaxations for matrix-weighted state-estimation problems in robotics,”IEEE Trans. Robot., vol. 40, pp. 4805–4824, 2024

    2024

  29. [37]

    J. M. Lee,Introduction to Riemannian Manifolds. Springer, 2018, vol. 2

    2018

  30. [38]

    Bhatia,Positive Definite Matrices

    R. Bhatia,Positive Definite Matrices. Princeton University Press, 2007

    2007

  31. [39]

    Order properties of bounded self-adjoint operators,

    R. V . Kadison, “Order properties of bounded self-adjoint operators,”Proc. Amer. Math. Soc., vol. 2, no. 3, pp. 505–510, 1951

    1951

  32. [40]

    MotionBenchMaker: A tool to generate and benchmark motion planning datasets,

    C. Chamzas, C. Quintero-Pena, Z. Kingston, A. Orthey, D. Rakita, M. Gleicher, M. Toussaint, and L. E. Kavraki, “MotionBenchMaker: A tool to generate and benchmark motion planning datasets,”IEEE Robot. Autom. Lett., vol. 7, no. 2, pp. 882–889, 2022

    2022

  33. [41]

    AORRTC: Almost-surely asymptotically optimal planning with RRT-Connect,

    T. S. Wilson, W. Thomason, Z. Kingston, and J. D. Gammell, “AORRTC: Almost-surely asymptotically optimal planning with RRT-Connect,”IEEE Robot. Autom. Lett., 2025

    2025

  34. [42]

    RRT*-Connect: Faster, asymptotically optimal motion planning,

    S. Klemm, J. Oberl ¨ander, A. Hermann, A. Roennau, T. Schamm, J. M. Zollner, and R. Dillmann, “RRT*-Connect: Faster, asymptotically optimal motion planning,” inProc. IEEE Int. Conf. Robot. Biomimetics (ROBIO), 2015, pp. 1670–1677

    2015

  35. [43]

    The open motion planning library,

    I. A. Sucan, M. Moll, and L. E. Kavraki, “The open motion planning library,”IEEE Robot. Autom. Mag., vol. 19, no. 4, pp. 72–82, 2012

    2012

  36. [44]

    Motions in microseconds via vectorized sampling-based planning,

    W. Thomason, Z. Kingston, and L. E. Kavraki, “Motions in microseconds via vectorized sampling-based planning,” inProc. IEEE Int. Conf. Robot. Autom. (ICRA), 2024, pp. 8749–8756

    2024

  37. [45]

    The Pinocchio C++ library: A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives,

    J. Carpentier, G. Saurel, G. Buondonno, J. Mirabel, F. Lamiraux, O. Stasse, and N. Mansard, “The Pinocchio C++ library: A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives,” inProc. IEEE/SICE Int. Symp. Syst. Integr. (SII), 201...

    2019

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