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arxiv: 2606.20871 · v1 · pith:L4OFJQYFnew · submitted 2026-06-18 · 💻 cs.RO

Geometric Entropy: When Trajectory Diversity Helps and Hurts in Imitation Learning

Qian Luo , Ruizhe Liu , Pei Zhou , Xunzhe Zhou , Yanchao Yang This is my paper

Pith reviewed 2026-06-26 16:55 UTC · model grok-4.3

classification 💻 cs.RO
keywords imitation learningtrajectory diversitygeometric entropyrobot manipulationdemonstration qualityinverted-U relationshipcontact-rich tasks
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The pith

Geometric diversity in demonstrations forms an inverted-U with imitation learning success.

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

The paper introduces Geometric Entropy as a way to measure the intrinsic shape variety of robot trajectories once differences in goal pose and workspace scale are removed. It shows that imitation learning performance rises with added diversity at first because the learner gains robustness, but then falls once the variety makes it unclear which strategy to copy. The location of the performance peak moves toward lower diversity as more data arrives, tasks simplify, or stronger model priors are used. For pretrained vision-language-action models the relationship turns steadily downward instead of curved. The metric therefore lets practitioners check a demonstration set before training to decide whether it sits in the helpful range.

Core claim

Geometric Entropy (H_G) is obtained by aligning each trajectory to a common target frame, removing extrinsic pose and scale variation so that only intrinsic shape diversity remains. Across several imitation-learning architectures and both simulated and real contact-rich manipulation tasks, success rate traces an inverted-U against H_G: moderate geometric diversity improves robustness while excess diversity produces strategy ambiguity that lowers performance. The entropy value that maximizes success decreases as task mastery increases through added data, easier tasks, or stronger priors, and becomes effectively monotonic for a pretrained vision-language-action model.

What carries the argument

Geometric Entropy (H_G), a task-agnostic scalar obtained by target-frame alignment that isolates intrinsic trajectory-shape diversity from extrinsic pose and scale factors.

If this is right

  • Success peaks at intermediate rather than minimal or maximal geometric diversity.
  • The diversity level that maximizes success drops as datasets enlarge or tasks become easier.
  • Pretrained vision-language-action models show steadily falling performance with rising geometric diversity.
  • H_G supplies a fast pre-training check that flags whether a demonstration set lies inside the learnable diversity band.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Dataset collection protocols could be designed to target the intermediate entropy band identified by the metric.
  • The same alignment-plus-entropy procedure might be tested on sequential tasks outside contact-rich manipulation to check whether the inverted-U pattern generalizes.
  • An online version of the metric could be used to decide when to stop adding new demonstrations or to trigger data filtering during training.

Load-bearing premise

Target-frame alignment removes all extrinsic variation so that the remaining quantity truly reflects only intrinsic trajectory-shape diversity and predicts learning outcomes.

What would settle it

A controlled experiment that varies only intrinsic trajectory shape while holding goal poses and scales fixed and finds success rates that are flat or strictly monotonic with diversity instead of inverted-U.

Figures

Figures reproduced from arXiv: 2606.20871 by Pei Zhou, Qian Luo, Ruizhe Liu, Xunzhe Zhou, Yanchao Yang.

Figure 1
Figure 1. Figure 1: Trajectory manifolds under varying approach radius [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: HG convergence on StackCube-v1. Across all (r, k) settings, HG stabilizes within the first 50–100 suc￾cessful trajectories, enabling reliable pre-training audits at practical collection scales. For each (r, k) configuration, we generate datasets of size N ∈ {100, 200, 500, 1000}, segment each episode into transit phases based on gripper events, and compute HG per phase, averaged using Eq. 3. All simulation… view at source ↗
Figure 3
Figure 3. Figure 3: Diffusion Policy success rate vs. Geometric Entropy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Real-robot success rate vs. HG for three tasks. StackCube (left) exhibits a clear Inverted-U, peaking at HG ≈ 3.3. PlacePanda (center) and OpenDrawer (right) show monotonic decline, consistent with high-mastery tasks where additional diversity is interference. Error bars denote variation over three seeds. C. Real-Robot Validation We validate that the mastery–entropy behavior is not an artifact of simulatio… view at source ↗
Figure 4
Figure 4. Figure 4: π0.5 success rate vs. HG on StackCube-v1. The approximately monotonically decreasing curve is consistent with a high-mastery regime where additional geometric di￾versity acts as interference. B. The Mastery–Entropy Principle Synthesizing results across tasks, data scales, and model architectures suggests an organizing principle: the optimal geometric entropy H∗ G is governed by the learner’s task mastery. … view at source ↗
Figure 6
Figure 6. Figure 6: Baseline metric convergence on StackCube-v1. Mean variance, LogDet, and participation ratio converge quickly but collapse distinct (r, k) settings (aliasing/mis￾ordering), e.g., (0, 0), (0.01, 0), and (0.03, 0) are nearly indistinguishable under variance despite increasing HG. kNN entropy shows pronounced sample-size drift in the D=150 descriptor space. marginal spread alone cannot distinguish “many tight … view at source ↗
read the original abstract

We study how trajectory-shape diversity in demonstrations affects imitation learning (IL) performance across models, tasks, and data scales. We introduce Geometric Entropy (H_G), a task-agnostic metric that quantifies the intrinsic diversity of transit trajectories after normalizing away extrinsic variation, such as goal pose and workspace scale, via target-frame alignment. Across multiple IL architectures and both simulated and real-robot contact-rich manipulation tasks, we observe a consistent inverted-U relationship between success and H_G: increasing geometric diversity improves robustness in low-diversity regimes but degrades performance once diversity induces strategy ambiguity. Moreover, the optimal entropy shifts toward lower values as task mastery increases through more data, easier tasks, or stronger priors, and for a pretrained vision-language-action model the trend becomes effectively monotonic decreasing. Practically, H_G enables fast pre-training auditing of demonstration datasets and offers a simple guideline for calibrating demonstrations toward the learnable regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Geometric Entropy (H_G) as a task-agnostic metric quantifying intrinsic trajectory-shape diversity in imitation learning demonstrations after target-frame alignment to normalize extrinsic factors such as goal pose and workspace scale. Across multiple IL architectures and both simulated and real-robot contact-rich manipulation tasks, it reports a consistent inverted-U relationship between H_G and task success: moderate geometric diversity improves robustness while higher diversity induces strategy ambiguity. The optimal H_G shifts lower with increased data, easier tasks, or stronger model priors, becoming monotonic for a pretrained VLA model. H_G is positioned as a practical tool for pre-training dataset auditing and demonstration calibration.

Significance. If the metric isolation holds and the inverted-U is robust, the work supplies an actionable, pre-training heuristic for demonstration selection in robotics IL that could improve performance in contact-rich settings without requiring additional training runs. The cross-architecture and real-robot consistency strengthens the empirical observation, though the result remains observational rather than derived from first principles.

major comments (2)
  1. [Target-frame alignment / H_G definition] Target-frame alignment section: the claim that alignment isolates intrinsic shape diversity (making H_G task-agnostic and predictive) lacks an explicit post-alignment invariance test or ablation (e.g., residual correlation of H_G with goal pose, scale, or contact geometry). This is load-bearing for the inverted-U interpretation, as performance drops at high H_G could instead reflect increasing extrinsic task difficulty.
  2. [Results / Experiments] Experimental reporting: the abstract and results claim consistent inverted-U observations across architectures, data scales, and real/simulated tasks, yet provide no quantitative details on statistical tests, multiple-comparison corrections, or exclusion criteria for trajectories. This weakens confidence that the relationship is not driven by post-hoc analysis choices.
minor comments (2)
  1. [Metric definition] Clarify the precise normalization steps and any free parameters in the H_G formula to allow direct reproduction.
  2. [Results] Add a table or figure explicitly showing the shift in optimal H_G across data regimes, task difficulties, and model strengths.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and are prepared to revise the manuscript accordingly to strengthen the claims.

read point-by-point responses

  1. Referee: [Target-frame alignment / H_G definition] Target-frame alignment section: the claim that alignment isolates intrinsic shape diversity (making H_G task-agnostic and predictive) lacks an explicit post-alignment invariance test or ablation (e.g., residual correlation of H_G with goal pose, scale, or contact geometry). This is load-bearing for the inverted-U interpretation, as performance drops at high H_G could instead reflect increasing extrinsic task difficulty.

    Authors: We agree that an explicit post-alignment invariance test would strengthen the interpretation that H_G isolates intrinsic trajectory-shape diversity. The alignment procedure normalizes goal pose and workspace scale by transforming to a common target frame, but the manuscript does not include a quantitative ablation (such as residual correlations with extrinsic factors). We will add this analysis in the revision, including correlation checks and an ablation on alignment variants, to confirm the metric's task-agnostic property. revision: yes

  2. Referee: [Results / Experiments] Experimental reporting: the abstract and results claim consistent inverted-U observations across architectures, data scales, and real/simulated tasks, yet provide no quantitative details on statistical tests, multiple-comparison corrections, or exclusion criteria for trajectories. This weakens confidence that the relationship is not driven by post-hoc analysis choices.

    Authors: We acknowledge that the current reporting lacks explicit statistical details. While the inverted-U pattern is observed consistently across the reported experiments, we did not include quantitative tests (e.g., significance of quadratic terms) or clarify exclusion criteria. In the revised manuscript we will add appropriate statistical analyses, multiple-comparison corrections where relevant, and explicit statements on trajectory inclusion/exclusion criteria. revision: yes

Circularity Check

0 steps flagged

No circularity; H_G is a fixed normalization metric and the inverted-U is an empirical observation

full rationale

The paper defines Geometric Entropy (H_G) through an explicit target-frame alignment procedure that normalizes pose and scale, then reports an observational inverted-U relationship between H_G and IL success rates across architectures and tasks. No derivation, prediction, or first-principles result reduces to the metric's own inputs by construction. The relationship is presented as data-driven rather than forced by the definition of H_G, and the alignment step is a preprocessing choice whose validity is external to the metric itself. No self-citation chains, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the load-bearing steps. The central claim remains falsifiable via the reported experiments and does not collapse to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the validity of the normalization step that defines H_G and on the assumption that observed success differences are driven by geometric diversity rather than correlated factors.

axioms (1)
  • domain assumption Target-frame alignment removes extrinsic variation (goal pose, workspace scale) without distorting intrinsic trajectory diversity
    Invoked to make H_G task-agnostic and comparable across demonstrations.

pith-pipeline@v0.9.1-grok · 5693 in / 1249 out tokens · 28505 ms · 2026-06-26T16:55:56.704870+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 6 linked inside Pith

  1. [1]

    Rt-1: Robotics transformer for real-world control at scale,

    A. Brohan, N. Brown, J. Carbajal, Y . Chebotar, J. Dabis, C. Finn, K. Gopalakrishnan, K. Hausman, A. Herzog, J. Hsuet al., “Rt-1: Robotics transformer for real-world control at scale,”arXiv preprint arXiv:2212.06817, 2022

    Pith/arXiv arXiv 2022

  2. [2]

    Rt-2: Vision-language-action models transfer web knowledge to robotic control,

    B. Zitkovich, T. Yu, S. Xu, P. Xu, T. Xiao, F. Xia, J. Wu, P. Wohlhart, S. Welker, A. Wahidet al., “Rt-2: Vision-language-action models transfer web knowledge to robotic control,” inConference on Robot Learning. PMLR, 2023, pp. 2165–2183

    2023

  3. [3]

    Open x-embodiment: Robotic learning datasets and rt-x models: Open x- embodiment collaboration 0,

    A. O’Neill, A. Rehman, A. Maddukuri, A. Gupta, A. Padalkar, A. Lee, A. Pooley, A. Gupta, A. Mandlekar, A. Jainet al., “Open x-embodiment: Robotic learning datasets and rt-x models: Open x- embodiment collaboration 0,” in2024 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2024, pp. 6892–6903

    2024

  4. [4]

    Droid: A large-scale in-the-wild robot manipulation dataset,

    A. Khazatsky, K. Pertsch, S. Nair, A. Balakrishna, S. Dasari, S. Karam- cheti, S. Nasiriany, M. K. Srirama, L. Y . Chen, K. Elliset al., “Droid: A large-scale in-the-wild robot manipulation dataset,”arXiv preprint arXiv:2403.12945, 2024

    Pith/arXiv arXiv 2024

  5. [5]

    Octo: An open-source generalist robot policy,

    Octo Model Team, D. Ghosh, H. Walke, K. Pertsch, K. Black, O. Mees, S. Dasari, J. Hejna, C. Xu, J. Luo, T. Kreiman, Y . Tan, L. Y . Chen, P. Sanketi, Q. Vuong, T. Xiao, D. Sadigh, C. Finn, and S. Levine, “Octo: An open-source generalist robot policy,” inProceedings of Robotics: Science and Systems, Delft, Netherlands, 2024

    2024

  6. [6]

    Openvla: An open-source vision-language-action model,

    M. Kim, K. Pertsch, S. Karamcheti, T. Xiao, A. Balakrishna, S. Nair, R. Rafailov, E. Foster, G. Lam, P. Sanketi, Q. Vuong, T. Kollar, B. Burchfiel, R. Tedrake, D. Sadigh, S. Levine, P. Liang, and C. Finn, “Openvla: An open-source vision-language-action model,” arXiv preprint arXiv:2406.09246, 2024

    Pith/arXiv arXiv 2024

  7. [7]

    Bc-z: Zero-shot task generalization with robotic imitation learning,

    E. Jang, A. Irpan, M. Khansari, D. Kappler, F. Ebert, C. Lynch, S. Levine, and C. Finn, “Bc-z: Zero-shot task generalization with robotic imitation learning,” inconference on Robot Learning. PMLR, 2022, pp. 991–1002

    2022

  8. [8]

    Robocat: A self- improving generalist agent for robotic manipulation,

    K. Bousmalis, G. Vezzani, D. Rao, C. Devin, A. X. Lee, M. Bauz ´a, T. Davchev, Y . Zhou, A. Gupta, A. Rajuet al., “Robocat: A self- improving generalist agent for robotic manipulation,”arXiv preprint arXiv:2306.11706, 2023

    arXiv 2023

  9. [9]

    Palm-e: an embodied multimodal language model,

    D. Driess, F. Xia, M. S. Sajjadi, C. Lynch, A. Chowdhery, B. Ichter, A. Wahid, J. Tompson, Q. Vuong, T. Yuet al., “Palm-e: an embodied multimodal language model,” inProceedings of the 40th International Conference on Machine Learning, 2023, pp. 8469–8488

    2023

  10. [10]

    Vima: General robot manipulation with multimodal prompts,

    Y . Jiang, A. Gupta, Z. Zhang, G. Wang, Y . Dou, Y . Chen, L. Fei- Fei, A. Anandkumar, Y . Zhu, and L. Fan, “Vima: General robot manipulation with multimodal prompts,” inFortieth International Conference on Machine Learning, 2023

    2023

  11. [11]

    Aw-opt: Learning robotic skills with imitation andreinforcement at scale,

    Y . Lu, K. Hausman, Y . Chebotar, M. Yan, E. Jang, A. Herzog, T. Xiao, A. Irpan, M. Khansari, D. Kalashnikovet al., “Aw-opt: Learning robotic skills with imitation andreinforcement at scale,” inConference on Robot Learning. PMLR, 2022, pp. 1078–1088

    2022

  12. [12]

    Data quality in imitation learning,

    S. Belkhale, Y . Cui, and D. Sadigh, “Data quality in imitation learning,”Advances in neural information processing systems, vol. 36, pp. 80 375–80 395, 2023

    2023

  13. [13]

    A reduction of imitation learning and structured prediction to no-regret online learning,

    S. Ross, G. Gordon, and D. Bagnell, “A reduction of imitation learning and structured prediction to no-regret online learning,” inProceedings of the fourteenth international conference on artificial intelligence and statistics. JMLR Workshop and Conference Proceedings, 2011, pp. 627–635

    2011

  14. [14]

    Dart: Noise injection for robust imitation learning,

    M. Laskey, J. Lee, R. Fox, A. Dragan, and K. Goldberg, “Dart: Noise injection for robust imitation learning,” inConference on robot learning. PMLR, 2017, pp. 143–156

    2017

  15. [15]

    What matters in learning from offline human demonstrations for robot manipula- tion,

    A. Mandlekar, D. Xu, J. Wong, S. Nasiriany, C. Wang, R. Kulkarni, L. Fei-Fei, S. Savarese, Y . Zhu, and R. Mart´ın-Mart´ın, “What matters in learning from offline human demonstrations for robot manipula- tion,” inConference on Robot Learning. PMLR, 2022, pp. 1678– 1690

    2022

  16. [16]

    Discriminator-weighted offline imitation learning from suboptimal demonstrations,

    H. Xu, X. Zhan, H. Yin, and H. Qin, “Discriminator-weighted offline imitation learning from suboptimal demonstrations,” inInternational Conference on Machine Learning. PMLR, 2022, pp. 24 725–24 742

    2022

  17. [17]

    Scizor: Self-supervised data curation for large-scale imitation learn- ing,

    Y . Zhang, Y . Xie, H. Liu, R. Shah, M. Wan, L. Fan, and Y . Zhu, “Scizor: Self-supervised data curation for large-scale imitation learn- ing,” inIEEE International Conference on Robotics and Automation (ICRA), 2026

    2026

  18. [18]

    Cupid: Curating data your robot loves with influence functions,

    C. Agia, R. Sinha, J. Yang, R. Antonova, M. Pavone, H. Nishimura, M. Itkina, and J. Bohg, “Cupid: Curating data your robot loves with influence functions,”arXiv preprint arXiv:2506.19121, 2025

    arXiv 2025

  19. [19]

    Curating demon- strations using online experience,

    A. S. Chen, A. M. Lessing, Y . Liu, and C. Finn, “Curating demon- strations using online experience,”arXiv preprint arXiv:2503.03707, 2025

    arXiv 2025

  20. [20]

    Learning from imperfect demonstrations with self-supervision for robotic manipulation,

    K. Wu, N. Liu, Z. Zhao, D. Qiu, J. Li, Z. Che, Z. Xu, and J. Tang, “Learning from imperfect demonstrations with self-supervision for robotic manipulation,” in2025 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2025, pp. 16 899–16 906

    2025

  21. [21]

    Mimicgen: A data generation system for scalable robot learning using human demonstrations,

    A. Mandlekar, S. Nasiriany, B. Wen, I. Akinola, Y . Narang, L. Fan, Y . Zhu, and D. Fox, “Mimicgen: A data generation system for scalable robot learning using human demonstrations,” inConference on Robot Learning. PMLR, 2023, pp. 1820–1864

    2023

  22. [22]

    Demogen: Synthetic demonstration generation for data-efficient visuomotor pol- icy learning,

    Z. Xue, S. Deng, Z. Chen, Y . Wang, Z. Yuan, and H. Xu, “Demogen: Synthetic demonstration generation for data-efficient visuomotor pol- icy learning,” in7th Robot Learning Workshop: Towards Robots with Human-Level Abilities, 2025

    2025

  23. [23]

    Manibox: Enhancing spatial grasping generalization via scalable simulation data generation,

    H. Tan, X. Xu, C. Ying, X. Mao, S. Liu, X. Zhang, H. Su, and J. Zhu, “Manibox: Enhancing spatial grasping generalization via scalable simulation data generation,”arXiv preprint arXiv:2411.01850, 2024

    arXiv 2024

  24. [24]

    Adversarial data collection: Human-collaborative perturba- tions for efficient and robust robotic imitation learning,

    S. Huang, Y . Liao, S. Feng, S. Jiang, S. Liu, H. Li, M. Yao, and G. Ren, “Adversarial data collection: Human-collaborative perturba- tions for efficient and robust robotic imitation learning,”arXiv preprint arXiv:2503.11646, 2025

    arXiv 2025

  25. [25]

    Fieldgen: From teleoperated pre-manipulation trajectories to field-guided data generation,

    W. Wang, K. Ye, X. Zhou, T. Chen, C. Min, Q. Zhu, X. Yang, P. Luo, Y . Shen, Y . Yanget al., “Fieldgen: From teleoperated pre-manipulation trajectories to field-guided data generation,”arXiv preprint arXiv:2510.20774, 2025

    arXiv 2025

  26. [26]

    Move: A simple motion-based data collection paradigm for spatial generalization in robotic manipulation,

    H. Wang, C. B. Chen, Y . Yue, D. Tao, T. Guo, S. Xie, D. Huang, S. Song, G. Yao, and G. Huang, “Move: A simple motion-based data collection paradigm for spatial generalization in robotic manipulation,” arXiv preprint arXiv:2512.04813, 2025

    arXiv 2025

  27. [27]

    Multi-modal imitation learning from unstructured demonstrations using generative adversarial nets,

    K. Hausman, Y . Chebotar, S. Schaal, G. Sukhatme, and J. J. Lim, “Multi-modal imitation learning from unstructured demonstrations using generative adversarial nets,”Advances in neural information processing systems, vol. 30, 2017

    2017

  28. [28]

    Behavior transformers: Cloningkmodes with one stone,

    N. M. Shafiullah, Z. Cui, A. A. Altanzaya, and L. Pinto, “Behavior transformers: Cloningkmodes with one stone,”Advances in neural information processing systems, vol. 35, pp. 22 955–22 968, 2022

    2022

  29. [29]

    Diffusion policy: Visuomotor policy learning via action diffusion,

    C. Chi, Z. Xu, S. Feng, E. Cousineau, Y . Du, B. Burchfiel, R. Tedrake, and S. Song, “Diffusion policy: Visuomotor policy learning via action diffusion,”The International Journal of Robotics Research, vol. 44, no. 10-11, pp. 1684–1704, 2025

    2025

  30. [30]

    Implicit behavioral cloning,

    P. Florence, C. Lynch, A. Zeng, O. A. Ramirez, A. Wahid, L. Downs, A. Wong, J. Lee, I. Mordatch, and J. Tompson, “Implicit behavioral cloning,” inConference on robot learning. PMLR, 2022, pp. 158– 168

    2022

  31. [31]

    Behavior generation with latent actions,

    S. Lee, Y . Wang, H. Etukuru, H. J. Kim, N. M. M. Shafiullah, and L. Pinto, “Behavior generation with latent actions,” inInternational Conference on Machine Learning. PMLR, 2024, pp. 26 991–27 008

    2024

  32. [32]

    Baku: An efficient transformer for multi-task policy learning,

    S. Haldar, Z. Peng, and L. Pinto, “Baku: An efficient transformer for multi-task policy learning,”Advances in Neural Information Process- ing Systems, vol. 37, pp. 141 208–141 239, 2024

    2024

  33. [33]

    Towards diverse behaviors: A benchmark for imitation learning with human demonstrations,

    X. Jia, D. Blessing, X. Jiang, M. Reuss, A. Donat, R. Lioutikov, and G. Neumann, “Towards diverse behaviors: A benchmark for imitation learning with human demonstrations,” inThe Twelfth International Conference on Learning Representations, 2024

    2024

  34. [34]

    Generative adversarial imitation learning,

    J. Ho and S. Ermon, “Generative adversarial imitation learning,” Advances in neural information processing systems, vol. 29, 2016

    2016

  35. [35]

    Is diversity all you need for scalable robotic manipulation?

    M. Shi, L. Chen, J. Chen, Y . Lu, C. Liu, G. Ren, P. Luo, D. Huang, M. Yao, and H. Li, “Is diversity all you need for scalable robotic manipulation?”arXiv preprint arXiv:2507.06219, 2025

    Pith/arXiv arXiv 2025

  36. [36]

    Dynamic programming algorithm optimiza- tion for spoken word recognition,

    H. Sakoe and S. Chiba, “Dynamic programming algorithm optimiza- tion for spoken word recognition,”IEEE transactions on acoustics, speech, and signal processing, vol. 26, no. 1, pp. 43–49, 1978

    1978

  37. [37]

    Maniskill3: Gpu parallelized robotics simulation and rendering for generalizable embodied ai,

    S. Tao, F. Xiang, A. Shukla, Y . Qin, X. Hinrichsen, X. Yuan, C. Bao, X. Lin, Y . Liu, T.-k. Chanet al., “Maniskill3: Gpu parallelized robotics simulation and rendering for generalizable embodied ai,” arXiv preprint arXiv:2410.00425, 2024

    arXiv 2024

  38. [38]

    π 0.5: a vision-language-action model with open-world generalization,

    P. Intelligence, K. Black, N. Brown, J. Darpinian, K. Dhabalia, D. Driess, A. Esmail, M. Equi, C. Finn, N. Fusaiet al., “π 0.5: a vision-language-action model with open-world generalization,”arXiv preprint arXiv:2504.16054, 2025

    Pith/arXiv arXiv 2025

  39. [39]

    Arx robotics,

    “Arx robotics,” https://arx-x.com/

  40. [40]

    Learning fine-grained bimanual manipulation with low-cost hardware,

    T. Z. Zhao, V . Kumar, S. Levine, and C. Finn, “Learning fine-grained bimanual manipulation with low-cost hardware,”arXiv preprint arXiv:2304.13705, 2023

    Pith/arXiv arXiv 2023