pith. sign in

arxiv: 2605.13063 · v1 · pith:Z72TIE6Mnew · submitted 2026-05-13 · 💻 cs.LG

Ergodic Trajectory Design by Learned Pushforward Maps: Provable Coverage via Conditional Flow Matching

Ehsan Aghazadeh , Masoud Malekzadeh , Ahmad Ghasemi , Hossein Pishro-Nik This is my paper

Pith reviewed 2026-05-14 20:02 UTC · model grok-4.3

classification 💻 cs.LG
keywords ergodic coverageconditional flow matchingpushforward mapstrajectory designUAV planningrobotic explorationdensity matchingoptimal transport
1
0 comments X

The pith

A learned pushforward map turns an analytic uniform ergodic path into trajectories whose time-averaged occupancy matches any target density with error controlled by training loss.

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

The paper establishes that ergodic coverage can be achieved by separating the uniform ergodicity property from density matching. An analytic latent trajectory supplies exact uniform occupancy on a simple domain, while a single conditional flow matching map learned offline transports that occupancy onto the prescribed target density. The resulting composed trajectory is asymptotically ergodic with respect to the learned pushforward, and the deviation from the target is bounded by the flow-matching loss together with a Lipschitz constant on the velocity field. This construction lets constraints enter as soft penalties during training and yields a reusable map that serves unlimited trajectories and multiple agents without retraining. A sympathetic reader cares because the approach replaces repeated online re-optimization with one offline training step whose quality is directly measurable from standard CFM diagnostics.

Core claim

The central claim is that composing an analytic latent trajectory, which is exactly ergodic with respect to uniform measure on an annular domain, with a conditional flow matching pushforward map produces trajectories that are asymptotically ergodic with respect to the target density; the approximation error is controlled by the flow-matching training loss, an acceleration-energy bound, and an O(1/sqrt(K)) ergodic convergence rate in the number of cycles K, so that the three results combine into a single end-to-end coverage bound that can be estimated from training diagnostics once an architectural Lipschitz bound on the learned velocity field is given.

What carries the argument

The epushforward map learned by conditional flow matching, which transports the exact uniform ergodic occupancy of the latent trajectory onto the target density while incorporating operational constraints as additive soft penalties.

If this is right

  • The composed trajectory converges to the target density at an O(1/sqrt(K)) rate in the number of cycles K.
  • Operational constraints such as no-fly zones or acceleration limits enter the design as soft penalties without requiring new analytic constructions.
  • A single trained map can be reused for an unbounded number of trajectories and across an entire multi-agent fleet without per-agent retraining.
  • The end-to-end coverage error is estimable directly from conditional flow matching training diagnostics once the velocity-field Lipschitz constant is known.

Where Pith is reading between the lines

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

  • The offline-training pattern could be extended to time-varying target densities by periodically updating the map while keeping the same latent trajectory.
  • Because the map is differentiable, gradient-based planners could further refine individual trajectories on top of the learned pushforward without losing the ergodicity guarantee.
  • The separation of ergodicity and density matching suggests the same latent trajectory could be reused across entirely different sensing modalities once new maps are trained.

Load-bearing premise

A single offline-trained conditional flow matching map can transport the exact uniform ergodic occupancy from the latent trajectory onto an arbitrary target density while respecting all operational constraints, with the approximation error bounded solely by the training loss and the Lipschitz constant of the learned velocity field.

What would settle it

Train the map on a known target density with a computable Lipschitz bound on v_theta, generate many composed trajectories, measure their actual ergodic deviation from the target, and check whether the deviation remains within the end-to-end bound predicted by the training loss; systematic violation of the bound falsifies the claim.

Figures

Figures reproduced from arXiv: 2605.13063 by Ahmad Ghasemi, Ehsan Aghazadeh, Hossein Pishro-Nik, Masoud Malekzadeh.

Figure 1
Figure 1. Figure 1: Two-stage pipeline. The latent trajectory provides structural ergodicity (i.i.d. cycles, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Coverage–energy Pareto on Milano. Left: single-disc (1D) NFZ. Right: multi-disc (MD) NFZ. Pearson ρ (higher better) plotted against the swept energy budget P∥v∥ 2 dt (log scale; lower￾right is best); shaded bands are ±1 std over three seeds. The OT-CFM family traces an empirical Pareto front above the evaluated baselines in both configurations; +E occupies the low-energy end and +NFZ the high-fidelity end.… view at source ↗
Figure 3
Figure 3. Figure 3: Experiment 1: Two-mode Gaussian mixture target. (a) Latent ergodic trajectory on the annulus Dδ (K = 300 cycles): the radial back-and-forth traversals with i.i.d. uniform heading angles produce a uniform time-averaged density by Proposition 1. (b) Target density ftarget: a symmetric two-mode Gaussian mixture centered at (±0.3, 0) with standard deviation 0.2, restricted to the annulus Dδ (so supp(ftarget) i… view at source ↗
Figure 4
Figure 4. Figure 4: Experiment 2: Binary 3:1 density target. (a) Target density: the lower half of the disc has 3× higher density than the upper half, corresponding to a UAV coverage scenario where the southern region has 3× higher service demand. (b) Achieved density from the learned map Gθ (IID evaluation, correlation ρ = 0.79). The sharp boundary at x2 = 0 is smoothed by the continuous transport map, which cannot produce a… view at source ↗
Figure 5
Figure 5. Figure 5: Experiment 3: Constraint flexibility with off-center no-fly zone (NFZ). Three variants on the same Gaussian mixture target (Experiment 1) with an NFZ centered at (0.5, −0.5) (radius 0.2, shown as a circle). (a) Unconstrained: density correlation ρ = 0.89, NFZ violation 0.7%, acceleration ratio 1.77×. (b) NFZ penalty only (λnfz = 50): correlation 0.76, NFZ violation 0.1%, acceleration 1.43×. The NFZ penalty… view at source ↗
Figure 6
Figure 6. Figure 6: Empirical convergence rate of the statistical error. Log-log plot of ∥Z traj K − Ziid∥RMSE, the grid-RMSE between the empirical K-cycle density and the IID-transport reference density Ziid = Gθ#π δ 0 (20,000 IID samples), measured on the Experiment 1 target. Averaging Ziid over IID samples isolates the purely statistical component, cancelling the Theorem 3 approximation floor. Error bars are one s.d. over … view at source ↗
Figure 7
Figure 7. Figure 7: Coverage–NFZ Pareto on Milano. Left: single-disc (1D). Right: multi-disc (MD). Pearson ρ (higher better) plotted against fraction of trajectory inside any NFZ disc (lower better; lower-right is best). Shaded bands are ±1 std over three seeds. N.3 Coverage–Energy Trade-off: Additional Coverage Axes [PITH_FULL_IMAGE:figures/full_fig_p042_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Coverage–energy trade-off across additional coverage axes. Columns (left to right): [PITH_FULL_IMAGE:figures/full_fig_p042_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Multi-agent reuse on Milano. Left: single-disc. Right: multi-disc. Aggregate EN vs. fleet size N, with the predicted E1 p 1/N reference (dashed). Per-agent power remains essentially flat across N. Amortization cost structure [PITH_FULL_IMAGE:figures/full_fig_p043_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Representative trajectories on the Milano target (purple heatmap) under the single-disc [PITH_FULL_IMAGE:figures/full_fig_p045_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representative trajectories on the Milano target (purple heatmap) under the multi-disc [PITH_FULL_IMAGE:figures/full_fig_p046_11.png] view at source ↗
read the original abstract

Designing continuous trajectories whose time-averaged occupancy provably matches a prescribed spatial density (the \emph{ergodic coverage} problem) is central to UAV-assisted data collection and sensing, robotic exploration, and mobile monitoring. For flying agents in particular, this challenge is acute: trajectories must balance coverage fidelity against tight energy budgets, no-fly zones, and acceleration limits. Existing methods either re-optimize each trajectory online (with cost growing in the horizon and re-running for every target, agent, and realization) or rely on bespoke analytical constructions that must be re-derived for each new constraint. We propose a \emph{epushforward} framework that decouples ergodicity from density matching: an analytic latent trajectory provides exact uniform ergodicity on a simple annular domain, and a single map, learned offline by optimal-transport conditional flow matching, transports this latent occupancy onto the prescribed target density. The composed trajectory is then asymptotically ergodic with respect to the learned pushforward distribution, with deviation from the target controlled by the flow-matching training loss. Once trained for a given target density and constraint set, the map serves an unbounded number of trajectories and a multi-agent fleet without per-agent retraining, and many differentiable operational constraints (no-fly zones, acceleration ceilings, or fairness penalties) enter as additive soft penalties in the training loss without re-deriving the design. We prove three results (an acceleration-energy bound, an $O(1/\sqrt{K})$ ergodic convergence rate in the number of trajectory cycles $K$, and an approximation-error bound) that combine into an end-to-end coverage bound estimable from CFM training diagnostics (certified given an architectural Lipschitz bound on $v_\theta$).

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

1 major / 2 minor

Summary. The manuscript proposes an 'epushforward' framework for ergodic trajectory design: an analytic latent trajectory supplies exact uniform ergodicity on an annular domain, while a single conditional flow matching (CFM) map, trained offline, transports this occupancy to a prescribed target density. Operational constraints enter as soft penalties in the CFM loss. Three results—an acceleration-energy bound, an O(1/√K) ergodic convergence rate in the number of cycles K, and an approximation-error bound—are combined into an end-to-end coverage guarantee whose deviation from the target is controlled by the CFM training loss (certified given an architectural Lipschitz bound on the learned velocity field v_θ). Once trained, the map generates trajectories for arbitrary numbers of agents without retraining.

Significance. If the three bounds can be rigorously closed, the approach would decouple ergodicity from density matching and allow constraint-aware coverage trajectories to be generated at scale from a single offline training run. The use of CFM to learn the transport map and the provision of explicit convergence rates are positive features. However, the practical and certified value of the coverage guarantee remains limited by the unresolved dependence on the Lipschitz constant of v_θ.

major comments (1)
  1. [Abstract] Abstract: The end-to-end coverage bound is stated to be 'estimable from CFM training diagnostics (certified given an architectural Lipschitz bound on v_θ)'. The approximation-error bound necessarily depends on this constant (via Gronwall-type estimates on the flow or integrated velocity discrepancies). No section derives, computes, or numerically evaluates such an L for the trained network, and the soft-penalty formulation for constraints can further increase L without a priori control. Consequently the claimed 'estimable from diagnostics' property does not hold.
minor comments (2)
  1. [Introduction] The invented term 'epushforward' is used in the abstract and title without an immediate formal definition; a precise definition should be supplied in the first paragraph of the introduction.
  2. [Preliminaries] Notation for the learned velocity field v_θ and the pushforward measure should be introduced consistently before the statement of the three theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our coverage guarantees. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The end-to-end coverage bound is stated to be 'estimable from CFM training diagnostics (certified given an architectural Lipschitz bound on v_θ)'. The approximation-error bound necessarily depends on this constant (via Gronwall-type estimates on the flow or integrated velocity discrepancies). No section derives, computes, or numerically evaluates such an L for the trained network, and the soft-penalty formulation for constraints can further increase L without a priori control. Consequently the claimed 'estimable from diagnostics' property does not hold.

    Authors: We agree that the manuscript does not currently derive, compute, or numerically report a concrete value (or bound) for the Lipschitz constant L of the trained velocity field v_θ. The theoretical development states the dependence on L explicitly and notes that the bound is estimable once an architectural L is available, but we do not close this step with an explicit calculation. We will revise the manuscript by (i) adding an appendix that derives a rigorous upper bound on L from the network architecture (product of spectral norms of the weight matrices for the chosen MLP with ReLU activations) and (ii) reporting both this architectural bound and a numerical estimate of the realized Lipschitz constant on the trained models in the experimental section. We will also add a short discussion of how the soft-penalty terms affect the Lipschitz constant and, where possible, provide a priori control via weight regularization. These changes will make the end-to-end coverage guarantee fully certifiable from training diagnostics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chains from independent analytic ergodicity and standard CFM approximation bounds

full rationale

The paper separates the ergodicity guarantee (exact uniform occupancy from an analytic latent trajectory on an annular domain, independent of data or fitting) from the density-matching step (a single offline-trained conditional flow-matching map whose pushforward deviation is controlled by the training loss). The three proved results—an acceleration-energy bound, O(1/√K) ergodic rate, and approximation-error bound—are combined into an end-to-end coverage statement that explicitly conditions on an external architectural Lipschitz constant for v_θ rather than deriving that constant from the fitted quantities or self-referential definitions. No equation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled through prior work by the same authors. The derivation therefore remains self-contained against external benchmarks once the Lipschitz constant is supplied separately.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of an analytic latent trajectory that is exactly ergodic on the annular domain, the ability of conditional flow matching to learn a sufficiently accurate transport map, and the validity of an architectural Lipschitz bound on the velocity network for certifying the final error.

free parameters (1)
  • Lipschitz constant of v_θ
    Architectural bound supplied by the user to certify the coverage error bound; its value is chosen or verified outside the training loop.
axioms (2)
  • domain assumption Conditional flow matching training converges to the optimal transport map between the latent uniform ergodic measure and the target density
    Invoked when the abstract states that deviation is controlled by the training loss.
  • standard math The analytic latent trajectory is exactly uniformly ergodic on the annular domain
    Stated as the starting point that requires no learning.
invented entities (1)
  • epushforward map no independent evidence
    purpose: Learned transport that composes with the latent trajectory to achieve target ergodicity
    New object introduced by the framework; no independent evidence supplied beyond the training procedure itself.

pith-pipeline@v0.9.0 · 5627 in / 1712 out tokens · 65594 ms · 2026-05-14T20:02:55.471184+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    and Murphey, T

    Abraham, I. and Murphey, T. D. Decentralized ergodic control: distribution-driven sensing and exploration for multiagent systems.IEEE Robotics and Automation Letters, 3(4):2987–2994, 2018. 9

    work page 2018

  2. [2]

    Caffarelli, L. A. The regularity of mappings with a convex potential.Journal of the American Mathematical Society, 5(1):99–104, 1992

    work page 1992

  3. [3]

    and ´Swie ¸ch, A

    Gangbo, W. and ´Swie ¸ch, A. Optimal maps for the multidimensional Monge–Kantorovich problem. Communications on Pure and Applied Mathematics, 51(1):23–45, 1998

    work page 1998

  4. [4]

    Moving aerial base station networks: A stochastic geometry analysis and design perspective.IEEE Trans

    Enayati, S., Saeedi, H., Pishro-Nik, H., and Yanikomeroglu, H. Moving aerial base station networks: A stochastic geometry analysis and design perspective.IEEE Trans. Wireless Communications, 18(6):2977– 2988, 2019

    work page 2019

  5. [5]

    and Carreras, M

    Galceran, E. and Carreras, M. A survey on coverage path planning for robotics.Robotics and Autonomous Systems, 61(12):1258–1276, 2013

    work page 2013

  6. [6]

    Safety-critical ergodic exploration in cluttered environments via control barrier functions

    Lerch, C., Dong, D., and Abraham, I. Safety-critical ergodic exploration in cluttered environments via control barrier functions. InIEEE International Conference on Robotics and Automation (ICRA), pp. 10205-10211, 2023

    work page 2023

  7. [7]

    Time optimal ergodic search

    Dong, D., Berger, H., and Abraham, I. Time optimal ergodic search. InRobotics: Science and Systems (RSS), 2023

    work page 2023

  8. [8]

    Lipman, Y ., Chen, R. T. Q., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. International Conference on Learning Representations, 2023

    work page 2023

  9. [9]

    Robust UA V trajectory design for non-uniform coverage

    Malekzadeh, M., Ghasemi, A., and Pishro-Nik, H. Robust UA V trajectory design for non-uniform coverage. IEEE Communications Letters, 30:188–192, 2026. DOI:10.1109/LCOMM.2025.3629065

    work page doi:10.1109/lcomm.2025.3629065 2026

  10. [10]

    and Mezi´c, I

    Mathew, G. and Mezi´c, I. Metrics for ergodicity and design of ergodic dynamics for multi-agent systems. Physica D, 240(4–5):432–442, 2011

    work page 2011

  11. [11]

    M., Pinosky, A., and Murphey, T

    Sun, M. M., Pinosky, A., and Murphey, T. Flow matching ergodic coverage. InRobotics: Science and Systems (RSS), 2025. arXiv:2504.17872

    work page arXiv 2025

  12. [12]

    M., Pinosky, A., and Murphey, T

    Sun, M. M., Pinosky, A., and Murphey, T. Flow matching ergodic coverage — official tuto- rials. GitHub repository, https://github.com/MurpheyLab/lqr-flow-matching/tree/main/ tutorials, 2025

    work page 2025

  13. [13]

    TartanAir: A dataset to push the limits of visual SLAM,

    Theile, M., Bayerlein, H., Nai, R., Gesbert, D., and Caccamo, M. UA V coverage path planning under varying power constraints using deep reinforcement learning. In2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1444–1449, 2020. doi:10.1109/IROS45743.2020.9340934

    work page doi:10.1109/iros45743.2020.9340934 2020

  14. [14]

    Improving and generalizing flow-based generative models with minibatch optimal transport.Transactions on Machine Learning Research, 2024

    Tong, A., Malkin, N., Huguet, G., Zhang, Y ., Rector-Brooks, J., Fatras, K., Wolf, G., and Bengio, Y . Improving and generalizing flow-based generative models with minibatch optimal transport.Transactions on Machine Learning Research, 2024

    work page 2024

  15. [15]

    Accessing from the sky: A tutorial on UA V communications for 5G and beyond.Proceedings of the IEEE, 107(12):2327–2375, 2019

    Zeng, Y ., Wu, Q., and Zhang, R. Accessing from the sky: A tutorial on UA V communications for 5G and beyond.Proceedings of the IEEE, 107(12):2327–2375, 2019

    work page 2019

  16. [16]

    A tutorial on UA Vs for wireless networks: Applications, challenges, and open problems.IEEE Communications Surveys & Tutorials, 21(3):2334–2360, 2019

    Mozaffari, M., Saad, W., Bennis, M., Nam, Y .-H., and Debbah, M. A tutorial on UA Vs for wireless networks: Applications, challenges, and open problems.IEEE Communications Surveys & Tutorials, 21(3):2334–2360, 2019

    work page 2019

  17. [17]

    Ergodic coverage in constrained environments using stochastic trajectory optimization

    Ayvali, E., Salman, H., and Choset, H. Ergodic coverage in constrained environments using stochastic trajectory optimization. In2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5204–5210, 2017. doi: 10.1109/IROS.2017.8206410

    work page doi:10.1109/iros.2017.8206410 2017

  18. [18]

    Teschl, G.Ordinary Differential Equations and Dynamical Systems. V ol. 140. American Mathematical Society, 2012

    work page 2012

  19. [19]

    Roberts, G. O. and Tweedie, R. L. Exponential convergence of Langevin distributions and their discrete approximations.Bernoulli, 2(4):341–363, 1996

    work page 1996

  20. [20]

    Neal, R. M. MCMC using Hamiltonian dynamics. InHandbook of Markov Chain Monte Carlo, pp. 47–95, Chapman and Hall/CRC, 2011

    work page 2011

  21. [21]

    Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    work page 2020

  22. [22]

    P., Kumar, A., Ermon, S., and Poole, B

    Song, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations.International Conference on Learning Representations, 2021. 10

    work page 2021

  23. [23]

    Deep unsupervised learning using nonequilibrium thermodynamics

    Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. InProceedings of the 32nd International Conference on Machine Learning (ICML), V ol. 37,Proceedings of Machine Learning Research, pp. 2256–2265, Lille, France, 2015

    work page 2015

  24. [24]

    Chen, R. T. Q., Rubanova, Y ., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 2018

    work page 2018

  25. [25]

    M., and Vanden-Eijnden, E

    Albergo, M., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26:1–80, 2025

    work page 2025

  26. [26]

    Diffusion policy: Visuomotor policy learning via action diffusion.The International Journal of Robotics Research, 44(10–11):1684–1704, 2025

    Chi, C., Xu, Z., Feng, S., Cousineau, E., Du, Y ., Burchfiel, B., Tedrake, R., and Song, S. Diffusion policy: Visuomotor policy learning via action diffusion.The International Journal of Robotics Research, 44(10–11):1684–1704, 2025

    work page 2025

  27. [27]

    Interpolating between optimal transport and mmd using sinkhorn divergences

    Feydy, J., Séjourné, T., Vialard, F.-X., Amari, S.-I., Trouvé, A., and Peyré, G. Interpolating between optimal transport and mmd using sinkhorn divergences. InThe 22nd International Conference on Artificial Intelligence and Statistics, pp. 2681–2690, PMLR, 2019

    work page 2019

  28. [28]

    Spectral normalization for generative adversarial networks

    Miyato, T., Kataoka, T., Koyama, M., and Yoshida, Y . Spectral normalization for generative adversarial networks. InICLR, 2018

    work page 2018

  29. [29]

    Convergence of continuous normalizing flows for learning probability distributions.arXiv:2404.00551, 2024

    Gao, Y ., Huang, J., Jiao, Y ., and Zheng, S. Convergence of continuous normalizing flows for learning probability distributions.arXiv:2404.00551, 2024

    work page arXiv 2024

  30. [30]

    Stein variational ergodic search

    Lee, D., Lerch, C., Ramos, F., and Abraham, I. Stein variational ergodic search. InRobotics: Science and Systems (RSS), 2024

    work page 2024

  31. [31]

    Miller, L. M. and Murphey, T. D. Trajectory optimization for continuous ergodic exploration. In2013 American Control Conference, pp. 4196–4201, 2013. doi: 10.1109/ACC.2013.6580484

    work page doi:10.1109/acc.2013.6580484 2013

  32. [32]

    Flow straight and fast: Learning to generate and transfer data with rectified flow

    Liu, X., Gong, C., and Liu, Q. Flow straight and fast: Learning to generate and transfer data with rectified flow. InICLR, 2023

    work page 2023

  33. [33]

    and Scaman, K

    Virmaux, A. and Scaman, K. Lipschitz regularity of deep neural networks: analysis and efficient estimation. Advances in Neural Information Processing Systems, 31, 2018

    work page 2018

  34. [34]

    J., Mohamed, S., and Lakshminarayanan, B

    Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., and Lakshminarayanan, B. Normalizing flows for probabilistic modeling and inference.Journal of Machine Learning Research, 22(57):1–64, 2021

    work page 2021

  35. [35]

    Grundlehren der mathematischen Wissenschaften, vol

    Villani, C.Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, vol. 338. Springer, 2009

    work page 2009

  36. [36]

    and Cuturi, M.Computational optimal transport: With applications to data science

    Peyré, G. and Cuturi, M.Computational optimal transport: With applications to data science. Now Foundations and Trends, 2019

    work page 2019

  37. [37]

    M., Brisolara, L

    Cabreira, T. M., Brisolara, L. B., and Ferreira Jr., P. R. Survey on coverage path planning with unmanned aerial vehicles.Drones, 3(1):4, 2019

    work page 2019

  38. [38]

    Joint trajectory and communication design for multi-UA V enabled wireless networks.IEEE Transactions on Wireless Communications, 17(3):2109–2121, 2018

    Wu, Q., Zeng, Y ., and Zhang, R. Joint trajectory and communication design for multi-UA V enabled wireless networks.IEEE Transactions on Wireless Communications, 17(3):2109–2121, 2018

    work page 2018

  39. [39]

    and Brenier, Y

    Benamou, J.-D. and Brenier, Y . A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem.Numerische Mathematik, 84(3):375–393, 2000

    work page 2000

  40. [40]

    Trajectory optimization for autonomous flying base station via reinforcement learning

    Bayerlein, H., De Kerret, P., and Gesbert, D. Trajectory optimization for autonomous flying base station via reinforcement learning. In2018 IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 1–5, IEEE, 2018

    work page 2018

  41. [41]

    M.Real analysis and probability

    Dudley, R. M.Real analysis and probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, 2002

    work page 2002

  42. [42]

    C.Partial differential equations

    Evans, L. C.Partial differential equations. V ol. 19, American Mathematical Society, 2022

    work page 2022

  43. [43]

    Springer, 2006

    Henrot, A.Extremum problems for eigenvalues of elliptic operators. Springer, 2006

    work page 2006

  44. [44]

    Polar factorization and monotone rearrangement of vector-valued functions.Communications on Pure and Applied Mathematics, 44(4):375–417, 1991

    Brenier, Y . Polar factorization and monotone rearrangement of vector-valued functions.Communications on Pure and Applied Mathematics, 44(4):375–417, 1991. 11

    work page 1991

  45. [45]

    Sinkhorn distances: Lightspeed computation of optimal transport.Advances in Neural Information Processing Systems, 26:2292–2300, 2013

    Cuturi, M. Sinkhorn distances: Lightspeed computation of optimal transport.Advances in Neural Information Processing Systems, 26:2292–2300, 2013

    work page 2013

  46. [46]

    M., Silverman, Y ., MacIver, M

    Miller, L. M., Silverman, Y ., MacIver, M. A., and Murphey, T. D. Ergodic exploration of distributed information.IEEE Transactions on Robotics, 32(1):36–52, 2016

    work page 2016

  47. [47]

    Mavrommati, A., Tzorakoleftherakis, E., Abraham, I., and Murphey, T. D. Real-time area coverage and target localization using receding-horizon ergodic exploration.IEEE Transactions on Robotics, 34(1):62–80, 2018

    work page 2018

  48. [48]

    J., Chancán, M., Dollar, A

    Seewald, A., Lerch, C. J., Chancán, M., Dollar, A. M., and Abraham, I. Energy-aware ergodic search: Continuous exploration for multi-agent systems with battery constraints. InIEEE International Conference on Robotics and Automation (ICRA), pp. 7048–7054, 2024

    work page 2024

  49. [49]

    M., Gaggar, A., Trautman, P., and Murphey, T

    Sun, M. M., Gaggar, A., Trautman, P., and Murphey, T. D. Fast ergodic search with kernel functions.IEEE Transactions on Robotics, 41:1841–1860, 2025

    work page 2025

  50. [50]

    Ergodic trajectory optimization on generalized domains using maximum mean discrepancy

    Hughes, C., Warren, H., Lee, D., Ramos, F., and Abraham, I. Ergodic trajectory optimization on generalized domains using maximum mean discrepancy. InIEEE International Conference on Robotics and Automation (ICRA), 2025

    work page 2025

  51. [51]

    and Abraham, I

    Hughes, C. and Abraham, I. Infinite-horizon ergodic control via kernel mean embeddings. arXiv:2604.01023, 2026

    work page arXiv 2026

  52. [52]

    A multi-source dataset of urban life in the city of Milan and the Province of Trentino.Scientific Data, 2:150055, 2015

    Barlacchi, G., De Nadai, M., Larcher, R., Casella, A., Chitic, C., Torrisi, G., Antonelli, F., Vespignani, A., Pentland, A., and Lepri, B. A multi-source dataset of urban life in the city of Milan and the Province of Trentino.Scientific Data, 2:150055, 2015

    work page 2015

  53. [53]

    Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor

    Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. InInternational Conference on Machine Learning (ICML), pp. 1861–1870, 2018

    work page 2018

  54. [54]

    Energy minimization for wireless communication with rotary-wing UA V

    Zeng, Y ., Xu, J., and Zhang, R. Energy minimization for wireless communication with rotary-wing UA V. IEEE Trans. Wireless Communications, 18(4):2329–2345, 2019. A Notation We collect here the notation used throughout the paper, grouped thematically. Standard symbols (expectation, norms, Wasserstein distances, pushforward) are included alongside paper-sp...

    work page 2019

  55. [55]

    Step 2: Apply ergodicity ofz(t).By the ergodicity ofz(t)with respect toπ δ 0: 1 T Z T 0 φ(G(z(t)))dt= 1 T Z T 0 ψ(z(t))dt T→∞ − − − − → Z Dδ ψ(z)π δ 0(z)dz

    by the pushforward identity R |ψ|dπ δ 0 = R |φ|d ftarget <∞. Step 2: Apply ergodicity ofz(t).By the ergodicity ofz(t)with respect toπ δ 0: 1 T Z T 0 φ(G(z(t)))dt= 1 T Z T 0 ψ(z(t))dt T→∞ − − − − → Z Dδ ψ(z)π δ 0(z)dz. Step 3: Change of variables via the pushforward.By the pushforward condition G#πδ 0 =f target and the change-of-variables formula: Z Dδ ψ(z...

    work page

  56. [56]

    ftarget has a C1 density bounded away from zero on a homeomorphic support

    is strictly below σ2 1 =∥A∥ 2 op whenever σ1 > σ2. For general C2 maps the bound is conservative for the same reason: the actual energy depends on the angle-averaged Eθ[∥JGeθ∥2], which may be substantially smaller than L2. Experiment 2 illustrates the gap empirically: the sup-Lipschitz estimate is ˆL≈2 (Table 3), giving a worst-case ratio bound L2 ≈4 , wh...

    work page 2048

  57. [57]

    Ergodic” in the Coverage Literature The term “ergodic

    give tighter bounds at higher cost. Our experiments do not use spectral normalization; the empirical ˆLv ∈[2,4] across all three experiments is sufficient for the approximation bound (7) to be informative, and we flag this as the natural route if certifiedL v becomes operationally required. 28 (b) From velocity Lipschitz to flow Lipschitz via Grönwall.Und...

    work page 2048

  58. [58]

    Guidelines: • The answer [N/A] means that the paper does not involve crowdsourcing nor research with human subjects

    Institutional review board (IRB) approvals or equivalent for research with human subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or ...

    work page