Pith. sign in

REVIEW 4 major objections 4 minor 78 references

Unified Linear Parametric Map Modeling and Perception-aware Trajectory Planning for Mobile Robotics

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims a sparse random projection can reduce a high-dimensional robot-map representation while provably preserving the residual energy that carries geometric information.

desk verdict The RMRP/RPATR pipeline is a genuinely useful integration, but the Residual Energy Preservation Theorem as stated does not follow from the cited embeddings, and the paper's central guarantee is unsupported. read the letter →

arxiv 2507.09340 v2 pith:L4KPZ7SE submitted 2025-07-12 cs.RO cs.LG

Hongyu Nie , Xu Liu , Zhaotong Tan , Sen Mei , Wenbo Su This is my paper
classification cs.ROcs.LG
keywords randomprojectionresidualenergypreservationoccupancymappingESDFmapterrainperception-awareplanningtrajectoryoptimizationunmannedaerialvehicles
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

This paper argues that a robot's environment map does not need to be stored as a dense grid. It proposes Random Mapping and Random Projection (RMRP), which lifts raw point clouds into a high-dimensional feature space and then reduces the dimension with a sparse random projection, producing one continuous, differentiable linear model that can answer occupancy, Euclidean distance, and terrain elevation queries. The paper's central theoretical claim is the Residual Energy Preservation Theorem: after projection, the energy of the component of the data lying outside the learned subspace is preserved up to a factor of $1 \pm C_2\epsilon$, so the geometric structure needed for planning survives dimensionality reduction. If correct, this would let robots build memory-lean parametric maps with closed-form gradients, run faster path search and trajectory optimization, and predict unseen obstacles in sensor blind spots.

What carries the argument

The load-bearing object is the Residual Energy Preservation Theorem (Eqs. 12-13). It combines three ingredients: an $\epsilon$-subspace embedding of the target subspace $S$ and its orthogonal complement $S^\perp$ (via the Clarkson-Woodruff dimension bound), a Davis-Kahan $\sin\Theta$ angle-leakage bound that controls how much of $x_\perp$ leaks into the projected subspace $RS$, and a sparse Hanson-Wright concentration inequality that keeps $\|R x_\perp\|$ near $\|x_\perp\|$ in a single draw. The theorem guarantees that the orthogonal-complement energy, which carries the regression/classification residual, is preserved, and it supplies the closed-form gradients $\partial y/\partial x = \eta^T J_g(x)$ and $\nabla\mathrm{Elevation}(x) = W(\beta \odot \cos(W^T x + b))$ used by the planners. The same machinery also yields the dimension-selection formula $k = \Theta(p \log(p/\delta)/\epsilon^2)$ and reduces multiplication cost from $O(dk)$ to $O(sk)$.

What would settle it

Take a concrete random projection with dimension $k$ from the formula, compute the projection residual $e_{\text{proj}}$ for a family of vectors $x$ whose orthogonal component $x_\perp$ has known norm, and check whether $\|e_{\text{proj}}\|$ stays within $[1 - C_2\epsilon,\, 1 + C_2\epsilon]\|x_\perp\|$ for the claimed $\epsilon$ and $\delta$; a single draw where the residual falls outside this band at the claimed probability would disprove the theorem. A cheaper test: compute $\|R^T R - I\|_2$ and the principal angle between $S$ and $RS$ for a small random $S$, and see whether the angle bound $\epsilon$ actually holds when only the subspace-embedding dimension condition is satisfied.

Watch

Extended reading notes

Core claim

The central discovery is that a sparse random projection onto $k \geq C p \log(p/\delta)/\epsilon^2$ dimensions preserves not just pairwise distances but also the residual energy of the data relative to a task-relevant subspace $S$. Writing each feature vector as $x = x_S + x_\perp$, the projection residual $e_{\text{proj}} = R x - P_{RS} R x$ obeys $(1 - C_2\epsilon)\|x_\perp\| \leq \|e_{\text{proj}}\| \leq (1 + C_2\epsilon)\|x_\perp\|$, with $C_2 \approx 3.9$ for $\epsilon \leq 0.3$. This lets the authors treat occupancy mapping as classification and ESDF/terrain fitting as regression on a single linear parametric map, whose analytic gradients drive both front-end path refinement and back-end trajectory optimization. The same model, trained offline on complete scenes, is then used online to complete occluded regions, enabling proactive rather than reactive obstacle avoidance for UAVs and pit-avoiding traversal for UGVs.

Load-bearing premise

The key assumption is that preserving the lengths of all vectors in the subspace and its complement also controls how much the projection tilts the subspace; the quoted dimension bound only guarantees length preservation, not this tilt, so if the tilt is not controlled the main inequality can fail.

Editorial extensions

If this is right

  • Occupancy, ESDF, and terrain maps collapse into one continuous parametric model; a robot stores a few weight vectors instead of a dense grid.
  • Because the map is differentiable, path refinement and trajectory optimization use exact analytic gradients instead of interpolated or finite-difference values, removing the resolution-dependence of existing planners.
  • The same RMRP model, trained on complete scenes, completes occluded regions online, letting a high-speed UAV plan around obstacles it has not yet observed.
  • For ground robots, the terrain gradient becomes a closed-form risk cost, so a UGV can steer around pits and steep slopes in real time.
  • The dimension-selection formula gives a principled way to choose the projected dimension, converting a heuristic dimensionality choice into a theorem-grounded one.

Reading between the lines

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

  • The Residual Energy Preservation Theorem is stated for a single fixed subspace $S$; if $S$ is learned online and shifts as data arrives, the lemma would need to be applied with a union bound over subspace updates, which would raise the required dimension roughly logarithmically in the number of updates.
  • The angle-leakage step appears to require a property stronger than standard subspace embedding; checking whether the sparse random projection actually satisfies $\|R^T R - I\|_2 \le \epsilon$ under the quoted dimension could be a quick empirical falsification before deploying the planner.
  • The predictive scene completion relies on the offline training scenes sharing geometric priors with the deployment environment; in environments with substantially different object geometry, the learned completion may hallucinate obstacles, so a confidence-based gating of the completion output would be a natural extension.
  • Because the map is linear in the projected features, the framework could be extended to dynamic environments by updating only $\eta$ (or $\beta$) online, as the paper's incremental learning already hints.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes RMRP, a two-stage parametric environment mapping method that first lifts point-cloud data into a high-dimensional space via a random mapping (RMM) and then applies a sparse random projection (SRP). The central theoretical contribution is the Residual Energy Preservation Theorem (Section V-D), which claims that the projection residual e_proj remains within a factor (1 ± C2 ε) of the norm of the component orthogonal to a chosen subspace S, provided k = Θ(p log(p/δ)/ε²). On top of this map, the authors build the RPATR planner: a front-end path refinement using analytical occupancy gradients, a back-end trajectory optimizer using a closed-form ESDF for UAVs, and a terrain-aware optimizer using analytical terrain gradients for UGVs. The experimental sections report mapping efficiency, planning speed, success rates, and real-world UAV/UGV flights.

Significance. If the Residual Energy Preservation Theorem were correct, the paper would offer a lightweight, continuous, differentiable map representation with a quantitative guarantee on residual geometry, unifying occupancy, ESDF, and terrain modeling. The empirical work is extensive and gives explicit credit to the system-level contributions: closed-form gradients for terrain and ESDF, predictive completion of sensor blind spots, and real-world validation on both UAV and UGV platforms at meaningful speeds. Code release is promised. However, the paper's central theoretical guarantee—the proof of Eqs. (12)–(13)—does not follow from the stated assumptions. The key angle-leakage step is invalid, and the dimension bound is incompatible with preserving the orthogonal complement when d ≫ p. Because the map and planner are explicitly justified by this theorem, the unsupported proof is load-bearing rather than cosmetic.

major comments (4)
  1. [Section V-B, Eq. (8)] The implication from the ε-subspace embedding (6) to ||R^T R − I||_2 ≤ ε is false: norm preservation on S and on S⊥ separately does not constrain the cross inner products ⟨Rs, Rv⟩ for s∈S and v∈S⊥. For example, in R^{2p}, take R = [0 I_p; I_p 0] and S = span(e_1,…,e_p); then R preserves the length of every vector in S∪S⊥ exactly, yet maps S onto S⊥, so the angle between S and RS is 90°. Hence inequality (9), and therefore the theorem's bounds (12)–(13), do not follow from the stated assumptions.
  2. [Section V-B and Appendix Eq. (46)] The Davis–Kahan/Wedin sin Θ theorem is invoked to compare S ⊂ R^d with RS ⊂ R^k, but these subspaces live in different ambient spaces when k ≠ d. The projector difference ||P_S − P_RS|| and the quantity sin Θ(S, RS) are not well-defined as used, so Eq. (8) is not a valid instance of the theorem. A subspace comparison would require an isometric embedding of S into R^k or a different formulation of the angle between subspaces of different dimensions.
  3. [Section V, Eqs. (5)–(6)] The dimension bound k = Θ(p log(p/δ)/ε^2) cannot ensure the event E_len on S⊥, whose dimension is d − p. A subspace embedding for S⊥ requires k ≥ C(d−p) log((d−p)/δ)/ε^2; for d ≫ p this contradicts the claimed bound. Thus the union-bound event E_len, which requires norm preservation for all v ∈ S ∪ S⊥, is not attainable at the stated projection dimension, and the theorem's probability guarantee is unsupported.
  4. [Section V-C, Eq. (10)] The sparse Hanson–Wright concentration is stated for a fixed vector x⊥, while the proof of the Residual Energy Preservation Theorem requires a uniform statement over all x⊥ ∈ S⊥ (or over the event E_len). No covering-net or union-bound argument over S⊥ is provided, so the transition from Eq. (10) to the theorem's conclusion is not justified.
minor comments (4)
  1. [Section IV-C, Eq. (2)] The learning-rate symbol is inconsistent: the update uses γ, while the text says η is the learning rate.
  2. [Algorithm 2] In the data branch, line 7 sets M_{k+1} ← M_k, so the map is never updated with the newly observed sample; this appears inconsistent with the intended online-update semantics.
  3. [Section VIII-B and Table I] Entries such as 'MAX2000.00' and 'MAX30,000' would be clearer if the table caption explicitly stated the time and node limits; as printed, they look like data values.
  4. [Throughout] There are typographical and formatting issues, e.g., 'UA Vs' spacing, 'arrivate' in Section VIII-D, and inconsistent use of 'RMRP' versus 'RPATR' in captions; a careful proofread is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

Residual Energy Preservation Theorem conditions on its own angle-leakage conclusion; RMM foundation is a load-bearing self-citation.

  1. self definitional [Section V-B, Eq. (8) and Section V-D, event Eang (Eqs. 11-13)]
    "Under the ε-subspace embedding assumption (i.e., ∥R⊤R − I∥2 ≤ ε), the Davis–Kahan sin Θ theorem implies ∥sin Θ(S, RS)∥2 = ∥PS − PRS∥2 ≤ ε. (8) ... Define the following events: ... Eang: the angle-leakage event ∥PRS R x⊥∥ ≤ C1 ε ∥x⊥∥ ... Under event Etot ... (12)"

    The theorem's guarantee (12)-(13) is proved by conditioning on Eang, but Eang is defined as exactly the angle-leakage bound that the theorem must establish. The only route to Eang, Eq. (8), is obtained by replacing the stated ε-subspace embedding condition (6) with the stronger statement ∥R⊤R−I∥2≤ε, introduced by the phrase '(i.e., ...)'. Norm preservation on S and S⊥ individually does not control the cross inner products that determine the operator norm, so the angle bound is effectively assumed rather than derived. The conclusion is thus built into the definition of the event/assumption.

  2. self citation load bearing [Section IV-B, Feasibility and Existence Analysis]
    "The theoretical validity of RMRP is established by proving the following core claim [4]: Claim: For any target vector T ... there exists a natural number P ... [4]. The proof of this claim rests upon two key lemmas also presented in [4]."

    Reference [4] is the authors' prior RA-L paper by co-author Xu Liu. The linear-separability/existence result that makes RMRP a valid parametric map is assumed from that paper rather than proved or independently verified here; all downstream map queries and planner gradients rely on this premise. The paper does provide external empirical validation of the full pipeline, so this self-citation is load-bearing but not the only support for the framework.

full rationale

The paper's planning stack is largely self-contained: the closed-form occupancy, ESDF, and terrain gradients follow from the parametric form by ordinary calculus, and the experiments are benchmarked against external planners. The central theoretical novelty, however, is the Residual Energy Preservation Theorem, and its proof reduces by construction to the event Eang, which is defined as the very angle-leakage inequality the theorem needs. The derivation of Eang from Eq. (8) silently strengthens the subspace-embedding hypothesis to ∥R⊤R−I∥2≤ε, a condition that already encodes the cross-subspace angle control that Eq. (8) is supposed to deliver. Consequently the theorem's probabilistic claim is not derived from the stated dimension bound (5); it is assumed through the definition of its own event. Additionally, the RMM foundation is imported from the authors' prior work, a self-citation that is load-bearing for the linear-separability premise. The score is 6 rather than higher because the empirical evaluations are externally benchmarked and the map-to-planning component is not circular; the circularity is concentrated in the theoretical guarantee and its self-cited foundation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new physical entities are postulated. The contributions are an algorithmic combination, a theorem label over known inequalities, and a set of learned model parameters.

free parameters (6)
  • linear model coefficients beta/eta = learned via AdamW or ridge, values not reported
    Central map parameters fitted to point-cloud data in Algorithms 1 and 2.
  • RMM dimension M = 100
    Set in Section VIII-A to balance accuracy and efficiency.
  • SRP dimension k = not reported
    The theorem gives a formula, but the deployed value and sparsity s are not given.
  • regularization alpha = 0.01
    Set in Section VIII-A to avoid overfitting.
  • trajectory cost weights lambda_s, lambda_c, lambda_d, lambda_g = not reported
    Hand-tuned trade-offs in front-end and back-end optimization.
  • occupancy threshold tau = not reported
    Used in Algorithm 2 to classify predicted points as occupied; value not specified.
assumptions (5)
  • domain assumption RMM feasibility: for any target and epsilon, a random mapping of sufficiently large dimension M admits a linear model with error below epsilon.
    Invoked in Section IV-B and reference [4]; the RMRP map's linear separability rests on this theorem from prior work by the same group.
  • standard math epsilon-subspace embedding for S and S_perp with k >= C p log(p/delta)/epsilon^2.
    Clarkson-Woodruff style guarantee cited in Section V-4; needed to preserve norms after SRP.
  • ad hoc to paper The sparse random projection also satisfies ||R^T R - I||_2 <= epsilon, so the Davis-Kahan sin Theta bound applies.
    This stronger operator-norm assumption is asserted in Section V-B but is not implied by the standard subspace embedding property nor by the quoted dimension bound.
  • domain assumption Offline training on complete 3D scenes transfers to online partially observed scenes.
    The predictive scene completion in Section VI-A and Algorithm 2 needs train and test distribution match; no domain-shift analysis is given.
  • standard math Sparse Hanson-Wright concentration holds for the Achlioptas {0, +/-1} projection with density 1-s.
    Invoked as Eq. (10) and cited to reference [65].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Unified Linear Parametric Map Modeling and Perception-aware Trajectory Planning for Mobile Robotics." pith.science (2026). https://pith.science/paper/L4KPZ7SE

@misc{pith2026250709340,
  author       = {Pith},
  title        = {Pith review of: Unified Linear Parametric Map Modeling and Perception-aware Trajectory Planning for Mobile Robotics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4KPZ7SE}},
  note         = {Machine review of arXiv:2507.09340}
}
read the original abstract

Autonomous navigation in mobile robots, reliant on perception and planning, faces major hurdles in large-scale, complex environments. These include heavy computational burdens for mapping, sensor occlusion failures for UAVs, and traversal challenges on irregular terrain for UGVs, all compounded by a lack of perception-aware strategies. To address these challenges, we introduce Random Mapping and Random Projection (RMRP). This method constructs a lightweight linear parametric map by first mapping data to a high-dimensional space, followed by a sparse random projection for dimensionality reduction. Our novel Residual Energy Preservation Theorem provides theoretical guarantees for this process, ensuring critical geometric properties are preserved. Based on this map, we propose the RPATR (Robust Perception-Aware Trajectory Planner) framework. For UAVs, our method unifies grid and Euclidean Signed Distance Field (ESDF) maps. The front-end uses an analytical occupancy gradient to refine initial paths for safety and smoothness, while the back-end uses a closed-form ESDF for trajectory optimization. Leveraging the trained RMRP model's generalization, the planner predicts unobserved areas for proactive navigation. For UGVs, the model characterizes terrain and provides closed-form gradients, enabling online planning to circumvent large holes. Validated in diverse scenarios, our framework demonstrates superior mapping performance in time, memory, and accuracy, and enables computationally efficient, safe navigation for high-speed UAVs and UGVs. The code will be released to foster community collaboration.

Figures

Figures reproduced from arXiv: 2507.09340 by the authors.

Figure 1
Figure 1. Applications of the RMRP perception-aware planning framework. (a) For UAVs, obstacle completion in sensor blind-spots enables proactive avoidance and safer, smoother trajectories. (b) For UGVs, analytical terrain gradients serve as a risk cost for trajectory optimization to safely circumvent depressions. We will soon release a video demonstration to provide more information about the experiments. autonomous navigati… view at source ↗
Figure 2
Figure 2. The proposed LPATR framework for autonomous flight integrates environmental perception and motion planning through a unified, lightweight, and continuous parametric model. The process begins with raw point cloud inputs, which are transformed into a unified differentiable map via a RMRP module. This module employs high-dimensional mapping and efficient random projection to convert the data into a continuous, differen… view at source ↗
Figure 3
Figure 3. presents occupancy grid maps generated on Dataset 1 using three methods: (a) Ground Truth, (b) RMM, and (c) RMRP. Qualitative analysis indicates that (c) exhibits strong visual consistency with (a) and (b), validating its capacity to preserve high classification accuracy while markedly reducing model dimensionality [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: validates the fidelity and efficiency of RMRP in terrain regression. The figure compares the modeling results of (a) Ground Truth, (b) RMM, and (c) RMRP. RMRP accurately reconstructs key terrain features, such as geometry and gradients, with lower overhead, providing r…
Figure 5
Figure 5. Figure 5: quantitatively evaluates the performance of RMM and RMRP in classification and regression tasks. The results confirm that RMRP achieves comparable classification accuracy and regression metrics (R², MSE) to RMM, while drastically reducing training and testing computati…
Figure 6
Figure 6. Figure 6: This figure showcases the performance of our gradient-optimized front-end search in a dense, cluttered map. Leveraging the occupancy field gradient, our method proactively guides trajectories into safer areas. This effectively avoids the path quality degradation of sta…
Figure 8
Figure 8. Figure 8: By incorporating the occupancy field gradient generated by the RMRP model for the prediction region, our optimizer actively pushes the trajectory away from regions with abrupt occupancy changes (i.e., obstacle boundaries). This effectively mitigates unnecessary corneri…
Figure 9
Figure 9. Figure 9: Traditional methods (right) can only plan based on current observa￾tions. Our method (left), however, employs the RMRP model for predictive completion of sensor blind spots. This allows the planner to acquire more complete environmental information in advance, enabling…
Figure 10
Figure 10. Figure 10: Conventional methods (top) rely on discrete maps and interpolated distance estimates, resulting in limited accuracy. Our method (bottom) obtains a closed-form ESDF solution through a parametric model, enabling the analytical computation of precise distances and guidin…
Figure 11
Figure 11. Figure 11: Scenario 1: Utilizing the RMRP model, Ours proactively completes the sensor blind spot behind the obstacle (blue point cloud) based on the observed data (yellow point cloud). This enables the generation of a safer, globally evasive trajectory, avoiding reactive emerge…
Figure 13
Figure 13. Figure 13: The figure clearly demonstrates that RMRP generates a smooth, accurate gradient field with near-zero numerical error. In contrast, the finite difference method produces a noisy gradient field with significant error hotspots. This fundamental difference in gradient qua…
Figure 12
Figure 12. Figure 12: Scenario 2: Completion and Planning in Lateral Blind Spots at Corners. The left image (Before) shows a risky trajectory closely adhering to the inside of the corner in the absence of predictive capabilities. In the right image (After), our RMRP model completes the cor…
Figure 14
Figure 14. Figure 14: The plot compares the safe distance between the drone and the blind spot center at five key instants, both without (beige) and with (light green) our method. The results clearly indicate that our method maintains a larger safety margin throughout the process, with the…
Figure 15
Figure 15. Figure 15: This figure illustrates the sensitivity of different planners’ trajectory optimization to map resolution in a dense obstacle environment. Traditional planners (e.g., Fast-Planner, Faster), which rely on discrete ESDF maps, and EGO-Planner, which bypasses ESDF but stil…
Figure 16
Figure 16. Figure 16: (a) Raw LiDAR Point Cloud Input. (b) Continuous and Differentiable Terrain Model Generated by RMRP [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]
Figure 17
Figure 17. Figure 17: Complete Planning Pipeline Based on Terrain-Aware RMRP Model. (a) Perception: Raw 3D Point Cloud. (b) Modeling: Continuous Terrain Map via RMRP. (c) Planning: Trajectory Optimization Comparison. C. Gradient of the Terrain Penalty Term To efficiently use the aforementi…
Figure 18
Figure 18. Figure 18: Experiment I (Zero Obstacle-Avoidance Weights): With disabled obstacle avoidance, conventional planners (except Ours and UnevenPlanner) ignored obstacles/depressions, taking the shortest path. UnevenPlanner inade￾quately avoided shallow pits. Conversely, our approach …
Figure 19
Figure 19. Figure 19: Experiment II (Default Weights): Conventional planners, reverting to default settings, treated the ground as an obstacle, hindering progress and failing to avoid depressions. UnevenPlanner showed limited depression avoidance, still traversing moderately deep pits. Con…
Figure 20
Figure 20. Figure 20: (a) Experimental UAV Platform. (b) Experimental UGV Platform. [PITH_FULL_IMAGE:figures/full_fig_p014_20.png]
Figure 22
Figure 22. Figure 22: Trajectory of UAV in Scenario 2 (Indoor Experiment). trajectory tracking. While traditional methods require com￾putationally intensive ESDF map calculations at appropriate resolutions for trajectory optimization (which relies on dis￾tance and gradient values), our app…
Figure 23
Figure 23. Figure 23: Performance comparison of different planners in two scenarios. The plots compare our proposed method against five others based on (a) total time (ttot), (b) total path length (ltot), and (c) total energy (Eeng). Solid lines represent the mean performance over multiple…
Figure 24
Figure 24. Figure 24: showcases the outdoor flight experiment within a forest environment. (a) illustrates the UAV trajectory and the online-generated forest map, represented as a closed-form ESDF, with elevation indicated by color. (b) depicts the real-world scene of the autonomous fast f…
Figure 25
Figure 25. Figure 25: This figure showcases the UGV’s real-world experiment in a terrain environment. (a) presents the raw point cloud data acquired by the Mid360 LiDAR; (b) shows the side-view profiles of multiple pits within the terrain, which are annotated with different colors; (c) dep…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

78 extracted references · 72 canonical work pages

  1. [1]

    Collaborative multi-robot search and rescue: Planning, coordination, perception, and active vision,

    J. P. Queralta, J. Taipalmaa, B. C. Pullinen, V . K. Sarker, T. N. Gia, H. Tenhunen, M. Gabbouj, J. Raitoharju, and T. Westerlund, “Collaborative multi-robot search and rescue: Planning, coordination, perception, and active vision,” Ieee Access, vol. 8, pp. 191 617–191 643, 2020

    work page 2020

  2. [2]

    An efficient spatial-temporal trajectory planner for autonomous vehicles in unstructured environments,

    Z. Han, Y . Wu, T. Li, L. Zhang, L. Pei, L. Xu, C. Li, C. Ma, C. Xu, S. Shen et al. , “An efficient spatial-temporal trajectory planner for autonomous vehicles in unstructured environments,” IEEE Transactions on Intelligent Transportation Systems , vol. 25, no. 2, pp. 1797–1814, 2023

    work page 2023

  3. [3]

    Autonomous power line inspection with drones via perception-aware mpc,

    J. Xing, G. Cioffi, J. Hidalgo-Carri ´o, and D. Scaramuzza, “Autonomous power line inspection with drones via perception-aware mpc,” in 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2023, pp. 1086–1093

    work page 2023

  4. [4]

    A unified framework for large-scale occu- pancy mapping and terrain modeling using rmm,

    X. Liu, D. Li, and Y . He, “A unified framework for large-scale occu- pancy mapping and terrain modeling using rmm,” IEEE Robotics and Automation Letters, vol. 7, no. 2, pp. 5143–5150, 2022

    work page 2022

  5. [5]

    Extensions of lipschitz mappings into a hilbert space,

    W. B. Johnson and J. Lindenstrauss, “Extensions of lipschitz mappings into a hilbert space,” in Contemporary Mathematics. American Mathe- matical Society, 1984, vol. 26, pp. 189–206, presented at the Conference in Modern Analysis and Probability (New Haven, Conn., 1982)

    work page 1984

  6. [6]

    Tighter bounds for random projections of manifolds,

    K. Clarkson, “Tighter bounds for random projections of manifolds,” in ACM Symposium on Computational Geometry , 2008, pp. 39–48

    work page 2008

  7. [7]

    On sparsity and sub-gaussianity in the johnson-lindenstrauss lemma,

    A. Garivier and E. Pilliat, “On sparsity and sub-gaussianity in the johnson-lindenstrauss lemma,” ArXiv preprint, 2024

    work page 2024

  8. [8]

    Machine learning friendly set version of john- son–lindenstrauss lemma,

    M. Klopotek, “Machine learning friendly set version of john- son–lindenstrauss lemma,” Knowledge and Information Systems, vol. 62, pp. 1961–2009, 2017

    work page 1961

Show all 78 references
  1. [9]

    Nearly tight oblivious subspace embeddings by trace inequalities,

    M. B. Cohen, “Nearly tight oblivious subspace embeddings by trace inequalities,” in SODA, 2016, pp. 278–287

    2016

  2. [10]

    Lower bounds for oblivious subspace embeddings,

    J. Nelson and H. L. Nguyen, “Lower bounds for oblivious subspace embeddings,” in ICALP, 2013, pp. 883–894

    2013

  3. [11]

    Osnap: Faster numerical linear algebra algorithms via sparser subspace embeddings,

    ——, “Osnap: Faster numerical linear algebra algorithms via sparser subspace embeddings,” in FOCS, 2012, pp. 117–126

    2012

  4. [12]

    On principal components regression, random projections, and column subsampling,

    M. Slawski, “On principal components regression, random projections, and column subsampling,” arXiv: Statistics Theory , 2017

    2017

  5. [13]

    Distance-preserving property of random projection for subspaces,

    G. Li and Y . Gu, “Distance-preserving property of random projection for subspaces,” in ICASSP, 2017, pp. 3959–3963

    2017

  6. [14]

    Restricted isometry property of gaussian random projection for finite set of subspaces,

    ——, “Restricted isometry property of gaussian random projection for finite set of subspaces,”IEEE Transactions on Signal Processing, vol. 66, pp. 1705–1720, 2017

    2017

  7. [15]

    Octomap: An efficient probabilistic 3d mapping framework based on octrees,

    A. Hornung, K. M. Wurm, M. Bennewitz, C. Stachniss, and W. Burgard, “Octomap: An efficient probabilistic 3d mapping framework based on octrees,” Autonomous robots, vol. 34, pp. 189–206, 2013

    2013

  8. [16]

    Online generation of collision-free trajectories for quadrotor flight in unknown cluttered environments,

    J. Chen, T. Liu, and S. Shen, “Online generation of collision-free trajectories for quadrotor flight in unknown cluttered environments,” in 2016 IEEE international conference on robotics and automation (ICRA). IEEE, 2016, pp. 1476–1483

    2016

  9. [17]

    Real-time 3d reconstruction at scale using voxel hashing,

    M. Nießner, M. Zollh ¨ofer, S. Izadi, and M. Stamminger, “Real-time 3d reconstruction at scale using voxel hashing,” ACM Transactions on Graphics (ToG), vol. 32, no. 6, pp. 1–11, 2013

    2013

  10. [18]

    Robust and effi- cient quadrotor trajectory generation for fast autonomous flight,

    B. Zhou, F. Gao, L. Wang, C. Liu, and S. Shen, “Robust and effi- cient quadrotor trajectory generation for fast autonomous flight,” IEEE Robotics and Automation Letters , vol. 4, no. 4, pp. 3529–3536, 2019

    2019

  11. [19]

    Learning-aided 3-d occupancy mapping with bayesian generalized kernel inference,

    K. Doherty, T. Shan, J. Wang, and B. Englot, “Learning-aided 3-d occupancy mapping with bayesian generalized kernel inference,” IEEE Transactions on Robotics , vol. 35, no. 4, pp. 953–966, 2019

    2019

  12. [20]

    Efficient, multifidelity perceptual repre- sentations via hierarchical gaussian mixture models,

    S. Srivastava and N. Michael, “Efficient, multifidelity perceptual repre- sentations via hierarchical gaussian mixture models,” IEEE Transactions on Robotics, vol. 35, no. 1, pp. 248–260, 2018

    2018

  13. [21]

    Gaussian process occupancy maps,

    S. T. O’Callaghan and F. T. Ramos, “Gaussian process occupancy maps,” The International Journal of Robotics Research , vol. 31, no. 1, pp. 42– 62, 2012

    2012

  14. [22]

    Deep, spatially coherent occu- pancy maps based on radar measurements,

    D. Bauer, L. Kuhnert, and L. Eckstein, “Deep, spatially coherent occu- pancy maps based on radar measurements,” in AmE 2019-Automotive meets Electronics; 10th GMM-Symposium . VDE, 2019, pp. 1–6

    2019

  15. [23]

    Deep occupancy maps: a continuous mapping technique for dynamic environments,

    R. Senanayake, T. Ganegedara, and F. Ramos, “Deep occupancy maps: a continuous mapping technique for dynamic environments,” 2017

    2017

  16. [24]

    Distance transforms of sampled functions,

    P. F. Felzenszwalb and D. P. Huttenlocher, “Distance transforms of sampled functions,” Theory of computing , vol. 8, no. 1, pp. 415–428, 2012

    2012

  17. [25]

    Improved updating of euclidean distance maps and voronoi diagrams,

    B. Lau, C. Sprunk, and W. Burgard, “Improved updating of euclidean distance maps and voronoi diagrams,” in 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems . IEEE, 2010, pp. 281– 286

    2010

  18. [26]

    V oxblox: Incremental 3d euclidean signed distance fields for on-board mav planning,

    H. Oleynikova, Z. Taylor, M. Fehr, R. Siegwart, and J. Nieto, “V oxblox: Incremental 3d euclidean signed distance fields for on-board mav planning,” in 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2017, pp. 1366–1373

    2017

  19. [27]

    Fiesta: Fast incremental euclidean distance fields for online motion planning of aerial robots,

    L. Han, F. Gao, B. Zhou, and S. Shen, “Fiesta: Fast incremental euclidean distance fields for online motion planning of aerial robots,” in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2019, pp. 4423–4430

    2019

  20. [28]

    Gpu-accelerated incremental euclidean distance transform for online motion planning of mobile robots,

    Y . Chen, S. Lai, J. Cui, B. Wang, and B. M. Chen, “Gpu-accelerated incremental euclidean distance transform for online motion planning of mobile robots,” IEEE Robotics and Automation Letters , vol. 7, no. 3, pp. 6894–6901, 2022

    2022

  21. [29]

    Sampling-free obstacle gradients and reactive planning in neural radiance fields,

    M. Pantic, C. Cadena, R. Siegwart, and L. Ott, “Sampling-free obstacle gradients and reactive planning in neural radiance fields,” in Workshop on” Motion Planning with Implicit Neural Representations of Geome- try” at 2022 IEEE International Conference on Robotics and Automati...

    2022

  22. [30]

    Continuous-time trajectory optimization for online uav replan- ning,

    H. Oleynikova, M. Burri, Z. Taylor, J. Nieto, R. Siegwart, and E. Gal- ceran, “Continuous-time trajectory optimization for online uav replan- ning,” in 2016 IEEE/RSJ international conference on intelligent robots and systems (IROS) . IEEE, 2016, pp. 5332–5339

    2016

  23. [31]

    Faster: Fast and safe trajec- tory planner for flights in unknown environments,

    J. Tordesillas, B. T. Lopez, and J. P. How, “Faster: Fast and safe trajec- tory planner for flights in unknown environments,” in 2019 IEEE/RSJ international conference on intelligent robots and systems (IROS) . IEEE, 2019, pp. 1934–1940

    2019

  24. [32]

    6d slam—3d mapping outdoor environments,

    A. N ¨uchter, K. Lingemann, J. Hertzberg, and H. Surmann, “6d slam—3d mapping outdoor environments,” Journal of Field Robotics , vol. 24, no. 8-9, pp. 699–722, 2007

    2007

  25. [33]

    Natural terrain classification using three-dimensional ladar data for ground robot mobility,

    J.-F. Lalonde, N. Vandapel, D. F. Huber, and M. Hebert, “Natural terrain classification using three-dimensional ladar data for ground robot mobility,” Journal of field robotics , vol. 23, no. 10, pp. 839–861, 2006

    2006

  26. [34]

    Autonomous rover navigation on unknown terrains: Func- tions and integration,

    S. Lacroix, A. Mallet, D. Bonnafous, G. Bauzil, S. Fleury, M. Herrb, and R. Chatila, “Autonomous rover navigation on unknown terrains: Func- tions and integration,” The International Journal of Robotics Research , vol. 21, no. 10-11, pp. 917–942, 2002

    2002

  27. [35]

    Multi-level surface maps for out- door terrain mapping and loop closing,

    R. Triebel, P. Pfaff, and W. Burgard, “Multi-level surface maps for out- door terrain mapping and loop closing,” in 2006 IEEE/RSJ international conference on intelligent robots and systems . IEEE, 2006, pp. 2276– 2282

    2006

  28. [36]

    Gaussian processes for machine learning,

    S. T. Ounpraseuth, “Gaussian processes for machine learning,” 2008

    2008

  29. [37]

    Multiresolution representations for large- scale terrain with local gaussian process regression,

    X. Liu, D. Li, and Y . He, “Multiresolution representations for large- scale terrain with local gaussian process regression,” in 2021 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2021, pp. 5497–5503

    2021

  30. [38]

    A note on two problems in connexion with graphs,

    E. W. Dijkstra, “A note on two problems in connexion with graphs,” in Edsger Wybe Dijkstra: His Life, Work, and Legacy , 2022, pp. 287–290

    2022

  31. [39]

    A formal basis for the heuristic determination of minimum cost paths,

    P. E. Hart, N. J. Nilsson, and B. Raphael, “A formal basis for the heuristic determination of minimum cost paths,” IEEE transactions on Systems Science and Cybernetics , vol. 4, no. 2, pp. 100–107, 1968

    1968

  32. [40]

    Anytime heuristic search,

    E. A. Hansen and R. Zhou, “Anytime heuristic search,” Journal of Artificial Intelligence Research , vol. 28, pp. 267–297, 2007

    2007

  33. [41]

    Online graph pruning for pathfinding on grid maps,

    D. Harabor and A. Grastien, “Online graph pruning for pathfinding on grid maps,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 25, no. 1, 2011, pp. 1114–1119

    2011

  34. [42]

    Path planning for autonomous vehicles in unknown semi-structured environments,

    D. Dolgov, S. Thrun, M. Montemerlo, and J. Diebel, “Path planning for autonomous vehicles in unknown semi-structured environments,” The international journal of robotics research , vol. 29, no. 5, pp. 485–501, 2010

    2010

  35. [43]

    Search-based motion planning for quadrotors using linear quadratic minimum time control,

    S. Liu, N. Atanasov, K. Mohta, and V . Kumar, “Search-based motion planning for quadrotors using linear quadratic minimum time control,” in 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS). IEEE, 2017, pp. 2872–2879

    2017

  36. [44]

    Apace: Agile and perception- aware trajectory generation for quadrotor flights,

    X. Chen, Y . Zhang, B. Zhou, and S. Shen, “Apace: Agile and perception- aware trajectory generation for quadrotor flights,” in 2024 IEEE Inter- NIE AND LI et al.: UNIFIED LINEAR PARAMETRIC MAP MODELING AND PERCEPTION-AW ARE TRAJECTORY PLANNING FOR MOBILE ROBOTICS 19 national C...

    2024

  37. [45]

    Raptor: Robust and perception- aware trajectory replanning for quadrotor fast flight,

    B. Zhou, J. Pan, F. Gao, and S. Shen, “Raptor: Robust and perception- aware trajectory replanning for quadrotor fast flight,” IEEE Transactions on Robotics, vol. 37, no. 6, pp. 1992–2009, 2021

    1992

  38. [46]

    Learning-based 3d occupancy prediction for autonomous navigation in occluded envi- ronments,

    L. Wang, H. Ye, Q. Wang, Y . Gao, C. Xu, and F. Gao, “Learning-based 3d occupancy prediction for autonomous navigation in occluded envi- ronments,” in 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2021, pp. 4509–4516

    2021

  39. [47]

    Learning perception- aware agile flight in cluttered environments,

    Y . Song, K. Shi, R. Penicka, and D. Scaramuzza, “Learning perception- aware agile flight in cluttered environments,” in 2023 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2023, pp. 1989–1995

    2023

  40. [48]

    Driving on point clouds: Motion planning, trajectory optimization, and terrain assessment in generic nonplanar environments,

    P. Kr ¨usi, P. Furgale, M. Bosse, and R. Siegwart, “Driving on point clouds: Motion planning, trajectory optimization, and terrain assessment in generic nonplanar environments,” Journal of Field Robotics , vol. 34, no. 5, pp. 940–984, 2017

    2017

  41. [49]

    Putn: A plane-fitting based uneven terrain navigation framework,

    Z. Jian, Z. Lu, X. Zhou, B. Lan, A. Xiao, X. Wang, and B. Liang, “Putn: A plane-fitting based uneven terrain navigation framework,” in 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2022, pp. 7160–7166

    2022

  42. [50]

    On variants of the johnson–lindenstrauss lemma,

    J. Matou ˇsek, “On variants of the johnson–lindenstrauss lemma,”Random Structures & Algorithms , vol. 33, no. 2, pp. 142–156, 2008

    2008

  43. [51]

    Dimensionality reduction via the johnson–lindenstrauss lemma: theoretical and empirical bounds,

    J. Fedoruk, B. Schmuland, J. Johnson, and G. Heo, “Dimensionality reduction via the johnson–lindenstrauss lemma: theoretical and empirical bounds,” The Journal of Supercomputing, vol. 74, pp. 3933–3949, 2018

    2018

  44. [52]

    Nearly tight oblivious subspace embeddings by trace inequalities,

    M. B. Cohen, “Nearly tight oblivious subspace embeddings by trace inequalities,” in Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms , 2016, pp. 278–287

    2016

  45. [53]

    The fast johnson-lindenstrauss transform and approximate nearest neighbors,

    N. Ailon and B. Chazelle, “The fast johnson-lindenstrauss transform and approximate nearest neighbors,” in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS) . IEEE, 2006, pp. 557– 563

    2006

  46. [54]

    Dense fast random projections and lean walsh transforms,

    E. Liberty, N. Ailon, and A. Singer, “Dense fast random projections and lean walsh transforms,” in International Colloquium on Automata, Languages and Programming, 2008, pp. 512–522

    2008

  47. [55]

    Decoupled weight decay regularization,

    I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” in International Conference on Learning Representations , 2019, pp. 1–8

    2019

  48. [56]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations , 2015, pp. 1– 15

    2015

  49. [57]

    Linear regression with random projec- tions,

    O.-A. Maillard and R. Munos, “Linear regression with random projec- tions,” Journal of Machine Learning Research , vol. 13, pp. 2735–2772, 2012

    2012

  50. [58]

    Accurate analysis of sparse random projections,

    A. Andoni, P. Indyk, L. Schmidt, D. P. Woodruff, and M. Thorup, “Accurate analysis of sparse random projections,” arXiv:2407.14518, 2024

    2024 arXiv

  51. [59]

    The fast johnson–lindenstrauss transform and approximate nearest neighbors,

    N. Ailon and B. Chazelle, “The fast johnson–lindenstrauss transform and approximate nearest neighbors,” SIAM Journal on Computing , vol. 39, no. 1, pp. 302–322, 2009

    2009

  52. [60]

    Low-rank approximation and regression in input sparsity time,

    K. L. Clarkson and D. P. Woodruff, “Low-rank approximation and regression in input sparsity time,” in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (STOC) . ACM, 2013, pp. 81–90

    2013

  53. [61]

    Optimal em- bedding dimension for sparse subspace embeddings,

    S. Chenakkod, M. Derezi ´nski, X. Dong, and M. Rudelson, “Optimal em- bedding dimension for sparse subspace embeddings,” arXiv:2311.10680, 2023

    2023

  54. [62]

    Why are the residuals in Rn−p?

    Cross Validated Discussion Thread, “Why are the residuals in Rn−p?” Online forum post, 2021

    2021

  55. [63]

    A useful variant of the davis– kahan theorem for statisticians,

    Y . Yu, T. Wang, and R. J. Samworth, “A useful variant of the davis– kahan theorem for statisticians,” Biometrika, vol. 102, no. 2, pp. 315– 323, 2015

    2015

  56. [64]

    Notes on matrix perturbation and davis–kahan sin θ theo- rem,

    D. J. Hsu, “Notes on matrix perturbation and davis–kahan sin θ theo- rem,” Lecture Notes, Columbia University, 2016, urlhttps://www.cs.columbia.edu/ djhsu/AML/lectures/davis-kahan.pdf

    2016

  57. [66]

    Hanson–wright inequality and sub- gaussian concentration,

    M. Rudelson and R. Vershynin, “Hanson–wright inequality and sub- gaussian concentration,” Electronic Communications in Probability , vol. 18, no. 82, pp. 1–9, 2013

    2013

  58. [67]

    Feller, An Introduction to Probability Theory and Its Applications , 3rd ed

    W. Feller, An Introduction to Probability Theory and Its Applications , 3rd ed. New York, NY: John Wiley & Sons, 1971, vol. 1, contains statement and proof of Boole’s inequality

    1971

  59. [68]

    Informed sam- pling for asymptotically optimal path planning,

    J. D. Gammell, T. D. Barfoot, and S. S. Srinivasa, “Informed sam- pling for asymptotically optimal path planning,” IEEE Transactions on Robotics, vol. 34, no. 4, pp. 966–984, 2018

    2018

  60. [69]

    Guided incremental local densification for accelerated sampling-based motion planning,

    A. Mandalika, R. Scalise, B. Hou, S. Choudhury, and S. S. Srinivasa, “Guided incremental local densification for accelerated sampling-based motion planning,” arXiv preprint arXiv:2104.05037 , 2021

    2021 arXiv

  61. [70]

    Intelligent bidirectional rapidly-exploring random trees for optimal motion planning in complex cluttered environ- ments,

    A. H. Qureshi and Y . Ayaz, “Intelligent bidirectional rapidly-exploring random trees for optimal motion planning in complex cluttered environ- ments,” Robotics and Autonomous Systems , vol. 68, pp. 1–11, 2015

    2015

  62. [71]

    Kinodynamic motion planning,

    B. R. Donald, P. G. Xavier, J. F. Canny, and J. H. Reif, “Kinodynamic motion planning,” Journal of the ACM (JACM), vol. 40, no. 5, pp. 1048– 1066, 1993

    1993

  63. [72]

    Ego-planner: An esdf- free gradient-based local planner for quadrotors,

    X. Zhou, Z. Wang, H. Ye, C. Xu, and F. Gao, “Ego-planner: An esdf- free gradient-based local planner for quadrotors,” IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 478–485, 2020

    2020

  64. [73]

    Faster: Fast and safe trajectory planner for navigation in unknown environments,

    J. Tordesillas, B. T. Lopez, M. Everett, and J. P. How, “Faster: Fast and safe trajectory planner for navigation in unknown environments,” IEEE Transactions on Robotics , vol. 38, no. 2, pp. 922–938, 2021

    2021

  65. [74]

    Autonomous uav exploration of dynamic environments via incremental sampling and probabilistic roadmap,

    Z. Xu, D. Deng, and K. Shimada, “Autonomous uav exploration of dynamic environments via incremental sampling and probabilistic roadmap,” IEEE Robotics and Automation Letters , vol. 6, no. 2, pp. 2729–2736, 2021

    2021

  66. [75]

    Lecture notes on subspace embeddings and ose length preservation,

    K. L. Clarkson and D. P. Woodruff, “Lecture notes on subspace embeddings and ose length preservation,” Lecture Notes, Carnegie Mellon University, Computer Science Department, 2020, available at https://www.cs.cmu.edu/∼dwoodruf/teaching.html

    2020

  67. [76]

    Johnson–lindenstrauss lemma and its applications,

    Z. Zhu, “Johnson–lindenstrauss lemma and its applications,” Lecture Notes, University of Wisconsin–Madison, 2009, available at http://pages. cs.wisc.edu/∼zhu/jl-lecture.pdf

    2009

  68. [77]

    Notes on matrix perturbation and davis–kahan sin θ the- orem,

    D. J. Hsu, “Notes on matrix perturbation and davis–kahan sin θ the- orem,” Lecture Notes, Columbia University, 2016, available at https: //www.cs.columbia.edu/∼djhsu/AML/lectures/davis-kahan.pdf

    2016

  69. [78]

    Why does ∥pW x∥ ≤ √ 2 sinθ ∥x∥ hold?

    Math StackExchange user, “Why does ∥pW x∥ ≤ √ 2 sinθ ∥x∥ hold?” Mathematics StackExchange, online Q&A, 2017

    2017

  70. [79]

    Sparse hanson–wright inequalities with applications,

    Y . L. Zhong, R. T. Antoine, M. E. Roux, X. T. Chen, and G. Bajrowski, “Sparse hanson–wright inequalities with applications,” arXiv:2410.15652, 2024

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.