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arxiv: 1907.07521 · v1 · pith:AWENP6XEnew · submitted 2019-07-17 · 💻 cs.RO

Stochastic Optimization for Trajectory Planning with Heteroscedastic Gaussian Processes

Luka Petrovi\'c , Juraj Per\v{s}i\'c , Marija Seder , Ivan Markovi\'c This is my paper

Pith reviewed 2026-05-24 20:21 UTC · model grok-4.3

classification 💻 cs.RO
keywords trajectory optimizationmotion planningGaussian processesstochastic optimizationcross-entropy methodheteroscedasticitycollision avoidancelocal minima
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The pith

Introducing heteroscedasticity into Gaussian process trajectory priors and using cross-entropy stochastic optimization produces motion plans that succeed more often in cluttered environments than GPMP2.

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

The paper introduces a motion planning algorithm that samples trajectories from a continuous-time Gaussian process made heteroscedastic to favor collision-free paths and then refines them with the cross-entropy method. Standard trajectory optimizers often settle into local minima, especially when the environment is complex. By treating planning as stochastic search over these priors, the approach explores more candidate solutions. Experiments indicate this yields higher success rates than the GPMP2 baseline at similar computation cost. A reader cares because better local-minima avoidance makes reliable robot navigation in real settings more practical.

Core claim

Trajectories represented as samples from a heteroscedastic continuous-time Gaussian process can be optimized stochastically with the cross-entropy method to achieve more thorough exploration of the solution space and higher success rates in complex environments than GPMP2 while maintaining comparable execution time.

What carries the argument

Heteroscedastic continuous-time Gaussian process priors optimized via the cross-entropy method.

If this is right

  • The stochastic search avoids many local minima that trap gradient-based methods.
  • Trajectory priors become more suitable for collision avoidance due to varying variance.
  • Success rates increase in cluttered scenes without increasing runtime significantly.
  • Planning remains efficient for high degree-of-freedom robots.

Where Pith is reading between the lines

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

  • Similar stochastic optimization could be applied to other continuous representations beyond Gaussian processes.
  • The approach might scale to online replanning if variance schedules can be learned from data.
  • Testing in dynamic environments would reveal whether the priors adapt well to moving obstacles.

Load-bearing premise

That the heteroscedastic Gaussian process model creates trajectory priors better suited for avoiding collisions than standard models.

What would settle it

A controlled comparison where the same stochastic optimizer is run once with heteroscedastic priors and once with homoscedastic priors, showing no difference in success rate, would falsify the value of the heteroscedasticity.

Figures

Figures reproduced from arXiv: 1907.07521 by Ivan Markovi\'c, Juraj Per\v{s}i\'c, Luka Petrovi\'c, Marija Seder.

Figure 1
Figure 1. Figure 1: Comparison of the homoscedastic and the proposed heteroscedastic GP priors which depend on the covariance of the white noise process [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a 4 × 4 maze where the proposed approach finds a collision-free solution, while GPMP2 converges to an infeasible local minimum. Slight undulation of the trajectory obtained by the proposed method is due to the criterium of finding a collision-free trajectory, unlike the GPMP2 criterium which explicitly encourages smoothness. TABLE I SUCCESS RATE (PERCENTAGE) / AVERAGE EXECUTION TIME (MILISECONDS… view at source ↗
Figure 3
Figure 3. Figure 3: A simulated WAM robotic arm in an environment featuring a table and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Trajectory optimization methods for motion planning attempt to generate trajectories that minimize a suitable objective function. Such methods efficiently find solutions even for high degree-of-freedom robots. However, a globally optimal solution is often intractable in practice and state-of-the-art trajectory optimization methods are thus prone to local minima, especially in cluttered environments. In this paper, we propose a novel motion planning algorithm that employs stochastic optimization based on the cross-entropy method in order to tackle the local minima problem. We represent trajectories as samples from a continuous-time Gaussian process and introduce heteroscedasticity to generate powerful trajectory priors better suited for collision avoidance in motion planning problems. Our experimental evaluation shows that the proposed approach yields a more thorough exploration of the solution space and a higher success rate in complex environments than a current Gaussian process based state-of-the-art trajectory optimization method, namely GPMP2, while having comparable execution time.

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 / 1 minor

Summary. The paper proposes a motion planning algorithm that represents trajectories as samples from a continuous-time heteroscedastic Gaussian process and optimizes them via the cross-entropy method (CEM) to mitigate local minima issues in cluttered environments. It claims this yields more thorough solution-space exploration and higher success rates than the GPMP2 baseline while maintaining comparable runtime.

Significance. If the attribution of gains to heteroscedasticity holds after proper controls, the work would strengthen the case for stochastic optimizers paired with uncertainty-aware priors in GP-based trajectory optimization, addressing a recognized weakness of gradient methods like GPMP2 in complex scenes. The approach builds directly on established GPMP2 and CEM techniques without introducing circularity.

major comments (1)
  1. [Experimental Evaluation] Experimental section (and abstract claim): the only baseline is GPMP2, which differs simultaneously in the GP model (homoscedastic vs. heteroscedastic) and the optimizer (gradient descent vs. CEM). No ablation compares the proposed heteroscedastic GP+CEM against a homoscedastic GP+CEM variant, so performance differences cannot be attributed specifically to heteroscedasticity rather than the optimizer change.
minor comments (1)
  1. [Abstract and Experiments] The abstract and experimental description mention 'higher success rate' and 'comparable execution time' but provide no quantitative metrics, environment details, or statistical significance tests; these should be expanded for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on our experimental design. We agree that the current comparison does not isolate the effect of heteroscedasticity from the optimizer change, and we will revise the manuscript to include the requested ablation.

read point-by-point responses

  1. Referee: [Experimental Evaluation] Experimental section (and abstract claim): the only baseline is GPMP2, which differs simultaneously in the GP model (homoscedastic vs. heteroscedastic) and the optimizer (gradient descent vs. CEM). No ablation compares the proposed heteroscedastic GP+CEM against a homoscedastic GP+CEM variant, so performance differences cannot be attributed specifically to heteroscedasticity rather than the optimizer change.

    Authors: We agree that the present experiments compare two methods that differ in both the GP prior and the optimizer, preventing direct attribution of gains to heteroscedasticity alone. We will add an ablation that fixes the optimizer to CEM and varies only the GP model (homoscedastic vs. heteroscedastic). The new results, together with updated discussion and abstract language, will clarify the contribution of each component. We will also report the additional runtime for the ablation to maintain the claim of comparable execution time. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces heteroscedastic GP priors as a modeling choice and combines them with the cross-entropy method for stochastic optimization, building on GPMP2 as an external baseline. No equations reduce by construction to fitted inputs, no self-definitional loops, and no load-bearing self-citations that render the central claim tautological. The experimental comparison to GPMP2 is independent of the derivation itself. This is the expected honest non-finding for a methods paper that does not rename known results or smuggle ansatzes via self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The ledger reflects assumptions typical in GP-based motion planning; no new entities introduced in the abstract.

free parameters (1)
  • heteroscedastic parameters
    Parameters controlling the varying variance in the GP are likely fitted or chosen but not detailed in abstract.
axioms (2)
  • domain assumption Continuous-time Gaussian processes can represent robot trajectories effectively
    This is foundational to the representation used.
  • domain assumption The cross-entropy method can be applied to optimize over GP trajectory samples
    Core to the stochastic optimization approach.

pith-pipeline@v0.9.0 · 5692 in / 1225 out tokens · 27038 ms · 2026-05-24T20:21:16.763966+00:00 · methodology

discussion (0)

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