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arxiv: 2606.12334 · v1 · pith:WAGSC4B4new · submitted 2026-06-10 · 💻 cs.LG · cs.RO

Fourier Features Let Agents Learn High Precision Policies with Imitation Learning

Bal\'azs Gyenes , Emiliyan Gospodinov , Jan Frieling , Enrico Krohmer , Nicolas Schreiber , Xiaogang Jia , Niklas Freymuth , Gerhard Neumann This is my paper

Pith reviewed 2026-06-27 10:40 UTC · model grok-4.3

classification 💻 cs.LG cs.RO
keywords imitation learningpoint cloudsFourier featuresrobotic manipulationspectral biashigh-precision policies3D encoders
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The pith

Fourier features let point-cloud policies learn high-precision manipulation by accessing high-frequency details.

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

The paper proposes mapping point clouds from Cartesian coordinates into high-dimensional Fourier space so that encoders gain direct access to high-frequency components. This addresses the hypothesis that neural networks suffer from spectral bias toward low-frequency functions when conditioned on slow-moving Cartesian inputs, which limits fine spatial reasoning in robotic tasks. Experiments across RoboCasa and ManiSkill3 benchmarks plus a real-robot setup show consistent gains for multiple encoder types. The method remains simple and robust to hyperparameter choices. If the claim holds, Fourier features become a general-purpose addition for point-cloud imitation learning that improves geometric detail capture without architectural overhaul.

Core claim

Mapping point clouds from Cartesian space into high-dimensional Fourier space equips the encoder with direct high-frequency features; this change produces significant performance gains on high-precision manipulation tasks across diverse architectures and benchmarks while remaining robust to hyperparameter variation.

What carries the argument

The mapping of point clouds from Cartesian space into high-dimensional Fourier space, supplying direct high-frequency features to the policy network.

If this is right

  • Policies achieve higher success rates on fine-grained manipulation tasks from the RoboCasa and ManiSkill3 suites.
  • The same Fourier mapping improves results across multiple point-cloud encoder architectures.
  • Performance gains hold on physical robot hardware as well as simulation.
  • The method remains effective without extensive hyperparameter tuning.

Where Pith is reading between the lines

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

  • The same Cartesian-to-Fourier transform could be applied to other 3D input modalities such as depth images or voxel grids to test generality beyond point clouds.
  • If spectral bias is the dominant bottleneck, Fourier features might reduce the need for deeper networks or larger datasets in geometric imitation tasks.
  • Tasks that combine point clouds with RGB could use Fourier features on the 3D branch to compensate for perspective and scale ambiguities in the image branch.

Load-bearing premise

The performance gap between point-cloud and image-based policies arises primarily from spectral bias in networks conditioned on Cartesian features.

What would settle it

A controlled task requiring only low-frequency spatial distinctions where adding Fourier features produces no measurable improvement over Cartesian inputs.

Figures

Figures reproduced from arXiv: 2606.12334 by Bal\'azs Gyenes, Emiliyan Gospodinov, Enrico Krohmer, Gerhard Neumann, Jan Frieling, Nicolas Schreiber, Niklas Freymuth, Xiaogang Jia.

Figure 1
Figure 1. Figure 1: Method Overview. Adding a Fourier feature mapping from Cartesian coordinates into a higher-dimensional feature space improves performance for any point cloud encoder used for diffu￾sion imitation learning. For high-precision policies, the network must learn to condition on fine details in the scene geometry e.g. to decide whether to insert the leg into the slot or reposition it, yet neural networks learn t… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of PointPatch encoder family. We group the input point cloud into neighborhoods N (i) (patch centers indicated in blue on the left). We map point coordinates into Fourier feature space to amplify subtle geometric differences between similar observations. The tokenizer extracts and aggregates features for each neighborhood to produce a set of tokens which are then forwarded to a goal-conditioned di… view at source ↗
Figure 3
Figure 3. Figure 3: Overview of all evalution tasks from RoboCasa, ManiSkill3, and Real World benchmarks. Left: 4 of 16 RoboCasa tasks used for evaluation. Middle: all evaluated ManiSkill3 tasks. Right: starting configurations for all real-world tasks. During policy sampling, i.e. during the reverse process, action samples are guided towards high-density regions of the data distribution by following the score function ∇a log … view at source ↗
Figure 4
Figure 4. Figure 4: Left: Setup for the real-world drawer experiments. Right: RGB views from left, right and gripper cameras and depth view from the left camera. sure unique features. If this is not possible, the input Carte￾sian coordinates can be concatenated with the Fourier fea￾tures, which always yields a unique mapping. 3.5. Data Augmentation The choice of wavelengths λk is essential, as too short wave￾lengths may cause… view at source ↗
Figure 5
Figure 5. Figure 5: Mean success rate across all tasks of 3D encoders with and without Fourier features on RoboCasa (left), ManiSkill3 (middle), and the real world (right). Methods using Fourier Features are marked via hatched bars, and methods are displayed in the order of the legend at the top. Across diverse tasks and architectures, Fourier features provide a consistent and meaningful benefit to task performance. we use th… view at source ↗
Figure 6
Figure 6. Figure 6: Success rates with and without Fourier features on point clouds of different sizes, achieved by voxel downsampling of the observations. Larger point clouds contain richer geometric detail, resulting in a larger benefit for Fourier features. For additional experiments, we use a reduced set of 8 Robo￾Casa tasks, utilizing the Pressing Buttons, Turning Levers, and Twisting Knobs task groups. Unless specified … view at source ↗
Figure 7
Figure 7. Figure 7: Absolute drop in success rate for each RoboCasa task resulting from removing Fourier features from the PointPatch policy architecture (no FF) or removing fine geometric information in the observation using Gaussian jitter (+ Noise(σ=5.0 cm)). Even when high frequency information is removed, Fourier features still provide a meaningful benefit, perhaps by improving the learning dynamics of the policy. PointP… view at source ↗
Figure 8
Figure 8. Figure 8: Graph Fourier spectra of the sensitivities of various archi￾tectures with respect to input point coordinates. During training, sensitivities increase by several orders of magnitude across all fre￾quencies, and Fourier features also increase sensitivity by several more orders of magnitude relative to the baseline. The peak near eigenvalue of 1 indicates the orthogonal response, i.e. the isolated contributio… view at source ↗
Figure 9
Figure 9. Figure 9: Parameter study of different Fourier feature wavelength configurations. Performance is robust to different numbers L of log-spaced wavelengths (left), as well as to the minimum wave￾length λmin (right) around our default of λmin=0.02, L=16 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Overview of RoboCasa Simulation Environments. Example kitchen scenes and tasks illustrating the diversity of household manipulation settings provided by RoboCasa. Category Task Description Insertion CoffeeServeMug Remove the mug from the holder and place it on the counter. CoffeeSetupMug Place the mug into the coffee machine’s mug holder. Pressing Buttons CoffeePressButton Press the button to pour coffee … view at source ↗
Figure 11
Figure 11. Figure 11: Overview of ManiSkill3 Simulation Environments. Example object-centric manipulation tasks illustrating the diversity of interactions supported by ManiSkill3 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qualitative comparison of PointPatch + FF (upper row) and PointPatch (lower row) policies on three RoboCasa tasks. Policies trained without Fourier features have difficulty learning the demonstration data and carrying out complex movements with precision. Time proceeds from left to right in each row. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Graph Spectral Sensitivity Comparison. We consider a toy problem where we study the point-wise gradient of the sum of model outputs of an untrained PointNet on a point cloud of a sphere. By projecting these gradients onto the basis of a Symmetric Normalized Laplacian constructed with Zelnik-Manor local scaling, we observe that the vanilla architecture (dashed) is inherently biased toward low-frequency geo… view at source ↗
read the original abstract

High-precision robotic manipulation requires fine-grained spatial reasoning that is often difficult to achieve with RGB-only policies due to depth ambiguity and perspective scale issues. Policies that leverage 3D information directly, such as those based on point clouds, offer a stronger geometric prior over purely image-based ones, yet their performance remains highly task-dependent. We hypothesize that this discrepancy may be due to the spectral bias of neural networks towards learning low frequency functions, which especially affects architectures conditioned on slow-moving Cartesian features. We thus propose to map point clouds from Cartesian space into high-dimensional Fourier space, effectively equipping the point cloud encoder with direct access to high-frequency features. We experimentally validate the use of Fourier features on challenging manipulation tasks from the RoboCasa and ManiSkill3 benchmarks and on a real robot setup. Despite their simplicity, we find that Fourier features provide significant benefits across diverse encoder architectures and benchmarks and are robust across hyperparameters. Our results indicate that Fourier features let policies leverage geometric details more effectively than Cartesian features, showing their potential as a general-purpose tool for point cloud-based imitation learning. We provide source code and videos on our project page: https://fourier-il.github.io/fourier-il

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The paper claims that neural networks exhibit spectral bias toward low-frequency functions when conditioned on Cartesian point-cloud coordinates, limiting high-precision robotic manipulation; mapping inputs to high-dimensional Fourier features provides direct access to high-frequency geometric details, yielding significant performance gains across encoder architectures on RoboCasa and ManiSkill3 benchmarks plus a real-robot setup. The method is presented as a simple, robust, general-purpose tool for point-cloud imitation learning, with source code released.

Significance. If the reported gains hold and the mechanism is confirmed, the approach offers a lightweight, architecture-agnostic improvement for geometric reasoning in IL without requiring new network designs. Releasing source code and videos strengthens reproducibility and allows direct verification of the empirical claims.

major comments (2)
  1. [Abstract / hypothesis paragraph] Abstract and hypothesis paragraph: the central claim attributes performance gains specifically to mitigation of spectral bias via high-frequency Fourier access, yet no direct test (Fourier analysis of learned mappings, frequency-content ablation, or comparison holding input dimensionality fixed) is described to establish this causal mechanism over alternatives such as increased input dimension or altered optimization dynamics.
  2. [Experimental sections (implied by abstract)] Experimental validation sections: while positive results are reported on multiple benchmarks and a real robot, the abstract provides no quantitative numbers, baseline details, statistical tests, or ablation tables; without these, it is impossible to assess effect sizes, rule out post-hoc tuning, or confirm robustness claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We address each major comment below and will revise the manuscript to improve clarity and strengthen the supporting evidence.

read point-by-point responses

  1. Referee: [Abstract / hypothesis paragraph] Abstract and hypothesis paragraph: the central claim attributes performance gains specifically to mitigation of spectral bias via high-frequency Fourier access, yet no direct test (Fourier analysis of learned mappings, frequency-content ablation, or comparison holding input dimensionality fixed) is described to establish this causal mechanism over alternatives such as increased input dimension or altered optimization dynamics.

    Authors: We agree that a direct test isolating the frequency mechanism from dimensionality or optimization effects would strengthen the causal interpretation. The current results demonstrate consistent gains across encoder architectures and tasks, but we did not perform Fourier analysis of the learned mappings or a fixed-dimensionality random-projection control. In the revision we will add an ablation comparing Fourier features against a high-dimensional random projection baseline (matched dimensionality, no explicit frequency structure) and will report the outcomes. We will also qualify the mechanistic claim in the abstract and hypothesis paragraph to reflect the current level of evidence. revision: yes

  2. Referee: [Experimental sections (implied by abstract)] Experimental validation sections: while positive results are reported on multiple benchmarks and a real robot, the abstract provides no quantitative numbers, baseline details, statistical tests, or ablation tables; without these, it is impossible to assess effect sizes, rule out post-hoc tuning, or confirm robustness claims.

    Authors: The main experimental sections and supplementary material already contain quantitative success rates, baseline comparisons, ablation tables, and robustness checks across hyperparameters and architectures. The abstract, however, is written at a high level and omits specific metrics. We will revise the abstract to include representative quantitative gains (e.g., success-rate improvements on RoboCasa and ManiSkill3) together with a brief statement on statistical robustness and the release of code for reproducibility. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical validation on benchmarks is independent of any derivation or fitted inputs

full rationale

The paper advances a hypothesis that spectral bias explains performance gaps between Cartesian and point-cloud policies, then proposes Fourier feature mapping and reports experimental gains on RoboCasa and ManiSkill3. No equations, parameter fits, or self-citations are invoked to derive the claimed benefits; the results rest on direct policy comparisons rather than any reduction of outputs to inputs by construction. The central claim therefore remains self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that neural networks suffer from spectral bias when conditioned on Cartesian coordinates; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Neural networks exhibit spectral bias towards low-frequency functions, especially when conditioned on slow-moving Cartesian features.
    Explicitly stated as the hypothesis motivating the Fourier mapping.

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discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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