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Fast Non-Parametric Heteroscedastic Imitation Learning With Geometric Priors

Maximilian Mühlbauer, Arne Sachtler, Markus Knauer, Cem Küçükgenç, Yanlong Huang, Alin Albu-Schäffer, João Silvério

arXiv:2610.08650v1cs.RO

TL;DR

Existing kernel-based imitation-learning methods do not fully address manifold geometry, uncertainty estimation, and adaptation needs. This paper presents a Riemannian non-parametric method with heteroscedastic uncertainty and task parameterization; in an insertion task, it succeeds across tested combinations, including an unseen 180° measurement-station rotation.

  • Problem

    Existing kernel-based approaches are limited in their treatment of manifold geometry, uncertainty estimation, and adaptation, despite the need for data-efficient learning and fast adaptation.

    Source: §I · p. 1

  • Method

    The paper develops a Riemannian non-parametric movement-modeling approach with aleatoric and epistemic uncertainties, adaptation schemes, task parameterization, and diagonal kernels.

    Source: §I · p. 2, §VIII · p. 9

  • Results

    In the insertion task, the approach succeeds for all tested combinations and produces smooth trajectories with the measurement station rotated by 180°, outside the training data.

    Source: B. Comparison to Vanilla… · p. 8

  • Takeaways & Limitations

    The authors identify shared control with humans in the loop as a suitable setting when explainability and fast computation are required.

    Source: §VII · p. 9

    Generalization is limited to areas close to available data; farther away, predictions converge to the process mean while epistemic uncertainty becomes high.

    Source: §VII · p. 9

Abstract

from arXiv · show

When learning probabilistic policies from human demonstrations, data-efficient learning and fast adaptations to new scenarios are key requirements. One popular way to achieve intuitive and reliable adaptations is through non-parametric, typically kernel-based, methods. However, existing solutions either fail to account for the geometry of manifolds common in robotics, limiting data efficiency, or, when geometry-aware, provide unreliable uncertainty estimates or require retraining to adapt. We propose a non-parametric approach leveraging geometric priors in scenarios of data scarcity and heteroscedastic uncertainties for probabilistic modeling. We utilize the method to formulate policies based on time or robot state, where non-separable diagonal kernels allow capturing uncertainty relations between degrees of freedom for same-sized in- and outputs. Fast updates, requiring less than 3 ms for a trajectory involving both position and orientation are possible through an optimized formulation. Our approach supports both manifold-valued input and manifold-valued output with large orientation changes. Using task parameterization, adaptation to different object poses is easily possible. We evaluate the approach on a set of toy examples and on real robot manipulation tasks both in autonomous execution and in shared control scenarios.

I. INTRODUCTION

The introduction motivates geometry-aware, non-parametric imitation learning that models both uncertainty types and supports adaptation, while identifying limitations in existing kernel methods. It presents Riemannian modeling, task parameterization, diagonal kernels, and computational improvements as responses to these gaps.

I. INTRODUCTION

  • Existing kernel methods are limited to Euclidean or single-tangent-space formulations, while other on-manifold approaches may be slow, hard to combine across product manifolds, or omit epistemic uncertainty.The paper’s approach uses Riemannian Gaussians to avoid discrete linearization points and their covariance distortions.Source: §I · p. 1
  • Non-separable diagonal kernels extend state-based Dynamical Systems to model uncertainty relations across same-sized inputs and outputs.Existing separable kernel methods predict only scalar-valued epistemic uncertainty.Source: §I · pp. 1–2
  • Cholesky decomposition improves KMP runtime over vanilla KMP and while retaining flexibility to add or remove data points.The paper notes that runtime grows with the number of data points and that point changes otherwise require a large matrix inversion.Source: §I · p. 2
  • The contributions include Riemannian movement primitives with aleatoric and epistemic heteroscedastic uncertainty, two adaptation schemes, task parameterization on SE(3), and faster computations.The method also integrates diagonal matrix-valued kernels to predict matrix-valued epistemic uncertainty.Source: §I · p. 2

A. Riemannian Manifolds, Optimization and Statistics

  • Pullback metrics from different charts on SE(3) induce distinct Riemannian geometries and Gaussian shapes; the Riemannian Gaussian is well-defined only for covariances small enough that Exp is a diffeomorphism.The log map supplies geodesic distances for the Riemannian Gaussian.Source: A. Riemannian Manifolds, Optimization… · p. 2

B. Probabilistic Learning from Demonstration

The section develops probabilistic movement modeling from demonstrations, combining data uncertainty with a Riemannian KMP formulation for manifold-valued trajectories. Its prediction includes Euclidean KMP as a special case, while antipodal queries can make the predicted mean imprecise.

B. Probabilistic Learning from Demonstration

  • KMPs model both aleatoric and epistemic uncertainty from probabilistic references, combining uncertainty information associated with demonstrations and predictions.The section contrasts GMMs, which model aleatoric uncertainty, with GPs, which model epistemic uncertainty.Source: B. Probabilistic Learning from… · pp. 2–3

A. Movement Primitive Prediction

  • The Riemannian derivation uses manifold-valued reference means and tangent-space covariances without linearizing the problem in one arbitrary tangent space.It permits general Riemannian manifolds when parallel transport can be expressed using left-multiplication of a rotation matrix.Source: A. Movement Primitive Prediction · p. 3
  • On R^n, the derived expectation matches Euclidean KMP, so the Riemannian prediction contains the Euclidean variant as a special case.The covariance is also derived from the Gaussian-conditioning formulation.Source: A. Movement Primitive Prediction · p. 3
  • Near antipodal points, such as equivalent −π and π orientation changes on SO(3), the predicted mean can be imprecise; recentering around the Fréchet mean can often avoid this.For SO(3), the method can instead use S3's double coverage to move the discontinuity away from query inputs; larger orientation changes may require multiple instances.Source: A. Movement Primitive Prediction · p. 3

B. Movement Primitive Adaptation

The section presents two ways to adapt KMPs to new situations: adding via-point references or applying uncertainty-aware null-space deformations. Their covariance and weight-based formulations govern how closely the prediction follows a requested change.

B. Movement Primitive Adaptation

  • The approaches suit different use cases: via points add explicit desired references, while null-space modulation makes changes relative to the learned covariance.The figure illustrates both a via point and a null-space action for a pose trajectory.Source: B. Movement Primitive Adaptation · pp. 3–4, Figure 3 · p. 4

a) Via-point based adaptation

  • Via-point adaptation extends the reference with desired inputs, outputs, and covariances; lower covariances pull predictions closer to the requested output.Adding points conventionally requires recomputing a costly matrix inverse, motivating faster update alternatives.Source: B. Movement Primitive Adaptation · p. 4

b) Modulation with uncertainty-aware deformations

  • Soft null-space adaptation penalizes deviations from desired weights, with the learned covariance guiding the deformation.The desired change is scaled in practice, and its log-map displacement can serve as an action space for covariance-guided exploration.Source: B. Movement Primitive Adaptation · p. 4

C. Task Parameterization

The evaluation reports that Riemannian KMP predicts orientations correctly where the STS KMP has discontinuity-related failures, while retaining good positional accuracy. The section also describes task parameterization for adapting learned movement models to new object poses and illustrates prediction on S2.

C. Task Parameterization

  • Task parameterization learns local movement models relative to SE(3) coordinate origins, which can be reset at execution time for new object positions or orientations.Models on M2 or M3 cannot have their origin translated without invalidating covariances, so their outputs are transformed and fused using a shared origin.Source: C. Task Parameterization · p. 4

D. Evaluation in Toy Examples

  • The Riemannian KMP predicts all tested orientations correctly, whereas STS KMP fails for some; all models achieve good position accuracy, and Riemannian KMP is slightly more accurate than Nadaraya–Watson.The comparison averages positional and rotational error over 200 query points; the table caption identifies the spiral-trajectory comparison.Source: Table I · p. 4, D. Evaluation in Toy Examples · p. 5
  • The S2 evaluation shows the method is not limited to manifolds with a Lie-group structure.The handwriting example projects the letter B onto S2 and depicts GMM means and covariance ellipsoids.Source: Figure 4 · p. 4, D. Evaluation in Toy Examples · p. 5

IV. MATRIX-VALUED KERNELS

Separable kernels assign every degree of freedom the same scalar uncertainty, while the proposed diagonal kernels model per-degree-of-freedom uncertainty within the Riemannian formulation. Their validity depends on the manifold and, for RBF kernels, suitable length scales.

A. Separable Kernels

  • Separable kernels produce the same scalar-valued epistemic uncertainty for every degree of freedom rather than measuring uncertainty per degree of freedom.They model output interactions without reference to the input space.Source: §IV · p. 5, A. Separable Kernels · p. 5

B. Diagonal Kernels

  • The proposed diagonal kernel multiplies a scalar-output kernel by a diagonal kernel to compute degree-of-freedom-aware covariance.Its diagonal entries are calculated from a kernel with inputs on R, and the construction is positive definite.Source: B. Diagonal Kernels · p. 5
  • The diagonal kernel is expressed in the tangent space through the manifold logarithm and is formulated for R^n, S^1, S^3, and their products.Valid length scales for the manifold kernels are determined through sampling.Source: B. Diagonal Kernels · p. 5
  • The construction does not yield a valid positive-definite kernel on S^2 because the required logarithm-map property fails there.For RBF kernels, the Riemannian distance does not ensure positive definiteness at every length scale; the method uses valid sampled length scales.Source: B. Diagonal Kernels · p. 5, A. Separable Kernels · p. 5

C. Evaluation in Toy Examples

The toy example shows that diagonal kernels represent degree-of-freedom-specific uncertainty and, when fused with a stabilizing policy, can preserve the certain learned component longer than separable kernels. The computational improvements use Cholesky factorization to speed prediction and measurement updates.

C. Evaluation in Toy Examples

  • With stabilizing-policy fusion, diagonal kernels keep the certain learned component active longer than separable kernels [12].The policy can proceed while the stabilizing policy corrects perturbed degrees of freedom, provided perturbations are not too large.Source: Figure 5 · p. 6, C. Evaluation in Toy Examples · p. 6
  • The diagonal kernel captures greater covariance along x2 than x1 when moving away from known points along x2, indicating higher uncertainty for that degree of freedom.The comparison is between the per-degree-of-freedom diagonal kernel and the scalar covariance from the separable kernel.Source: C. Evaluation in Toy Examples · pp. 5–6

V. COMPUTATIONAL IMPROVEMENTS

  • Cholesky factorization avoids explicit inverses to improve prediction speed and measurement additions and removals in the manifold-aware formulation.The method also introduces a mechanism to restore the decomposition when removing points from the reference data.Source: §V · p. 6

A. Cholesky-factorized KMP (cKMP)

Cholesky factorization replaces direct matrix inversion in KMP prediction and supports more efficient updates as measurements are added or removed. Appending measurements reuses the existing factorization, while removal restores its triangular form when needed.

A. Cholesky-factorized KMP (cKMP)

  • Cholesky-factorized KMP solves prediction systems by forward and backward substitution, adapting the Cholesky-factorized GP approach to manifold observations with Log and Exp.The formulation significantly improves prediction times over vanilla KMP.Source: A. Cholesky-factorized KMP (cKMP) · p. 6

B. Appending Measurements

  • Appending measurements updates the existing Cholesky factorization, giving a computational advantage over recomputing the inverse and running faster than iKMP.The update costs O(N^2) for the original factor size N and O(D^3) for added measurements of dimension D.Source: B. Appending Measurements · pp. 6–7, Figure 6 · p. 7

C. Removing Measurements

  • Removing measurements at the end requires only deleting the corresponding rows and columns, while removing an earlier measurement requires restoring the factor’s lower-triangular form.The figure reports indistinguishable trajectories across calculation methods, indicating numerical equivalence.Source: C. Removing Measurements · p. 7, Figure 6 · p. 7

D. Evaluation in Toy Examples

The evaluations report faster computation with cKMP and test task-parameterized KMP on bottle placement and ring insertion. The Riemannian model succeeds on an unseen 180° station orientation where the vanilla implementation’s orientation prediction fails.

D. Evaluation in Toy Examples

  • cKMP remains significantly faster than vanilla KMP because its covariance calculation is quicker, despite precomputing α for the vanilla and incremental formulations.The timing evaluation covers fitting, adding and removing points, and prediction for letter writing and a 500-pose trajectory.Source: D. Evaluation in Toy Examples · p. 7, Table IV · p. 7

VI. EVALUATION ON A ROBOTIC SYSTEM

  • The robotic evaluation combines Riemannian prediction with task parameterization, matrix-valued kernels, trajectory adaptation, and epistemic uncertainty across distinct experiments.The experiments include autonomous execution and shared control scenarios.Source: §VI · p. 7

A. Task Parameterization on Different Manifolds

  • Fusing a bottle-pick model with high angular covariance and a constrained-placement model with lower angular covariance supports picking from different angles while constraining placement.Both models are queried where data are available, so aleatoric uncertainty determines the fusion and stiffness.Source: A. Task Parameterization on Different… · pp. 7–8, Figure 7 · p. 8

B. Comparison to Vanilla Task-Parameterized KMPs

  • The Riemannian task-parameterized KMP succeeds for all tested ring-insertion combinations, including a smooth trajectory at the unseen 180° station orientation, while vanilla KMP fails after adding 90° training samples.The vanilla implementation’s quaternion treatment neglects geometry, producing meaningless orientations even after renormalization.Source: B. Comparison to Vanilla… · p. 8, Figure 8 · p. 8

C. Dynamical Systems with Matrix-valued Kernels

The evaluations connect per-degree-of-freedom uncertainty to perturbed policy behavior and demonstrate continuous trajectory adaptation and uncertainty-aware handling of missing data. The discussion and conclusion describe the method’s supported scope and remaining boundaries.

C. Dynamical Systems with Matrix-valued Kernels

  • Under a rotational perturbation, the diagonal Matérn kernel accelerates motion along y sooner than the separable kernel because it models per-DoF epistemic covariance.The stabilizing policy corrects rotation around z similarly with both kernels; epistemic uncertainty also lets it activate under perturbations, unlike aleatoric-only GMM uncertainty.Source: C. Dynamical Systems with… · pp. 8–9

D. Adapting Kernelized Trajectory Fixtures

  • Continuous adaptation updates 100 poses in under 10 ms, compared with 40 ms without the computational improvements, while via points make precise changes and null-space modifications preserve learned covariance.The update rate is deliberately limited to 2 Hz to avoid closing the control loop through adaptation.Source: D. Adapting Kernelized Trajectory… · p. 9, Figure 11 · p. 9

E. Kernelized Trajectory Fixtures With Missing Data

  • With missing demonstration data, KMP predicts increased covariance while the GMM incorrectly predicts high certainty, enabling downstream algorithms to reduce robot stiffness in uncertain regions.The covariance can be used by downstream algorithms to reduce stiffness.Source: E. Kernelized Trajectory Fixtures With… · p. 9, Figure 12 · p. 9

VII. DISCUSSION

  • The method models motion and orientation on Riemannian manifolds, supports arbitrary orientation changes, and can leverage different manifold properties through extended task parameterization.The discussion identifies explainability, fast computation, few demonstrations, fast adaptation, and precision requirements as relevant application conditions.Source: §VII · p. 9

VIII. CONCLUSION

  • The conclusion presents a diagonal kernel for per-DoF epistemic uncertainty and Cholesky-based computation with two adaptation schemes for real-time trajectory modification.Higher-dimensional inputs such as images require preprocessing; future work proposes kernels that predict off-diagonal epistemic uncertainty.Source: §VIII · pp. 9–10

APPENDIX A FUNCTION-SPACE DERIVATION OF RIEMANNIAN KMPS

The appendix derives Riemannian KMP prediction in function space through on-manifold Gaussian conditioning. It relates the prediction to conditioning while specifying how manifold tangent spaces and covariance assumptions shape the derivation.

APPENDIX A FUNCTION-SPACE DERIVATION OF RIEMANNIAN KMPS

  • The function-space derivation reformulates KMP prediction as Gaussian conditioning and reports the same result as the earlier weight-space derivation.The appendix first revisits Gaussian conditioning using on-manifold optimization.Source: Appendix A · p. 11

A. On-Manifold Gaussian Conditioning

  • On-manifold Gaussian conditioning iteratively updates the output estimate with Exp after a Gauss–Newton step, using parallel transport to move updates between tangent spaces.The derivation uses Log and Exp for manifold-valued observations and transports covariance to the output tangent space.Source: A. On-Manifold Gaussian Conditioning · p. 11

B. Movement Primitive Prediction through Conditioning

  • The movement-primitive conditioning derivation assumes the observation covariance and kernel result lie in their respective tangent spaces, so its equations do not require parallel transport.The covariance is block-diagonal across training observations, and the prediction is formulated as a noisy function with process mean e.Source: B. Movement Primitive Prediction… · p. 11

APPENDIX B SUPPLEMENTAL DERIVATIONS

The appendix derives the Riemannian null-space prediction by adding a penalty that keeps weights near desired values, then solving the resulting cost for its optimum.

APPENDIX B SUPPLEMENTAL DERIVATIONS

B. Restoring the Lower-triangular Form of L

Removing a measurement before the end can leave the permuted Cholesky factor with entries beyond its lower triangle; Givens rotations restore its triangular form.

B. Restoring the Lower-triangular Form of L

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