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Manifold Gaussian Processes for Regression
Roberto Calandra, Jan Peters, Carl Edward Rasmussen, Marc Peter Deisenroth
TL;DR
Standard GP covariance functions can be too restrictive for complex, non-smooth functions, and unsupervised feature transformations may not suit the regression objective. The paper proposes Manifold Gaussian Processes, which jointly learn a supervised transformation and GP regression; proof-of-concept evaluations report success on step functions, robot-contact effects, and other challenging functions while matching standard GPs on simple sinusoidal functions.
Problem
Standard GP covariance functions encode smoothness assumptions that can be inadequate for complex functions, while unsupervised input transformations may be suboptimal for the overall regression task.
Method
The mGP jointly learns a parametrized input transformation M into a feature space and GP regression G from that space to the observed outputs.
Results
The mGP successfully modeled highly non-linear functions, including step functions and robot locomotion effects from ground contacts, where standard GPs fail.
Takeaways & Limitations
The mGP provides enhanced modeling capabilities for robot modeling, reinforcement learning, and Bayesian optimization.
Abstract
from arXiv · showhide
Off-the-shelf Gaussian Process (GP) covariance functions encode smoothness assumptions on the structure of the function to be modeled. To model complex and non-differentiable functions, these smoothness assumptions are often too restrictive. One way to alleviate this limitation is to find a different representation of the data by introducing a feature space. This feature space is often learned in an unsupervised way, which might lead to data representations that are not useful for the overall regression task. In this paper, we propose Manifold Gaussian Processes, a novel supervised method that jointly learns a transformation of the data into a feature space and a GP regression from the feature space to observed space. The Manifold GP is a full GP and allows to learn data representations, which are useful for the overall regression task. As a proof-of-concept, we evaluate our approach on complex non-smooth functions where standard GPs perform poorly, such as step functions and robotics tasks with contacts.
1. Introduction
Standard GP covariance functions impose assumptions such as smoothness and stationarity that can be inadequate for discontinuous or otherwise complex functions. The mGP instead jointly learns a supervised feature transformation and GP regression to obtain representations suited to the regression task.
- Squared exponential covariance functions assume properties such as smoothness and stationarity, which may be inadequate for functions violating common smoothness assumptions.Ground contacts in robot locomotion are given as an example where generic covariance functions may fail.
- Combining standard covariance functions can produce complex kernels, but the result remains limited by the properties of its component covariance functions.
- Input transformations followed by standard covariance functions are equivalent to GP regression with a covariance function that explicitly depends on the transformation.Examples include output warping, input transformations, periodic covariance functions, normalization, and PCA.
- Common input transformations often use heuristics or unsupervised objectives, which may be suboptimal for the overall regression task.Unsupervised and supervised objectives need not align.
- The mGP jointly learns a data transformation into a feature space and GP regression from that feature space to the observed space.It remains a full GP and retains straightforward prior-mean incorporation and faithful model-uncertainty representation.
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The paper contrasts standard supervised regression, latent-space decompositions, and feature-space transformations, emphasizing that mGP jointly learns the transformation and regression mappings. Related work includes transformations with purposes narrower than the general transformations allowed by mGP.
- The Manifold GP jointly learns the input transformation M and regression task G.
- A latent-space decomposition represents F as G ◦ M, with M mapping inputs to a latent space and G mapping that space to outputs.
- Snoek et al. transformed inputs with a Beta distribution to account for skewness, whereas mGP allows a more general class of transformations.
2. Manifold Gaussian Processes
This section introduces Manifold Gaussian Processes as full GPs that jointly learn an input transformation and regression mapping, using a supervised objective to shape the feature representation. The model retains standard GP uncertainty properties while defining a flexible covariance through the learned feature space.
- Model formulation: The mGP represents regression as a composition F = G ◦M, where M maps X into H and G maps H to the observed output space Y.This decomposition introduces an auxiliary feature space for nonlinear regression.
- Model motivation: Manifold Gaussian Processes jointly learn a feature mapping M and GP regression G using the same supervised objective for the overall regression F = G ◦M.The mapping transforms inputs into a feature space H, which serves as the domain for GP regression.
- Gaussian Process background: Standard GP regression is defined by a mean function and covariance function, with hyperparameters selected by minimizing the negative log marginal likelihood.The predictive distribution uses the training data, kernel matrix, test covariance, and measurement-noise variance.
- Model formulation: The induced covariance is ˜k(xp, xq) = k(M(xp), M(xq)), so the GP kernel operates on the Q-dimensional feature space H = M(X).The resulting function is a valid GP covariance construction according to the cited formulation.
- Training: Training jointly optimizes transformation parameters θM and GP hyperparameters θG by minimizing the negative log marginal likelihood.Gradients for θM use the chain rule, with the transformation-dependent term entering through H = M(X).
- Input transformation: The approach can use any deterministic parametric transformation M, with the paper focusing on multilayer neural networks whose weights and biases are trained through backpropagation.Each layer applies a transfer function to an affine transformation of its input.
3. Experimental Results
Across discontinuous functions, multiple length-scales, and bipedal locomotion, mGPs use learned transformations to improve regression with standard GP covariance functions. The experiments report better modeling of discontinuities, frequency structure, and robot dynamics than the evaluated alternatives.
- Experimental scope: mGPs were evaluated on discontinuous functions, multiple natural length-scales, and bipedal robot locomotion with challenging ground contacts.The study compares standard GPs, unsupervised feature extraction followed by GP regression, and mGP variants.
- Step Function: On the step function, the nonlinear mGP outperformed the other models on both NLML and NLPP per data point.The log-sigmoid transformation captured discontinuities better, whereas linear transformations did not substantially change the function.
- Multiple Length-Scales: For the multiple-length-scale function, both mGP variants transformed the input toward a more compact frequency support and outperformed the evaluated alternatives.The authors attribute this to transformations that aim to equalize natural frequencies.
- Bipedal Robot Locomotion: In the robot task, mGP models fit the data better than the other models while standard GPs still predicted the knee angle relatively well.The dataset comes from the bio-inspired bipedal walker Fox and includes joint angles and ground-contact signals.
- Bipedal Robot Locomotion: The mGP showed larger predictive variance during fast leg-swing movements but captured the data's structure and regularity better.The increased variance occurs in areas associated with fast movement.
4. Discussion
The mGP combines supervised feature learning with GP regression and captures challenging robot-locomotion structure while remaining competitive on smooth functions.
- Feature Learning: The mGP is designed to exploit GP feature learning despite prior concerns that GPs are unsuited for extracting complex features.Related work includes deep GPs, which can also be used in supervised regression.
- Bipedal Robot Locomotion: The mGP captures robot-locomotion structure better than GPs using SE-ARD or NN covariance functions.Its predictive variance is larger during fast leg-swinging movements.
- Bipedal Robot Locomotion: Table 3 reports that the mGP models bipedal locomotion better than the other models in training and test evaluation.The standard SE-ARD and NN GPs predict knee angle relatively well.
- Bipedal Robot Locomotion: The mGP learns a smooth feature-space representation of bipedal robot locomotion, with color indicating phase during a single step.
- Smooth Functions: On a smooth sinusoidal function, the mGP performs as well as a standard GP, with no relevant difference reported in Table 4.This indicates that the model’s expressiveness does not prevent it from handling easy regression tasks.
- Limitations: Increasing mapping parameters increases covariance flexibility, but when parameters exceed dataset size, the mGP becomes prone to over-fitting.For step functions, sparse data near discontinuities led to learned jumps slightly offset from their actual locations.
5. Conclusion
The conclusion presents mGP as a jointly and supervisedly learned feature mapping plus GP regression for complex functions. It reports successful modeling of discontinuities and robot ground-contact effects where standard GPs fail.
- Manifold Gaussian Processes decompose regression into jointly supervised learning of a feature-space mapping and GP regression to observed outputs.The model remains a valid GP with a more expressive covariance function.
- The mGP successfully models highly non-linear functions, including step functions and ground-contact effects in robot locomotion, where standard GPs fail.