Manifold Gaussian Processes for Regression (1402.5876v4)

Abstract: 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.

• The paper presents a dual learning approach that simultaneously optimizes data transformation and GP regression to better capture non-smooth functions.
• It proposes a flexible covariance design that adapts to multi-scale and discontinuous data, outperforming traditional GP models.
• Empirical evaluations on synthetic and robotic datasets validate its superior predictive accuracy and robustness compared to benchmark methods.

Analysis of "Manifold Gaussian Processes for Regression"

The paper "Manifold Gaussian Processes for Regression" addresses limitations encountered in conventional Gaussian Process (GP) models when tasked with modeling complex, non-differentiable functions. Conventional GPs, while powerful nonparametric Bayesian tools for regression, typically rely on smoothness assumptions embedded within their covariance functions, like the standard squared exponential covariance function. Such assumptions often restrict their applicability in cases where the data exhibit complex or sharp discontinuities, such as ground contacts in robotic locomotion.

Core Contributions and Methodology

The paper introduces Manifold Gaussian Processes (mGP), a novel supervised regression framework that concurrently learns a data transformation and conducts GP regression. This dual-step process involves transforming the input data into a feature space where the GP is then applied to map from this feature space to the observed space. Key highlights of the mGP framework include:

• Joint Learning of Data Representation and Regression: By learning an appropriate feature space representation alongside GP regression, the mGP framework alleviates the limitations of fixed covariance functions. This joint optimization targets the overall regression objective, potentially leading to more robust modeling of complex functions.
• Model Flexibility: The mGP inherently enables the formulation of more flexible covariance functions, accommodating data irregularities through transformations determined from data-driven, supervised learning—as opposed to solely relying on predefined structures.
• Comparison against Traditional Methods: The authors formally compare mGP against traditional GPs using both SE-ARD and neural network covariance functions. They also evaluate other input transformation approaches like PCA and random embeddings followed by GP regression.

Numerical Results and Implications

The authors demonstrate mGP's efficacy through experiments on synthetic datasets characterized by sharp discontinuities and multi-scale behaviors, as well as a real-world robotics dataset involving a bipedal walker. Key findings include:

1. Robustness in Non-Smooth Functions: mGP models functions with sharp discontinuities better than standard GP, as observed in the step function results. The learned feature mapping captures discontinuities more effectively, smoothing the regression task.
2. Handling Multiple Length-Scales: mGP outperforms traditional GPs in modeling functions with inherent multi-scale properties by transforming the input space to align with GP covariance assumptions, leading to improved predictive accuracy and lower error metrics.
3. Application to Robotics: In settings involving physical interactions, such as robotic locomotion, the mGP framework exhibits superior performance in terms of both Negative Log Marginal Likelihood and Negative Log Predictive Probability, illustrating its practical applicability in real-world complex systems.

Theoretical and Practical Implications

The mGP method opens new avenues for regression tasks where conventional GP models underperform due to rigid prior assumptions. It bridges the gap between feature learning in neural networks and probabilistic modeling in GPs, suggesting potential for speech synthesis, complex robotic system modeling, reinforcement learning, and beyond. Future works could delve into integrating probabilistic mappings into the transformation layer to further align the benefits of a fully Bayesian treatment with the predictive prowess of GPs.

Conclusion

This work provides significant contributions to machine learning and probabilistic modeling by presenting a framework that effectively couples data-driven feature representations with Gaussian Processes. The results underscore the benefits of integrating supervised learning objectives into covariance function design, facilitating enhanced modeling capacity for complex, structured data. While promising, the approach invites further exploration of optimization strategies and applications to unlock its full potential in diverse machine learning domains.