Gaussian Processes on Distributions based on Regularized Optimal Transport

« In recent years, the intersection of Gaussian Processes (GPs) and Optimal Transport (OT) has emerged as a fertile ground for innovation in statistical learning. Gaussian Processes, known for their flexibility and robustness, are a cornerstone in the field of machine learning, providing a probabilistic approach to learning in kernel-based models. On the other hand, Optimal Transport offers a powerful framework for comparing probability distributions, with applications ranging from economics to image processing.

The development presented in François Bachoc et al.’s paper[1] introduces a significant advancement in statistical analysis and algorithm design by creating a new kernel using Gaussian Processes, enhanced by techniques from Regularized Optimal Transport. This report offers a comprehensive overview of this novel kernel, detailing its conceptual foundation and potential applications in various scientific and technological fields.

In this project, we are collaborating with Safran, a leading aerospace manufacturer, to pioneer innovative approaches to designing aircraft engine components, specifically focusing on the blades of a propeller. Our objective is to leverage advanced machine learning techniques, particularly in regression analysis, to optimize the efficiency of these critical components.

The target variable in our machine learning model is the efficiency metric of the blade, a quantifiable measure of performance under various operational conditions. Intriguingly, the feature variable is the probabilistic distribution of the blade’s structure, represented as a point cloud in a three-dimensional space (R3). This representation is not just a mere geometrical depiction but encapsulates the intricate variability and physical characteristics of the blade. Through analyzing this relationship between shape distribution and efficiency, our goal is to identify underlying patterns and gain insights that could inform the engineering of propeller blades with enhanced efficiency and durability. This research represents a significant step in aerospace engineering, merging advanced statistical methods with engineering challenges, and contributes to the ongoing evolution of aircraft propulsion technology.

To approach François Bachoc et al.’s paper with a comprehensive understanding, we will first elucidate the methods of kernels in statistical learning, with a specific focus on kernel ridge regression. Additionally, we will revisit the genesis of optimal transport and its more recent formulation in an entropic framework, along with its resolution using the Sinkhorn algorithm. Subsequently, we will provide a detailed exposition of the kernel developed within the paper under examination. »

[1] François Bachoc et al. Gaussian Processes on Distributions based on Regularized Optimal Transport.2022. arXiv: 2210.06574 [stat.ML].