Geometric deep learning for likelihood-free statistical inference

深度学习在解释标准（欧几里德）数据（如1D时间序列数据、2D图像数据和3D视频或体积数据）方面取得了显著的成功，现在在许多情况下超过了人类的准确性。然而，标准的深度学习技术在应用于其他领域定义的数据时会发生灾难性的失败，例如在网络，图形，3D对象或其他流形（如球体）上定义的数据。这催生了几何深度学习领域（Bronstein et al. 2017; Bronstein et al. 2021）。许多科学分析的基础是统计推断，特别是贝叶斯方法。最近，基于模拟的推理技术（cf.无似然性推断（likelihood-free inference）已经出现，并且正在快速发展，用于显式似然性不可用的场景，或者只是为了在时间关键的应用中加速推断（例如，在用于快速电磁跟踪的引力波探测中）。有关简要综述，请参见Cranmer et al. 2020。这些技术建立在强大的机器学习模型的概率分布（例如Papamakarios等人。2021）。当前项目的重点是开发集成的几何深度学习和基于模拟的推理技术（参见。无似然性推断）用于在复杂域（例如球形流形和图形）上定义的数据。这将涉及开发几何仿真，成像和推理技术，作为总体推理管道的一部分。这种管道的一个关键组成部分将是几何散射网络表示（Mallat 2012;麦克尤恩et al. 2021）。所开发的技术将应用于宇宙学、医学成像、电子物理学等领域;我们将与其他人合作，将其应用于上述领域。

Deep learning has been remarkably successful in the interpretation of standard (Euclidean) data, such as 1D time series data, 2D image data, and 3D video or volumetric data, now exceeding human accuracy in many cases. However, standard deep learning techniques fail catastrophically when applied to data defined on other domains, such as data defined over networks, graphs, 3D objects, or other manifolds such as the sphere. This has given rise to the field of geometric deep learning (Bronstein et al. 2017; Bronstein et al. 2021).The bedrock of much scientific analysis is statistical inference, in particular Bayesian approaches. Recently, simulation-based inference techniques (cf. likelihood-free inference) have emerged, and are rapidly evolving, for scenarios where an explicit likelihood is not available or simply to speed up inference in time-critical applications (e.g. in gravitational wave detection for rapid electromagnetic follow-up). For a brief review see Cranmer et al. 2020. These techniques build on powerful machine learning models for probability distributions (e.g. Papamakarios et al. 2021).The focus of the current project is to develop integrated geometric deep learning and simulation-based inference techniques (cf. likelihood-free inference) for data defined over complex domains, such as spherical manifolds and graphs. This will involve developing geometric emulation, imaging, and inference techniques as part of a overarching inference pipeline. A key component of such a pipeline will be geometric scattering network representations (Mallat 2012; McEwen et al. 2021). The techniques developed will have application in cosmology, medical imaging, geophysics and beyond; we will collaborate with others to apply them in the aforementioned fields.