Geometric deep learning for likelihood-free statistical inference

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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.