Approximate Manifold Sampling Robust Bayesian Inference for Machine Learning

在机器学习、统计学和统计物理中产生的许多应用中，从集中在低维子流形上的概率密度进行有效采样是至关重要的。由于问题的极端各向异性和高维性，以及变量之间的相关性，这项任务特别具有挑战性。我们提出了一类新的定制MCMC算法，旨在有效地从这些密度中进行采样，并展示了它们相对于通用和专用采样器的计算优势。此外，我们还为积分器和马尔可夫片段的开发做出了贡献，这是一种用于贝叶斯推理和机器学习的特定类型的通用顺序算法，可以利用积分器空间的几何形状，对步长和积分步数的选择具有高度的健壮性，并且自然有助于并行化。在此基础上，我们提出了一种特别适合近似流形采样的序贯算法。

Efficient sampling from probability densities concentrated around a lower-dimensional submanifold is crucial in numerous applications arising in machine learning, statistics, and statistical physics. This task is particularly challenging due to the extreme anisotropy and high-dimensionality of the problem, and the correlation between the variables. We propose a novel family of bespoke MCMC algorithms designed to sample efficiently from these densities and show their computational superiority to general purpose and specialized samplers. Furthermore, we contribute to the development of integrator and Markov snippets, which are a particular class of general-purpose sequential algorithms for Bayesian inference and machine learning that can leverage the geometry of the space with integrators, is highly robust to the choice of the step size and the number of integration steps, and naturally lends itself to parallelisation. Building on these foundations, we present a sequential algorithm that is particularly well-suited to approximate manifold sampling.