Stochastic Numerics for Sampling on Manifolds

数字时代导致高度结构化数据的可用性越来越高，例如社交媒体图表和网络、在线零售和流媒体平台的评级和推荐系统数据以及高分辨率医学图像。这些数据的特点是具有重要的约束条件（不是每个人，只有朋友和家人在网络中形成一个群体；图像的形状在图像的旋转下是不变的），并且与存储和分析这些数据相关的绝对规模和复杂性需要使用概率模型来模拟数据生成的方式。对于高度结构化的数据，概率模型的成功实际应用的基础是从几何约束空间（称为流形）中采样或生成随机数据。基于理论保证的最先进的有效采样，在这个新生的领域中，仅限于流形光滑而没有边界的情况，或者采样分布属于特别适合理论分析的一类。这排除了人工智能和统计应用中经常遇到的许多重要问题，包括低秩矩阵补全（预测Netflix电影的用户评分）和分析物体形状（从医学图像中计算代表性肿瘤形状）。为此，这个及时项目的首要目标是开发和分析使用遍历随机微分方程从一般类型的流形和分布中抽样的方法。该项目定位于随机、数值分析和几何的界面，将对流形上SDEs的数值方法的进步做出重大贡献，从而开辟了有效分析复杂几何数据的可能性。

The digital era has led to the increasing availability of highly-structured data such as social media graphs and networks, ratings and recommender system data from online retail and streaming platforms, and high-resolution medical images. Such data are characterised by non-trivial constraints (not everyone but only friends and family form a group within a network; shape of the imaged brain is unchanged under rotations of the image), and the sheer scale and complexity associated with storing and analysing such data necessitate the use of probabilistic models to mimic the manner in which the data were generated. Fundamental to successful practical use of probabilistic models for highly-structured data is sampling, or generating random data, from geometrically constrained spaces known as manifolds. State-of-the-art in efficient sampling, backed by theoretical guarantees, within this nascent area is restricted to cases where the manifold is smooth without a boundary or the sampling distribution belongs to a class that is particularly amenable for theoretical analysis. This excludes many important problems one routinely encounters in AI and statistical applications, including low-rank matrix completion (predicting user ratings for Netflix movies) and analysing shapes of objects (computing a representative tumour shape from medical images). To this end, the overarching goal of this timely project is to develop and analyse methods to sample from a general class of manifolds and distributions using ergodic stochastic differential equations. Positioned at the interface of stochastics, numerical analysis and geometry, the project will make a major contribution to the advancement of numerical methods for SDEs on manifolds and thus open up the possibility to efficiently analyse complex, geometric data.