Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a universal causal geometric DL framework in which the user specifies a suitable pair of metric spaces $\mathscr{X}$ and $\mathscr{Y}$ and our framework returns a DL model capable of causally approximating any ``regular'' map sending time series in $\mathscr{X}^{\mathbb{Z}}$ to time series in $\mathscr{Y}^{\mathbb{Z}}$ while respecting their forward flow of information throughout time. Suitable geometries on $\mathscr{Y}$ include various (adapted) Wasserstein spaces arising in optimal stopping problems, a variety of statistical manifolds describing the conditional distribution of continuous-time finite state Markov chains, and all Fréchet spaces admitting a Schauder basis, e.g. as in classical finance. Suitable spaces $\mathscr{X}$ are compact subsets of any Euclidean space. Our results all quantitatively express the number of parameters needed for our DL model to achieve a given approximation error as a function of the target map's regularity and the geometric structure both of $\mathscr{X}$ and of $\mathscr{Y}$. Even when omitting any temporal structure, our universal approximation theorems are the first guarantees that Hölder functions, defined between such $\mathscr{X}$ and $\mathscr{Y}$ can be approximated by DL models.
Designing monitoring strategies for deployed machine learning algorithms: navigating performativity through a causal lens
University of California, San Francisco（加利福尼亚大学旧金山分校）；U.S. Food and Drug Administration, Center for Devices and Radiological Health（美国食品药品监督管理局医疗器械和放射健康中心）
SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry
University of Macau（澳门大学）；Fudan University（复旦大学）；Hangzhou Dianzi University（杭州电子科技大学）；Renmin University of China（中国人民大学）；Hanyang University（翰阳大学）；Tsinghua University（清华大学）；Zhejiang University（浙江大学）；University of Liverpool（利物浦大学）；Southeast University（东南大学）；Central South University（中南大学）
Robustness of Algorithms for Causal Structure Learning to Hyperparameter Choice
University of Essex（埃塞克斯大学）；Institute for Analytics and Data Science, University of Essex（埃塞克斯大学分析与数据科学研究所）；Institute for Social and Economic Research, University of Essex（埃塞克斯大学社会与经济研究所）
Ablate and Rescue: A Causal Analysis of Residual Stream Hyper-Connections
Stanford University（斯坦福大学）；Georgia Institute of Technology（佐治亚理工学院）；University of California, Berkeley（加州大学伯克利分校）；Independent（独立）；University of Oxford（牛津大学）
Meek Separators and Their Applications in Targeted Causal Discovery
Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard（麻省理工学院和哈佛大学布罗德研究所埃里克与温迪·施密特中心）；LIDS, Massachusetts Institute of Technology（麻省理工学院信息与决策系统实验室）
