Causal Inference for Extremes via Tropical Geometry

监测和预测洪水、金融崩溃或工程风险等极端事件对社会至关重要。然而，根据定义，极端事件是罕见的，涉及大的，不太可能的值，而传统的统计技术是基于平均值和大量的观察。该项目将创建新的，快速的方法，以揭示极端事件袭击系统（如河流，计算机网络或金融网络）时的原因和潜在的级联故障。具体的应用包括洪水风险预测，追踪地下水道中污染物的来源，以及模拟飞机跑道溢出的风险。这项研究将提高社会监测、预测和预防此类不良事件的能力。极值统计关注随机变量的最大值和分布尾部之间的关系，而不是平均值和相关性。这一领域的独特挑战是缺乏数据和缺乏平滑的可能性，这严重限制了统计学习和推理。本研究的目标A旨在通过可证明准确的算法来推进极值统计的因果推理，这些算法可以处理具有数千个变量和缺失数据的数据集。本研究的目标B旨在解决具有校正线性（ReLU）激活的深度神经网络的识别挑战，这是逆向工程问题的一个困难变体。这些问题是密切相关的，这两个将通过热带代数和凸几何在本提案中解决。PI和合著者对水文数据的初步研究表明，所提出的方法在隐藏的河流网络中达到了最先进的水平，这是极端情况因果推理的基准问题。这项研究将提高社会监测和预测金融、工程和自然灾害极端事件的能力。该奖项将同时促进极值统计和热带几何学的发展，拓宽其应用范围，并在其交叉点创造新的跨学科、数据驱动的研究。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Monitoring and predicting extreme events such as flooding, financial collapses or engineering risks are of huge importance to societies. However, extreme events by definition are rare and concern large, unlikely values, while traditional statistical techniques are based on averages and large numbers of observations. This project will create new, fast methodologies to uncover the causes and potential cascading failures when an extreme event hits a system, such as a river, computer network, or financial network. Concrete applications include flood risks predictions, tracing the source of contaminants in underground waterways, and modeling risks of airplanes runway overrun. This research will advance society’s ability to monitor, predict and prevent such adverse events. Extreme value statistics concerns the maxima of random variables and relations between the tails of distributions rather than averages and correlations. Unique challenges to this field are lack of data and lack of smoothness in the likelihood, which severely limits statistical learning and inference. Goal A of this research aims to advance causal inference for extreme value statistics with provably accurate algorithms that can handle datasets with thousands of variables and missing data.Goal B of this research aims to solve the Identification Challenge for deep neural networks with rectified linear (ReLU) activation, a difficult variant of the reverse-engineering problem. These problems are intimately connected and both will be tackled in this proposal via tropical algebraic and convex geometry. Preliminary work by the PI and co-authors on hydrology data have shown that the proposed methods achieve the state-of-the-art in the Hidden River Network, the benchmark problem in causal inference for extremes. The proposed research will advance society’s ability to monitor and predict extreme events in finance, engineering, and natural disasters. It will simultaneously advance both extreme value statistics and tropical geometry, widening their applications and create new interdisciplinary, data-driven research at their intersections.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.