Application driven Topological Data Analysis

现代科学技术以前所未有的速度产生数据。一个主要的挑战是，这些数据通常是复杂的、高维的，可能包括时间和/或空间信息。数据的“形状”可能很重要，但很难使用标准的机器学习或统计技术提取和量化它。例如，肿瘤附近血管的图像与健康血管的图像看起来非常不同；单靠统计数据无法量化这种形状，因为形状才是最重要的。本提案的重点是研究数据的形状，通过发展新的数学和算法，并建立在现有的数据科学技术，以获得和解释数据的形状。能够研究形状的数学理论领域是拓扑学。计算复杂形状的形状（其拓扑结构）的能力只有通过先进的数学和算法才能实现。该领域被称为拓扑数据分析（TDA），使人们能够使用拓扑来研究数据的形状，例如血管网络中的环路。特别是，TDA中的一种称为持久同源的算法，在多个尺度上提供了数据形状（例如，孔洞等特征）的拓扑摘要。持久同源的一个关键成功之处在于能够提供健壮的结果，即使数据有噪声。在将这些算法应用于大规模的真实世界数据时，存在理论和计算方面的挑战。这个项目的目的是建立在当前的持久同源工具的基础上，从理论上、计算上对其进行扩展，并使其适应实际应用。我们的核心团队由纯数学和应用数学家、计算机科学家和统计学家组成，他们的专业知识涵盖了最前沿的纯数学、数学建模、算法设计和数据分析。这个核心团队将与我们在一系列科学和工业领域的合作者密切合作。我们提出的一些应用挑战包括：我们能否通过观察血管图像的形状来检测肿瘤？我们可以通过观察分子的形状来设计新材料吗？我们如何设计这样的分子？我们能发现安全数据中的异常吗？更重要的是，我们如何加速算法来实时获取数据的拓扑特征？

Modern science and technology generates data at an unprecedented rate. A major challenge is that this data is often complex, high dimensional, may include temporal and/or spatial information. The "shape" of the data can be important but it is difficult to extract and quantify it using standard machine learning or statistical techniques. For example, an image of blood vessels near a tumor looks very different than an image of healthy bloodvessels; statistics alone cannot quantify this shape because it is the shape that matters. The focus of this proposal is to study the shape of data, through the development of new mathematics and algorithms, and build on existing data science techniques in order to obtain and interpret the shape of data. A theoretical field of mathematics that enables the study of shapes is topology. The ability to compute the shape (its topology) of complicated shapes is only possible with advanced mathematics and algorithms. The field known as topological data analysis (TDA), enables one to use topology to study the shape of data, such as loops in a blood vessel network. In particular, an algorithm within TDA known as persistent homology, provides a topological summary of the shape of the data (e.g., features such as holes) at multiple scales. A key success of persistent homology is the ability to provide robust results, even if the data are noisy. There are theoretical and computational challenges in the application of these algorithms to large scale, real-world data.The aim of this project is to build on current persistent homology tools, extending it theoretically, computationally, and adapting it for practical applications. Our core team is composed of experts in pure and applied mathematicians, computer scientists, and statisticians whose combined expertise covers cutting edge pure mathematics, mathematical modeling, algorithm design and data analysis. This core team will work closely with our collaborators in a range of scientific and industrial domains. Some of the application challenges we have set out include:Can we detect a tumor by looking at the shape of images of blood vessels? Can we design new materials by looking at the shape of molecules using topology? How can we design such molecules? Can we detect anomalies in security data? And importantly, how can we accelerate algorithms to obtain topological characteristics of data in real time?