AF: Small: Homological Methods for Big Enough Data

AF：小：足够大数据的同调方法

• 批准号: 1525978
• 负责人: Donald Sheehy
• 金额: $ 34.1万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1525978&HistoricalAwards=false
• 关键词: AF; Small; Homological; Methods; Big

How do we know if big data is big enough? As algorithms make more and more decisions from data, we also need these algorithms to assure us that the decisions were well-informed, i.e. that enough data went into them. The theory of homological sensor networks, a branch of topological data analysis, was originally created to test if a collection of sensors covers a domain of interest, but the same theory can test if a data set "covers" an underlying decision space. Homological methods can complement, extend, and even replace statistical methods to give confidence in the completeness of a data set. Because they are topological, they can give robust signatures or summaries of data that are invariant to a wide range of implicit or explicit transformations. This project aims to extend the theoretical and algorithmic foundations of the homological sensor networks to be applicable in data analysis. Broader impacts include strengthening connections between theoretical computer science (TCS) and applied algebraic topology, and widening the range of data analyses to which topological methods and tools apply.The PI will train both undergraduate and graduate researchers by incorporating advanced concepts in combinatorial topology in undergraduate and graduate curricula. The PI will also educate the larger TCS and data analysis communities through expository videos and open source software.The specific aim of the proposal is to extend guarantees on homological sensor networks to apply to non-smooth sets, k-coverage, and dynamic coverage. A second specific aim is to push these algorithmic results back into the theoretical foundations of the sampling theories that underlie data analysis problems, by extending the so-called Persistent Nerve Theorem and defining new classes of near-homeomorphisms to capture the realities of unknown transformations in data while still providing theoretical guarantees. The third specific aim is to develop algorithms that extract information from what was traditionally called "topological noise" as simple experiments reveal that although it doesn't carry topological information, it does carry useful geometric information that may be used for classification and inference.

我们如何知道大数据是否足够大？随着算法根据数据做出越来越多的决策，我们还需要这些算法来确保我们的决策是明智的，即有足够的数据进入其中。同源传感器网络理论是拓扑数据分析的一个分支，最初是为了测试传感器集合是否覆盖感兴趣的领域而创建的，但相同的理论可以测试数据集是否“覆盖”底层决策空间。同源方法可以补充、扩展甚至取代统计方法，以增强对数据集完整性的信心。因为它们是拓扑性的，所以它们可以提供对各种隐式或显式变换不变的稳健的数据签名或摘要。该项目旨在扩展同源传感器网络的理论和算法基础，使其适用于数据分析。更广泛的影响包括加强理论计算机科学（TCS）和应用代数拓扑之间的联系，并扩大拓扑方法和工具应用的数据分析范围。PI 将通过在本科生中融入组合拓扑的高级概念来培训本科生和研究生研究人员和研究生课程。 PI 还将通过说明视频和开源软件对更大的 TCS 和数据分析社区进行教育。该提案的具体目标是扩展同源传感器网络的保证，以应用于非平滑集、k 覆盖和动态覆盖。 第二个具体目标是通过扩展所谓的持续神经定理并定义新的近同态类来捕获数据分析中未知变换的现实，将这些算法结果推回数据分析问题背后的采样理论的理论基础。数据同时仍提供理论保证。 第三个具体目标是开发从传统上所谓的“拓扑噪声”中提取信息的算法，因为简单的实验表明，虽然它不携带拓扑信息，但它确实携带可用于分类和推理的有用几何信息。