Coarse geometry and cohomology of large data sets

数字经济是以数据为基础的。当前开发智能、客户主导、交互式、实时系统的趋势要求能够高效、快速地处理和解释大量数据，并具有相应要求的一定程度的准确性。最近的两个重要事态发展很好地强调了这一点。由IBM支持的智慧地球计划设想了“仪器化、互联的数据系统”，在这个系统中，物理环境的主要元素都配备了不断交换信息的传感器。其次，英国政府新的透明度驱动将使大量数据集向公众开放，创造“一个建立创新应用程序的机会，这将带来巨大的经济效益”。由于所涉及的数据集规模大、维度高，因此在数据分析中需要综合几何方法。本提案将扩展最近的重要理论成果，以创建一套用于数据分析的几何和拓扑工具，特别强调灵活性，效率，并与潜在的实际应用紧密结合。这是一个理想和非常激动人心的时间来启动这种性质的项目，从上述倡议和许多其他可能的应用的角度来看，其结果很可能产生直接和重要的后果。该提案的中心主题是研究不同尺度的大型数据集的几何特性，这对应于查看数据集时不同程度的“清晰度”。例如，在搜索大量包含人物照片的数字照片时，人们需要的分辨率与试图识别特定人物时不同。这一建议为发展纯数学方法提供了一个非常令人兴奋的机会，使它们可以直接应用于重要的、困难的和及时的实际问题。提议的工作是冒险的，跨学科的，汇集了纯粹和应用数学家，OR，计算机科学，统计学和能源系统的专家。长期实际应用的潜力将在更广泛的智能飞机计划的背景下，在两个特定的应用领域进行测试。该项目的一个主要目标是开发依赖于尺度的粗糙几何的几何和上同调工具，特别强调在有限度量空间，更具体地说，在数据集上的应用。我们将非常强调可以发展成有效的数据分析工具的方法，研究将通过从理论到更实际的应用中产生的具体问题来了解。我们将用两个重要的案例来检验理论思想和结果：一是英国政府开放数据计划产生的数据集。其次，在智能电网的背景下，我们将考虑由大量传感器监测电网性能的各个方面产生的数据，目的是提供供需之间的准确匹配。

The digital economy is founded on data. The current trend to develop intelligent, customer-led, interactive, real-time systems requires the ability to handle and interpret vast amounts of data efficiently, quickly, and with a degree of accuracy corresponding to the requirements. This point is underlined very well by two important recent developments. The Smarter Planet initiative, supported by the IBM, envisages 'instrumented, interconnected data systems' where main elements of the physical environment are equipped with sensors constantly exchanging information. Secondly, the new transparency drive of the UK government will make huge data sets available to the public, creating 'an opportunity to build innovative applications which will bring significant economic benefit'. The need for synthetic geometric methods in data analysis arises because of the large size and of high dimensionality of the sets involved. This proposal will extend recent important theoretic results to create a set of geometric and topological tools for data analysis, placing special emphasis on flexibility, efficiency, and on close alignment with potential practical applications. This is an ideal and a very exciting time to launch a project of this nature, and its results are very likely to have direct and important consequences from the point of view of initiatives mentioned above and many other possible applications. A central theme of the proposal is the study of geometric properties of large data sets at various scales, which corresponds to varying degree of 'sharpness' with which a data set is viewed. For example, in searching large numbers of digital photographs for those that contain pictures of of people one requires a different resolution than when trying to identify a specific person. This proposal offers a very exciting opportunity for developing pure mathematical methods to the point where they can be directly applied to important, difficult and timely practical problems. The proposed work is adventurous, interdisciplinary, and brings together pure and applied mathematicians, experts in OR, computer science, statistics, and energy systems. Potential for long-term practical applications will be tested in two specific areas of applications within the context of the wider Smarter Plane initiative. A main objective of the project is to develop geometric and cohomological tools of scale-dependent coarse geometry with special emphasis on applications to finite metric spaces and more specifically, to data sets. We will place strong emphasis on methods that can be developed into efficient tools for data analysis, and the research will be informed by specific problems arising from applications which range from the theoretical to the more practical. We will test the theoretical ideas and results two important cases: one, data sets arising from the UK Government's Open data initiative. Secondly, within the context of smart grids, we will consider data generated by large number of sensors monitoring various aspects of the performance of a power grid with the objective to provide an accurate matching between supply and demand.