FRG: Topological methods in data analysis

DMS-0101364 Gunnar Carlsson本项目的总体目标是开发灵活的拓扑方法，以便分析难以使用经典线性方法分析的数据。从欧氏空间中的高度弯曲的流形或奇异代数簇采样得到的数据是我们的方法将是有用的典型例子。 我们打算开发和完善两个软件，这两个软件是由我们的研究小组成员编写的，ISOMAP（Tenenbaum）和PLEX（de Silva-Carlsson）。 ISOMAP是一个高维数据集的降维和参数化工具，PLEX是一个同源计算工具，我们将使用它来定位和分析数据集中的奇异点，以及在标准方法不起作用的情况下估计维数。 我们计划扩展这两种工具的适用范围，在ISOMAP的情况下，通过研究嵌入到空间与非欧几里德度量，并在PLEX的情况下，通过建立在迈耶-Vietoris谱序列作为一种工具。ISOMAP和PLEX都将适用于并行计算。我们也将开始与拓扑学有关的统计问题的理论研究。 例如，我们将在各种抽样假设下，开始研究从欧几里得空间抽样的子集的高维同调。 研究的主要对象将是家庭的切赫复杂的距离函数在欧几里得空间中的随机选择的有限集的点一起构建。 该项目的目标是开发工具，用于理解使用标准方法不容易理解的数据集。 这类数据可能包含奇点，或者可能是强烈弯曲的。 数据也是高维的，在这个意义上，每个数据点有许多坐标。 例如，我们可能有一个数据集，它的每个点都是一个图像，每个像素都有一个坐标。 许多标准工具依赖于线性近似，这在强弯曲或奇异问题中不能很好地工作。 我们所考虑的这类工具在某种程度上是拓扑的，因为它们测量所涉及的空间的更多定性性质，如连通性或空间中的洞的数量等，这类方法能够识别描述空间所需的参数的数量，而无需实际将其参数化。 这些方法还具有识别奇异点（如两个非平行平面或非平行线相交的点）的能力，而无需实际构建空间上的坐标。 我们还将进一步发展和完善我们已经构建的方法，这些方法实际上可以为许多高维数据集找到良好的参数化。 这两个项目都将涉及把迄今为止用于解决理论问题的手工计算的许多方法改编成计算机。 我们还将开始在一个设置，其中包括错误和采样的拓扑工具的理论发展。

DMS-0101364Gunnar CarlssonThe overall goal of this project is to develop flexible topological methods which will allow the analysis of data which is difficult to analyze using classical linear methods. Data obtained by sampling from highly curved manifolds or singular algebraic varieties in Euclidean space are typical examples where our methods will be useful. We intend to develop and refine two pieces of software which have been written by members of our research group, ISOMAP (Tenenbaum) and PLEX (de Silva-Carlsson). ISOMAP is a tool for dimension reduction and parameterization of high dimensional data sets, and PLEX is a homology computing tool which we will use in locating and analyzing singular points in data sets, as well as estimating dimension in situations where standard methods do not work well. We plan to extend the range of applicability of both tools, in the case of ISOMAP by studying embeddings into spaces with non-Euclidean metrics, and in the case of PLEX by building in the Mayer-Vietoris spectral sequence as a tool Both ISOMAP and PLEX will be adapted for parallel computing. We will also begin the theoretical study of statistical questions relating to topology. For instance, we will initiate the study of higher dimensional homology of subsets sampled from Euclidean space under various sampling hypotheses. The key object of study will be the family of Cech complexes constructed using the distance function in Euclidean space together with a randomly chosen finite set of points in Euclidean space. The goal of this project is to develop tools for understanding data sets which are not easy to understand using standard methods. This kind of data might include singular points, or might be strongly curved. The data is also high dimensional, in the sense that each data point has many coordinates. For instance, we might have a data set whose points each of which is an image, which has one coordinate for each pixel. Many standard tools rely on linear approximations, which do not work well in strongly curved or singular problems. The kind of tools we have in mind are in part topological, in the sense that they measure more qualitative properties of the spaces involved, such as connectedness, or the number of holes in a space, and so on. This group of methods has the capability of recognizing the number of parameters required to describe a space, without actually parameterizing it. These methods also have the capability of recognizing singular points (like points where two non-parallel planes or non-parallel lines intersect), without actually having to construct coordinates on the space. We will also be further developing and refining methods we have already constructed which can actually find good parameterizations for many high dimensional data sets. Both projects will involve the adaptation for the computer of many methods which have heretofore been used in by-hand calculations for solving theoretical problems. We will also initiate the theoretical development of topological tools in a setting which includes errors and sampling.