CAREER: Multiscale methods for high-dimensional data, graphs and dynamical systems

该奖项是根据2009年美国复苏和投资法（公法111-5）资助的。研究者解决了由集合的多尺度几何的数学分析和这些集合上的函数空间的多尺度分解引起的基本问题。这些嵌入在高维空间中的集合的几何复杂性是有噪声的，并且在不同的位置通常具有不同的维度，因此需要在多个分辨率下进行分析。他发展了稳健的数学和统计定量多尺度分析方法，产生了对噪声极其稳健的内在维度估计器，即使在小样本量下也有高概率成功。他开发了高维点云的多尺度处理算法，就像对低维函数所做的那样，通过几何多尺度变换。他通过构建数据驱动的多尺度字典来研究和近似数据上的函数，推广了在低维欧几里德空间中熟悉的方法。他开发了有效的算法来计算这些多尺度分解和字典扩展。许多重要的应用问题直接需要对高维数据集、动态系统或大图进行分析，或者将它们置于高维数据集、动态系统或大图的框架上。例如，在不同的学科和现实世界的情况下，仪器或模拟产生大量的数据，就会出现这样的问题。研究者开发了分析高维数据集和具有大量变量的统计模型的新方法，假设数据的自由参数的数量（在某种意义上代表了内在维度或复杂性）与环境空间相比是小的。他进一步将这些方法应用于能够从示例中学习这些数据的函数的算法，或者学习复杂动力系统的模型（例如分子动力学，蛋白质折叠，神经元活动等），或者学习社会网络中的行为模式。该职业奖由数学科学部/数学与物理科学局应用数学项目和网络基础设施办公室资助。

MaggioniDMS-0847388 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigatoraddresses fundamental problems arising from the mathematicalanalysis of multiscale geometries of sets, and multiscaledecompositions of function spaces on such sets. The complexityof the geometry of these sets, which are embedded inhigh-dimensional spaces, are noisy, and often have differentdimensions at different locations, is analyzed at multipleresolutions. He develops robust mathematical and statisticalquantitative multiscale methods of analysis, yielding estimatorsof intrinsic dimensionality that are extremely robust to noiseand succeed with high probability even for small sample size. Hedevelops algorithms for multiscale processing of high-dimensionalpoint clouds much as is done for low-dimensional functions, bymeans of a geometric multiscale transform. He studies andapproximates functions on data by constructing data-drivenmultiscale dictionaries, generalizing approaches familiar inlow-dimensional Euclidean spaces. He develops efficientalgorithms for computing these multiscale decompositions anddictionary expansions. Many important application problems directly require theanalysis of high-dimensional data sets, dynamical systems, orlarge graphs, or they are posed on a framework ofhigh-dimensional data sets, dynamical systems, or large graphs.Such problems arise for instance in different disciplines andreal-world situations where instruments or simulations producelarge amounts of data. The investigator develops novel methodsfor the analysis of high-dimensional data sets and of statisticalmodels with a large number of variables, under the assumptionthat the number of free parameters (which in some senserepresents the intrinsic dimensionality or complexity) of thedata is small compared to that of the ambient space. He furtherapplies these methods to algorithms that are able to learnfunctions on such data from examples, or learn models of complexdynamical systems (e.g. molecular dynamics, protein folding,neuronal activity, etc.), or learn behavior patterns in socialnetworks. This Career award is funded by the Applied Mathematicsprogram in the Division of Mathematical Sciences/Directorate ofMathematical and Physical Sciences and by the Office ofCyberinfrastructure.