Exploiting Special Structures in High-Dimensional Data Classification

在高维数据分类中利用特殊结构

• 批准号: 0505424
• 负责人: Elizaveta Levina
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505424&HistoricalAwards=false
• 关键词: Exploiting; Special; Structures; Dimensional; Data

ABSTRACTProposed research develops new practical methodology and algorithms aswell as theoretical results for high-dimensional data classification. Themain issue is that when the number of measured variables exceeds by farthe number of observations, estimating the full covariance matrixaccurately is impossible. The investigator has previously shown thatignoring the dependence completely in such a situation is a better option,but it means discarding a lot of information. This problem will beresolved by developing new sparse covariance estimators which will onlyretain important dependence information, and studying their behaviortheoretically in the context of discriminant analysis. New practical,computationally efficient algorithms for computing these estimators fromdata will be developed on the basis of clustering and graph partitioningmethods and compared to existing regularization techniques. Theoretically,asymptotically optimal or near optimal classification performance isexpected to be demonstrated. Another way to reduce the size of theproblem and get to the underlying structure is to reduce the datadimension using recently developed nonlinear manifold projection methods,which aim to discover a nonlinear low-dimensional embedding preservingmost of the information contained in the data. Using these methodsrequires estimating intrinsic data dimension and neighborhood scale on amanifold, and new rigorous estimators for both are proposed, along with ananalysis of their statistical properties. Careful estimation of these twoparameters will improve on the current mostly heuristic methods used inmachine learning and increase applicability of manifold projection methodsfor high-dimensional data classification.This proposal addresses the new challenges posed by the massive amounts ofdata collected in the modern world by developing new theoretical andpractical tools for dealing with high-dimensional data, particularly withthe situation when the number of measurements taken for one observation islarge relative to the number of observations. New sparse estimators ofdependence structure in such data are developed, which only contain theinformation relevant for data classification. The new estimators can alsobe used in any problem where large covariance matrices need to beestimated from limited amount of data, and hence will have an impact on awide range of modern applications, such as classification and analysis ofgene expression data, analysis of complex chemical and physicalexperiments, remote sensing, and medical imaging, among others.

摘要所提出的研究开发了新的实用方法和算法以及高维数据分类的理论结果。主要问题是，当测量变量的数量远远超过观测值的数量时，准确估计完整的协方差矩阵是不可能的。研究人员之前已经表明，在这种情况下完全忽略依赖性是一个更好的选择，但这意味着丢弃大量信息。 这个问题将通过开发新的稀疏协方差估计器来解决，该估计器将只保留重要的依赖信息，并在判别分析的背景下从理论上研究它们的行为。将在聚类和图划分方法的基础上开发用于根据数据计算这些估计量的新的实用的、计算高效的算法，并与现有的正则化技术进行比较。理论上，渐进最优或接近最优的分类性能有望得到证明。 减少问题规模并了解底层结构的另一种方法是使用最近开发的非线性流形投影方法来减少数据维度，其目的是发现保留数据中包含的大部分信息的非线性低维嵌入。使用这些方法需要估计流形上的内在数据维度和邻域尺度，并且提出了两者的新严格估计器，以及对其统计特性的分析。 对这两个参数的仔细估计将改进当前机器学习中使用的大多数启发式方法，并提高流形投影方法在高维数据分类中的适用性。该提案通过开发处理高维数据的新理论和实用工具，特别是在一次观测所采取的测量次数较多的情况下，解决了现代世界收集的大量数据所带来的新挑战。 相对于观测值的数量来说很大。 开发了此类数据中依赖结构的新稀疏估计器，其仅包含与数据分类相关的信息。 新的估计器还可以用于需要从有限数量的数据中估计大型协方差矩阵的任何问题，因此将对广泛的现代应用产生影响，例如基因表达数据的分类和分析、复杂的化学和物理实验的分析、遥感和医学成像等。