Mathematical Sciences: Exploiting Hidden Sparsity in Statistical Estimation

数学科学：利用统计估计中隐藏的稀疏性

• 批准号: 9209130
• 负责人: David Donoho
• 金额: $ 37.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9209130&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Exploiting; Hidden; Sparsity

The aim of this project is to develop new methods for recovering curves, spectra, signals and images from indirect, noisy observations. This work is based on the discovery that the minimax principle requires one to act as if the object to be recovered is sparse - mostly zero - when viewed in the appropriate transform domain, so that nonlinear methods derived from the minimax principle can exploit this sparsity much better than traditional linear methods. The basic theory of recovering sparse sequences in noise will be expanded and applications to specific scientific settings will be developed using the wavelet transform, Fourier transform and wavelet-vaguelette decomposition. Results are expected for tomography, inversion of Abel transforms and time series spectral analysis; also foundational arguments for the white noise model will be constructed, with particular reference to nonlinearity. When data are recorded for high dimensions, as for example in the form of pictures or images, recovering an exact description or identifying the parameters of the process that produced the data can be exceedingly difficult. This work will consider some remarkable new mathematical and statistical techniques which should enable such a reconstruction efficiently and with as much accuracy as the quality of the data permit. Both the mathematical theory and the practical implementation will be undertaken as part of this project.

这个项目的目的是开发新的方法，从间接的、有噪声的观测中恢复曲线、光谱、信号和图像。这项工作是基于这样一个发现，即极小极大原理要求人们在适当的变换域中看起来好像要恢复的对象是稀疏的--几乎是零的，因此从极小极大原理导出的非线性方法可以比传统的线性方法更好地利用这种稀疏性。在噪声中恢复稀疏序列的基本理论将得到扩展，并将利用小波变换、傅立叶变换和小波-模糊分解开发特定科学环境的应用程序。层析成像、Abel变换的逆变换和时间序列频谱分析的结果是预期的；此外，还将构建白噪声模型的基本论点，特别是参考非线性。当记录高维数据时，例如以图片或图像的形式，恢复准确的描述或识别产生数据的过程的参数可能非常困难。这项工作将考虑一些值得注意的新的数学和统计技术，这些技术应能够在数据质量允许的情况下以尽可能高的精度有效地进行这种重建。数学理论和实际实施都将作为该项目的一部分进行。