Mathematical Sciences: Exploiting Hidden Sparsity in Statistical Estimation

该项目的目的是开发新的方法， 从间接数据中恢复曲线、光谱、信号和图像， 嘈杂的观察 这项工作是基于发现， 极小极大原则要求人们把对象当作 恢复是稀疏的-大多为零-当在 适当的变换域，使非线性方法推导 根据极大极小原则，可以更好地利用这种稀疏性 传统的线性方法。 恢复的基本理论 噪声中的稀疏序列将得到扩展， 具体的科学设置将开发使用小波 变换、傅里叶变换和小波变换 分解 结果预计将用于层析成像，反演 阿贝尔变换和时间序列谱分析;以及 白色噪声模型的基本论点是 构建，特别是参考非线性。 当数据记录为高维时，例如 以图片或图像的形式， 描述或识别过程的参数， 制作数据可能非常困难。 这项工作将 考虑一些新的数学和统计学 应该能够有效地进行这种重建的技术 并在数据质量允许的情况下尽可能准确地进行。 无论是数学理论还是实际应用 将作为该项目的一部分。

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.