Mathematical Sciences: Exploiting Hidden Sparsity in Statistical Estimation

该项目的目的是开发从间接、噪声观测中恢复曲线、光谱、信号和图像的新方法。这项工作是基于这样的发现：极大极小原理要求人们在适当的变换域中观察时，表现得好像要恢复的对象是稀疏的——大部分是零，因此，从极大极小原理派生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。将扩展在噪声中恢复稀疏序列的基本理论，并利用小波变换、傅立叶变换和小波-小模糊分解开发特定科学环境的应用。层析成像、阿贝尔变换反演和时间序列谱分析有望取得成果；此外，白噪声模型的基本论点将被构建，特别是非线性的参考。当数据被记录为高维时，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。这项工作将考虑到一些值得注意的新的数学和统计技术，这些技术应该能够有效地进行这种重建，并在数据质量允许的情况下尽可能准确。数学理论和实际实施都将作为这个项目的一部分进行。

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.