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.
