New Statistical Challenges Posed by Multiscale and Adaptive Representations

Stanford University, Stanford CA

Investigators

Abstract

NEW STATISTICAL CHALLENGES POSED BY MULTISCALE AND ADAPTIVE REPRESENTATIONS D.L. Donoho & I.M. Johnstone, PI's. Project period: 07/01/00 -- 06/30/05 The project will address the following specific topics: 1. Estimation in tomography: the curvelet tight frame of representation seems, according to preliminary calculations, to achieve faster rates of convergence than traditional tomographic methods. 2. Estimation and testing in time-frequency analysis: a new tight frame of chirplets has been built by deploying curvelets in the time-frequency plane. The new representation has promise for detection and estimation of chirps in the presence of noise. 3. Recent work in computational vision asks ``what is the best basis for representing natural images?'' It is proposed that many empirical results in the rapidly growing literature on this topic might be explained and improved using analytically-constructed representations: ridgelets and curvelets. 4. Geometry-driven diffusions are popular in applied image processing -- but their quantitative performance is not well-understood. It is proposed to develop a quantitative statistical theory using recent tools such as multiscale ridgelets. 5. Extensions of Sparsity-Based ideas. Problems of finding `edgels in white noise' and `subspaces in white noise' offer challenging and timely directions in which to generalize existing sparsity ideas. 6. Testing Sparse Means. An adaptive approach is suggested for testing if the mean of a random vector is nonzero, when the vector might exhibit an unknown degree of sparsity. 7. Asymptotics of top eigenvalues of large covariance matrices. A program is set out to develop statistical theory for principal components of large data matrices based on recent progress in random matrix theory. Consistent with the principle of ``reproducible research'', software and figures from this project will be made available in future releases of the public domain WaveLab system. In recent years, research in wavelets and time-frequency methods has broadened to construct new systems of representation, including systems custom-tailored for specific phenomena. Examples include wavelet packets and cosine packets, and very recently, systems like edgelets, ridgelets, chirplets, warplets, and curvelets. In parallel, research in statistical analysis and cognate fields allows data themselves to dictate the design of their own optimal systems of representation. Principal components (i.e. Karhunen-Loeve decomposition) is the oldest example of such data-adaptive representation; many newer ideas have been proposed recently, such as independent components analysis. The proposers have been active in both domains, creating new image and signal representations and developing statistical theory to underpin adaptive signal representations. The current project will (a) pursue two opportunities arising from the recent introduction of curvelets, (b) address two active applied research areas, computational vision and geometry driven diffusions, and (c) attack some issues which are argued to be at the core of new developments in statistical decision theory. Topic (a) may have implications for applied work in tomography, image and signal processing, and (c) may impact applied uses of principal components in domains such as climate and global change studies.